Modern Analysis and Applications: The Mark Krein Centenary Conference - Volume 2: Differential Operators and Mechanics (Operator Theory: Advances and Applications)

Operator Theory: Advances and Applications Vol. 191 Editor: I. Gohberg (FKÝPÜKÚÛ2H¹EG School of Mathematical Science...

Author: Vadim Adamyan | Yu.M. Berezansky | Israel Gohberg | Myroslav L. Gorbachuk | Valentyna Gorbachuk | Anatoly N. Kochubei | Heinz Langer | Gennadi Popov

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Operator Theory: Advances and Applications Vol. 191

Editor: I. Gohberg (FKÝPÜKÚÛ2H¹EG School of Mathematical Sciences Tel Aviv University Ramat Aviv Israel

(FKÝPÜKÚÛ%PÚÜF D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France)

+POPÜÚÜWÚOF$FTKSPÜW(FKÝPÜKÚÛ%PÚÜF L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

Modern Analysis and Applications The Mark Krein Centenary Conference Volume 2: Differential Operators and Mechanics

Vadim Adamyan Yurij Berezansky Israel Gohberg Myroslav Gorbachuk Valentyna Gorbachuk Anatoly Kochubei Heinz Langer Gennadiy Popov Editors

Birkhäuser Basel · Boston · Berlin

Editors: Vadim M. Adamyan Department of Theoretical Physics Odessa National I.I. Mechnikov University Dvoryanska st. 2 2GHVVD8NUDLQH e-mail: [email protected]

Yurij Berezansky Institute of Mathematics Ukrainian National Academy of Sciences Tereshchenkivska st. .\LY8NUDLQH e-mail: [email protected]

Israel Gohberg Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 5DPDW$YLY,VUDHO e-mail: [email protected]

Myroslav Gorbachuk Valentyna Gorbachuk Insitute of Mathematics Ukrainian National Academy of Sciences Tereshchenkivska st. .\LY8NUDLQH e-mail: [email protected]

Anatoly Kochubei Insitute of Mathematics Ukrainian National Academy of Sciences Tereshchenkivska st. .\LY8NUDLQH e-mail: [email protected]

Heinz Langer Institute of Analysis and 6FLHQWL½F&RPSXWLQJ Technical University of Vienna :LHGQHU+DXSWVWUDVVH° 9LHQQD$XVWULD e-mail: [email protected]

Gennadiy Popov Department of Mathematical Physics Odessa National I.I. Mechnikov University Dvoryanska st. 2 2GHVVD8NUDLQH e-mail: [email protected]

0DWKHPDWLFDO6XEMHFW&ODVVL½FDWLRQ(.$$$(11 51*%+-.5:4 /LEUDU\RI&RQJUHVV&RQWURO1XPEHU Bibliographic information published by Die Deutsche Bibliothek. 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD½H detailed bibliographic data is available in the Internet at http://dnb.ddb.de

,6%1%LUNKlXVHU9HUODJ$*%DVHO- Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of WKHPDWHULDOLVFRQFHUQHGVSHFL½FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI LOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR½OPVRULQRWKHUZD\VDQGVWRUDJH in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin 32%R[&+%DVHO6ZLW]HUODQG Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF Printed in Germany ,6%19RO27 H,6%1 ,6%19RO27 H,6%1 ,6%16HW

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Yu. Berezansky and V. Gorbachuk The World Dimension of the Heritage of a Ukrainian Mathematician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Part 1: Plenary Talks S. Aizikovich, V. Alexandrov and I. Trubchik Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

D. Alpay, I. Gohberg, M. Kaashoek, L. Lerer and A. Sakhnovich Krein Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Yu.M. Berezansky Spectral Theory of the Infinite Block Jacobi Type Normal Matrices, Orthogonal Polynomials on a Complex Domain, and the Complex Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

G.L. Brovko, O.A. Ivanova and A.S. Finoshkina On Geometrical and Analytical Aspects in Formulations of Problems of Classic and Non-classic Continuum Mechanics . . . . . . . . . . .

51

F. Gesztesy and M. Mitrea Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schr¨ odinger Operators on Bounded Lipschitz Domains . . . . . . . . . . . . . . .

81

M.L. Gorbachuk and V.I. Gorbachuk On Behavior of Weak Solutions of Operator Differential Equations on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 M.A. Grekov and N.F. Morozov Some Modern Methods in Mechanics of Cracks . . . . . . . . . . . . . . . . . . . . . .

127

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Contents

R. Kushnir and B. Protsiuk A Method of the Green’s Functions for Quasistatic Thermoelasticity Problems in Layered Thermosensitive Bodies under Complex Heat Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 S.M. Mkhitaryan On the Application of the M.G. Krein Method for the Solution of Integral Equations in Contact Problems in Elasticity Theory . . . . . .

155

G. Popov and N. Vaysfel’d The Stress Concentration in the Neighborhood of the Spherical Crack Inside the Infinite Elastic Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 J. Rovnyak and L.A. Sakhnovich Pseudospectral Functions for Canonical Differential Systems . . . . . . . . .

187

I. Selezov On Wave Hyperbolic Model for Disturbance Propagation in Magnetic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

Part 2: Research Papers A. Ashyralyev High-accuracy Stable Difference Schemes for Well-posed NBVP . . . . . . 229 A. Belyaev The Factorization of the Flow, Defined by the Euler-Poisson’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253

V.P. Burskii On a Moment Problem on a Curve Connected with Ill-posed Boundary Value Problems for a PDE and Some Other Problems . . . . . 273 F. Chaban and H. Shynkarenko The Construction and Analysis of a Posteriori Error Estimators for Piezoelectricity Stationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

H.O. Cordes Remarks about Observables for the Quantum Mechanical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 G. Eskin and J. Ralston Remark on Spectral Rigidity for Magnetic Schr¨ odinger Operators . . . . 323 S. Gefter and T. Stulova On Holomorphic Solutions of Some Implicit Linear Differential Equations in a Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331

Contents

vii

V.I. Gerasimenko Groups of Operators for Evolution Equations of Quantum Many-particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 V.A. Grishin, V.V. Reut and E.V. Reut Box-like Shells with Longitudinal Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 O.Ye. Hentosh Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces of Two Anticommuting Variables . . . . . . . . . . . . . . . . . . . . . .

365

N.D. Kopachevsky and V.I. Voytitsky On the Modified Spectral Stefan Problem and Its Abstract Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 O. Kryvyy The Discontinuous Solution for the Piece-homogeneous Transversal Isotropic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 M.V. Markin On the Carleman Ultradifferentiability of Weak Solutions of an Abstract Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 J. Michor and G. Teschl On the Equivalence of Different Lax Pairs for the Kac–van Moerbeke Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 V.A. Mikhailets and A.A. Murach Elliptic Problems and H¨ ormander Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .

455

G. Poletaev Connection of Solutions of Abstract Paired Equations in Rings with Factorization Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

479

V.G. Popov, O.V. Litvin and A.P. Moysyeyenok The Dynamic Problems About the Definition of Stress State Near Thin Elastic Inclusions Under the Conditions of Perfect Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 R.M. Trigub Fourier Multipliers and Comparison of Linear Operators . . . . . . . . . . . . . 499 V.M. Vorobel and V.V. Reut Forced Vibrations of the Infinite Shell of the Square Cross Section . . .

515

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Preface This is the second of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary of the birth of Mark Krein, one of the greatest mathematicians of the 20th century, was held in Odessa, Ukraine, on April 9–14, 2007. The conference focused on the main ideas, methods, results, and achievements of M.G. Krein. This second volume is devoted to the theory of differential operators and mechanics. It opens with the description of the conference and a number of survey papers about the work of M.G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of differential operators. The first volume of these proceedings, entitled Operator Theory and Related Topics, concerns other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences.


Operator Theory: Advances and Applications, Vol. 191, xi–xv c 2009 Birkh¨ auser Verlag Basel/Switzerland

The World Dimension of the Heritage of a Ukrainian Mathematician International Conference “Modern Analysis and Applications” (MAA – 2007) (April 9–14, 2007, Odessa)

Yu. Berezansky and V. Gorbachuk This forum has been dedicated to the centennial birthday anniversary of one of the most prominent mathematicians of the twentieth century Mark Grigorievich Krein, a corresponding member of the Academy of Sciences of the Ukr. SSR (1907–1989).

The organizers of the conference are the National Academy of Sciences of Ukraine, the Ministry of Education and Science of Ukraine, the Odessa City Council, the Institute of Mathematics of the National Academy of Sciences of Ukraine, the Odessa National I.I. Mechnikov University, the Odessa National Maritime University, the South-Ukrainian State K. Ushinsky Pedagogical University, the Institute of Mathematics, Economics, and Mechanics, and Faculties of Mathematical Physics and Theoretical Physics of the Odessa National University. A substantial assistance for conducting the conference came from the following sponsors: Swedish Institute, European Science Foundation, Odessa City Council, Manufactured Goods Market Limited, JV Dipolos Limited, Imexbank (Joint Stock Commercial Bank), Southern Bank, Porto–Franco Bank. There were 252 scientists who participated in the work of the conference, coming from 29 countries: Algeria, Armenia, Austria, Belgium, Brazil, Canada, Denmark, Finland, France, Germany, Great Britain, Israel, Japan, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, Qatar, Russia, Slovakia, South African Republic, Spain, Sweden, Switzerland, Turkey, Ukraine, USA. Many of the participating scientists are well known to the broad mathematical community. They are H. Langer (Austria), P. Sobolevskii (Brazil), P. Lancaster (Canada), O. Staffans (Finland), V. Zagrebnov (France), H. Dym, I. Gohberg, Yu. Lyubich, V. Matsaev, V. Milman (Israel), T. Ando (Japan), A. Dijksma (the Netherlands), B. Pavlov (the New Zealand), M. M¨ oller (South African Republic), A. Laptev, A. Lindquist (Sweden), C. Tretter (Switzerland), M. Agranovich, S. Kislyakov, A. Shkalikov (Russia), V. Adamyan, D. Arov, Yu. Berezansky, M. Gorbachuk,

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I. Kats, E. Khruslov, A. Kochubei, M. Kopachevsky, V. Marchenko, L. Pastur, G. Popov, F. Rofe-Beketov, A. Samoilenko, Yu. Samoilenko (Ukraine), F. Gesztesy, Yu. Latushkin, L. Nirenberg, J. Ralston, J. Rovnyak (USA), and others. The conference meetings were held in the building of the National Academy of Public Administration under the President of Ukraine. The topics discussed were concerned with problems in both modern Mathematical Analysis and Mechanics. There were 8 Mathematical Analysis sections including the following: the geometry of Banach and Krein spaces; evolution equations in Banach spaces; the theory of operators on Hilbert and Krein spaces and its applications; non-selfadjoint operators and algebraic problems of Functional Analysis; spectral theory of differential and difference operators; direct and inverse problems; applications of spectral theory to problems in analysis; control theory and the theory of stochastic processes; nonlinear analysis; harmonic analysis, the analysis of functions of infinitely many variables, generalized functions. The problems in Mechanics were discussed in the corresponding section. Opening talks containing recollections about the life and the work of M.G. Krein, importance of his scientific heritage for the development of the modern mathematics and for the applications were given by the Head of the Southern Scientific Center, academician of the National Academy of Sciences, S.A. Andronati, Director of the Institute of Mathematics, academician of the National Academy of Sciences of Ukraine, A.M. Samoilenko, Rector of the Odessa National I.I. Mechnikov University, academician of the Higher School Academy, V.A. Smintina, as well as the students and collaborators of M.G. Krein, – V.M. Adamyan, D.Z. Arov, Yu.M. Berezansky, I. Gohberg, M.L. Gorbachuk, H. Langer. All 25 plenary talks discussed the influence of ideas and scientific results of M.G. Krein on subsequent developments of different directions in the modern analysis, and applications of the ideas and the results in various branches of science and technology. M.G. Krein was a phenomenally gifted mathematician. His way to science was equally extraordinary. His life as a scientist and a person reflects dramatic events of Soviet totalitarian times. Let us shortly describe milestones of the life and work of this brilliant scientist, a world figure in mathematics. He was born in Kiev on April 3, 1907, in a modest income family that was raising seven children. M.G. Krein had exhibited his extraordinary mathematical abilities in his teens. Already when he was 14, he attended lectures and scientific seminars conducted by D.O. Grave and B.M. Delone at Kiev University and Kiev Polytechnic Institute. At the age of 17, influenced by the work of M. Gorky “My Universities”, he decided that it was the time to start his own “universities” and, together with his friend, he went to Odessa to join one of circus troupes, for he had a dream of becoming an acrobat. However, the fate had its way, and saved to the world, in the person of M.G. Krein, not an acrobat but a prominent mathematician whose influence on the development of mathematics can not be overestimated. In Odessa, he met N.G. Chebotarev, a famous algebraist and a

The World Dimension of the Heritage of a Ukrainian Mathematician

xiii

wonderful person, who conducted research in Odessa University. Having discovered Krein’s mathematical gifts, Nikolai Grigorievich Chebotarev achieved in getting, from the Department of Education, a permission for admitting M.G. Krein, who was 19, to a Doctorate Program, although Krein did not have even a high school diploma, to say nothing about a university degree. In his “Mathematical Autobiography”, N.G. Chebotarev talking about 17 year old M.G. Krein, recalled that he “without having graduated from a high school, had brought a personal work with a very distinguished content”. N.G. Chebotarev was very proud of his first student and regarded him as “one of the best mathematicians in Ukraine”. In 1931, M.G. Krein has obtained a Professor position at the Odessa University. The degree of Doctor in Physics and Mathematics was awarded to him at the age of 31 by Moscow State University without even a requirement of submitting a thesis. Shortly after that, in 1939, he was elected a corresponding member to the Academy of Sciences of the Ukrainian Soviet Socialist Republic. The early flourishing of M.G. Krein’s talent as a scientist was accompanied by as early opening of his pedagogical talent. At the age of 25, he started, in Odessa University, a scientific seminar that soon became one of the leading centers for the research in functional analysis, a young branch of mathematics at the time. During this period, M.G. Krein’s mathematical interests include oscillation matrices and kernels, geometry of Banach spaces, the Nevanlinna–Pick interpolation problem, extension of positive definite functions and their applications, spectral theory of linear operators. At the same time, M.G. Krein was working in the Mathematics Research Center at the Khar’kov University (1934–1940), just before the World War II in Kiev where he headed the Department of Functional Analysis at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR (one of the researchers working there during the period of 1940–1941 was great S. Banach with whom M.G. Krein had scientific contacts since his trip to L’vov in 1940). Many of his results obtained at that time now became classic and can be found in main monographs and textbooks on functional analysis. During the World War II, M.G. Krein headed the Chair of Theoretical Mechanics at Kuibyshev Industrial Institute (Russia). In 1944, he returned to Odessa. He loved this city, knew history of its streets, was fond of the local “Odessa language”. However, shortly after that, M.G. Krein was laid off the Odessa University. His closest friend B.Ya. Levin, a mathematician, could not also work there any more. This was a consequence of the antisemitic policies conducted by the Stalin regime and the corruption of the University administration. A principled scientific position of these scientists, their opposition to pushing through illiterate doctorate theses, was regarded as an indication of Zionism. During the period of 1944–1954, M.G. Krein worked in the Department of Theoretical Mechanics at the Odessa Marine Engineering Institute. Starting in 1954 and until the retirement, Mark Grigorievich headed the Chair of Theoretical Mechanics in Odessa Institute of Civil Engineering. After his retirement he worked as a consultant in the Institute of Physical Chemistry of the Academy of Sciences of Ukr. SSR.

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Regardless the difficulties pertaining these times, he has founded a number of important directions in mathematics and mechanics, became a world famous scientist. Together with theoretical value of his results, their importance in applications has also increased, especially those related to parametric resonance. V. Veksler, a renowned physicist, have remarked that “without works of M.G. Krein, we would not have a synchrophasotron”. In a popular book by N. Wiener, a father of cybernetics, “I Am a Mathematician”, the name of M.G. Krein goes together with the name of A.M. Kolmogorov, which was a way to acknowledge the value of their researches in the prediction and control theory during and shortly after the war. Because of the political situation that was in the USSR at that time, M.G. Krein was never elected an academician of the Academy of Sciences, although he became Honor member of the American Academy of Arts and Sciences (1968), a foreign member of the US National Academy of Sciences (1979). He was awarded with a Wolf Prize in Mathematics (1982), which is an analogue of the Nobel Prize, and the State Prize of the Ukrainian SSR for Science and Technology (1987). M.G. Krein is an author of more than 300 papers and monographs all of which with no exception were published abroad in translation, some of them several times. These works are of an excellent analytic level and quality, broad in topic, and have opened a number of new directions in mathematics, while significantly enriched traditional directions. The following is an incomplete list of branches in mathematics where M.G. Krein’s research became fundamental and, to an extent, have determined the direction for a later development: oscillation kernels and matrices, the moment problem, orthogonal polynomials and approximation theory, cones and convex sets in Banach spaces, operators on spaces with two norms, extension theory for Hermitian operators, the theory of extension of positive definite functions and spiral arcs, the theory of entire operators, integral operators, direct and inverse spectral problems for inhomogeneous strings and Sturm–Liouville equations, the trace formula and the scattering theory, the method of directing functionals, stability theory for differential equations, Wiener–Hopf and Toeplitz integrals and singular integral operators, the theory of operators on spaces with an indefinite metric, indefinite extension problems, non-selfadjoint operators, characteristic operator-valued functions and triangular models, etc. The participants of the conference dedicated to the 100th birthday anniversary of the famous mathematician unanimously agreed with the initiative of Yu.M. Berezansky, academician of the National Academy of Sciences of Ukraine, to install a memorial plaque on the building of Odessa National I.I. Mechnikov University. A.M. Samoilenko, academician of the National Academy of Sciences of Ukraine, announced that Krein Prize would be founded in 2007 to be awarded for outstanding merits in the area of functional analysis. With an initiative of I.Ts. Gohberg (Israel), a student of Mark Grigorievich Krein, the publisher “Birkh¨ auser” will publish a complete collection of works of the conference.

The World Dimension of the Heritage of a Ukrainian Mathematician

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On 11 of July, 2007 the Presidium of National Academy of Science of Ukraine founded Krein Prize for outstanding science works in the area of functional analysis and theory of functions.

The above is a picture of the memorial plaque installed in January, 2008 in honour of M.G. Krein on the building of Odessa National I.I. Mechnikov University. The text on the plaque is in Ukrainian and its translation is as follows: In this building from 1926 to 1948 worked the outstanding mathematician of XX century Krein Mark Grigorievich (1907–1989)

Part 1 Plenary Talks


Operator Theory: Advances and Applications, Vol. 191, 3–17 c 2009 Birkh¨ auser Verlag Basel/Switzerland

Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth Sergey Aizikovich, Victor Alexandrov and Irina Trubchik On the occasion of M.G. Krein’s 100th birthday

Abstract. An article is devoted to the method of reduction of one class of dual integral equations, which appear while solving the mixed problems of elastic theory, to the solution of infinite algebraic equation systems in accordance with the method described by Alexandrov V.M. For example, such dual equations arise in solving the contact problems of elasticity theory for the layer inhomogeneous in depth or inhomogeneous half-space. The approximate solution of such equations is reduced to the solution of finite algebraic equation systems. It is proved that the solution, constructed in such way, is asymptotically exact as for small, as for large values of dimensionless geometrical parameter of the problem. The equations generated by the Fourier and Hankel integral transforms are provided as an example. Mathematics Subject Classification (2000). Primary 45E05; Secondary 74E05. Keywords. Asymptotic solution, dual integral equations, contact problems, inhomogeneous media.

1. Introduction The continuous variation of mechanical properties in one of the coordinates is typical for many bodies; it depends on the conditions of their creation and exploitation. The 80s’ development of modern technologies has increased the interest to contact problems for the continuously inhomogeneous bodies, which allows to get material’s coatings with mechanical properties continuously variable in depth, This work was completed with the support of the Russian Foundation for Basic Research, project nos. 05-08-18270a, 06-08-01595a, 07-08-00730-a.

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S. Aizikovich, V. Alexandrov and I. Trubchik

radius and angular coordinate (depending on the geometry of the foundation). The tribology development has led to the expansion of theoretical researches of nonclassical contact problems of elasticity theory. In particular, one of the main tribology problems is the contact problem for the bodies with coatings. The characteristic property of inhomogeneous materials is the presence of additional sources of stress concentration. The stress concentration in homogeneous bodies appears in areas of sharp variations in body’s geometry and loading. There is additional stress in inhomogeneous materials, which arises in areas of sharp changes of physical and mechanical properties of the material (elastic modulus, Poisson’s ratio and others), that is along the adjoin surfaces of homogeneous elements. The detailed survey of the main results of statics problems for multilayered media is given in the monograph [1] by Nikishin V.S. The reduction of contact problems to the integral equation (IE), while considering the model in the inhomogeneous media, is complicated with the necessity of solving the boundary problem for differential equation system with variable coefficient for the construction of the transforms’ kernel. One of the simplest models of the foundation is the homogeneous layer or wedge coupled with nondeformable foundation. It was found that integrable singularities can appear in the vicinity of mutual top of two joint wedges. The type of these integrable singularities depends on material’s characteristics and local geometry of the joint. Moreover, kernel transforms of IE for inhomogeneous media in average case are constructed by numerical methods, which is opposite to homogeneous media (half-space, layer). [2]. The numerically constructed kernel transform is approximated by an expression of a special type, so that it is possible to obtain a closed solution of the approximate integral equation. It is shown that the resulting approximate solution is asymptotically accurate as for small as for large values of dimensionless parameter λ of each problems. It gives the opportunity to investigate the influence of arbitrary changing laws of inhomogeneity on the stress-strain state distributed within the depth of the layer or half-space.

2. The reduction of contact problems of inhomogeneous media to the dual IE The equation of spatial elasticity theory in cylindrical coordinate system is used to reduce mixed problem to the dual IE. When Hook’s law ratios are substituted in these equations, we get the system of differential equations with variable coefficient for the functions – transforms of integral transformation of Fourier, Hankel or Mellin (depending on the media geometry). It is convenient to write this system in vector form: → − → − dY = AY (2.1) dx

Bilateral Asymptotic Solution

5

→ − Here the dimension of the introduced vector-function Y is twice as large as the rank of the vector-function of the displacement transform, as the first derivatives of these functions should be taken into account in boundary conditions. The form and the rank of the matrix A depends on the geometry of the inhomogeneous medium, and they are provided in works [3]–[5]. The solution of the vector differential equation (2.1) is constructed with the method suggested in [2]. The method is based on the isolation of the exponential components in Green matrix construction. It allows getting the stable numerical algorithm. The solution of the equation (2.1) is constructed in the form of the linear combination of vectors of fundamental system of solutions − → → − Y (α, x) = ai (α) Ψ F i (α, x), r

(2.2)

i=1

→ − r – the vector-function rank Y . Fundamental system of solutions in (2.2) is represented in the form: → − →F − Ψ i (α, x) = Ti (α, x) Ψ i (α, x), where Ti – diagonal matrixes, whose diagonals hold the vectors’ components → − → − t i (α, x) (i = 1, 2, . . . , r). Vectors Ψ i are eigenvectors of the matrixes A for ho→ − mogeneous media, t i – are vectors of modulating functions, dependant on the inhomogeneity of the media, ai (α) – certain coefficient found in boundary conditions separately for each α. This way of solution allows explicit extraction of → − components in (2.2) Ψ i in an explicit form. Its rapid growth and oscillation complicates the process of numerical implementation of the solution. Components of → − the vectors Ψ i may be as exponential, as trigonometric function (as well as hypergeometric and cylindrical functions), depending on the geometry of the media. → − Vectors of modulating functions t i (α, x) are obtained from the set of Cauchy problems for distinct values α → − dti → − = Bi t i , (i = 1, . . . , r) , x1 ≤ x ≤ x2 . dr The elements of the matrix Bi have the form 6 −1 dΨji j k k . Ajk ti Ψi − Bij = Ψi dr k=1

− → The initial conditions for t i are determined by the form of the vector equation solution (2.1) for the case of homogeneous media. → − The components of the solution vector Y (αx) are k

Y =

r i=1

ai tki Ψki .

(2.3)

6


The application of inverse integral transform in (2.3) allows obtaining of the sought values of displacement and stress. The efficiency of this method can be illustrated with the example of equations generated by the Fourier and Hankel transforms. Other dual equations were considered by the authors in [5]. This paper is a continuation of [6] and [3]. In particular the component Y 1 corresponds here to the kernel transform of the vertical displacement function and denoted as K(u).

3. General characteristics of the transforms’ kernels of IE for someone class of problems When the following conditions are satisfied min

z∈(−∞;0]

Θ(z) ≥ c1 > 0,

max

z∈(−∞;0]

Θ(z) ≤ c2 < ∞, Θ(z) =

E(z) 2(1 − ν 2 (z))

(3.1)

where E(z) is Young modulus and ν(z) is Poisson’s ratio, it can be demonstrated (Aizikovich and Aleksandrov, [7]), that the kernel transform L(u) has the following properties 3

K(u) = A + B |u| + Cu2 + O(|u| ), u → 0, K(u) = 1 + D |u|

−1

lim K(u) =

z→0

−3

+ Eu−2 + O(|u|

Θ(0) =A Θ(−H)

), u → ∞

(3.2) (3.3)

where A, B, C, D, E are some constants. For multilayered media the property of a compliance function similar to (3.2) was noticed by Privarnikov [8]. The property (3.2) means that the value K(0) for the problem under consideration is independent of the way in which elastic moduli vary in the half-space from z = 0 to z → −∞ and it is determined only by their values for z = 0 and z = −H. Graphically it looks as follows: if the set of curves describing the certain laws of the elastic moduli variation with depth have identical values on the surface of the half-space and as z → −∞, then the graphs of corresponding transforms K(u) of the problem will issue from a common point K(0) = A and converge at one point K(∞) = 1. Let us introduce the following definitions: Definition 3.1. The function K(αλ) belongs to the class ΠN if it has the form K(αλ) = LΠN (αλ) ≡

N α2 + A2i λ−2 ; (Bi − Bk )(Ai − Ak ) = 0 for i = k. (3.4) α2 + Bi2 λ−2 i=1

Here Ai , Bi (i = 1, . . . , N ) are certain complex constants. Definition 3.2. The function K(αλ) belongs to the class ΣM if it has the form K(αλ) = LΣM (αλ) ≡

M Ck λ−1 |α| . α2 + Dk2 λ−2

k=1

(3.5)


7

Definition 3.3. The function K(u) belongs to the class SN,M , it has the form K(αλ) = LΠN (αλ) + LΣM (αλ).

(3.6)

We show that the expressions of the form (3.6) can approximate K(u) with the properties (3.2) and (3.3) using the following lemma ([9], [10]): Lemma 3.4. Let an even real function φ(u) be continuous on the whole real axis and vanish at infinity, then it can be approximated in C(−∞, ∞) a series of functions of the form −1 φk = u2 + Dk2 . Theorem 3.5. Provided that if the function K(u) possesses the properties (3.2) and (3.3), it allows approximation by the expressions of the form (3.6). Proof. We select constants Ai and Bi (i = 1, 2, . . . , N ) in (3.5) such that N

A2i Bi−2 = A.

(3.7)

i=1

We consider the function Ls (u) =

K(u) − LN Π (u) . |u|

(3.8)

On the basis of properties (3.1) and condition (3.7), it follows that Ls (u) satisfies the condition of 3.4. This means that the following representation holds Ls (u) =

∞

bk (u2 + D˜k2 )−1 ,

(3.9)

k=1

or from the conditions (3.8) and (3.9), K(u) = LN Π (u) + |u|

∞

bk (u2 + D˜k2 )−1 .

(3.10)

k=1

For a numerical realization the improving approximation of K(u) by functions of the class ΠN can be achieved successfully by using the following algorithm. 2 2 2 We map function K(u)

by mapping γ = u /(u + c ) from interval (0; ∞) into segment (0; 1) (u = c γ/(γ − 1)). Here c is a positive constant, which should be selected to build the optimal approximation of function K(u). As initial value c can be taken c = u∗ , where u∗ such as ∗ K(u ) = 0, 5 max K(u) + min K(u) . u∈[0;∞)

u∈[0;∞)

Here c is a parameter of mapping, which moves the point u = c of axes (0; ∞) into the point u = 0, 5 of segment (0; 1). We approximate the functions K(γ)

8


and K −1 (γ) on segment (0; 1) by N th-order Bernstein’s polynomials (or by Chebyshev’s nodes), and thus obtain

N N

LN (γ) = ai u i , L−1 bi u i (3.11) N (γ) = i=0

i=0

here ai , bi are coefficients of Bernstein’s polynomials which can be defined as follows. If f (x) is a continuous function determined on segment (0; 1) then Bernstein’s polynomial BN (x) for this function has the form according to Goncharov [11], BN (x) =

N m=0

f(

m m m )C x (1 − x)N −m N N

m CN

where are the binomial coefficients. Then, N

2 −N ∗ 2i γ + c2 LN (γ) = ai γ i=0

N

−N L−1 b∗i γ 2i γ 2 + c2 N (γ) =

(3.12)

i=0

where a∗i , b∗i are defined from (3.11) after the change of variable u = 2coefficients 2 2 γ γ +c . N −1 N

LN (γ) ∗ 2i ∗ 2i ai γ bi γ . (3.13) LN (γ) = = L−1 (γ) i=0 i=0 N For each inhomogeneity law, the parameter c is selected separately in order to LN (u) will approximate the function K(u) more exactly for given N . By determining the roots of the numerator and denominator in (3.13), we find the desired values of Ai , Bi (i = 1, 2, . . . , N ). Such method permits avoiding the presence of an N -triple root in the denominator of the approximation found.

4. The asymptotic solution of a class of dual integral equation Lets consider general class of dual integral equations, which the contact problems for inhomogeneous half-space reduced to. Suppose that we are given an integral transformation b g(x) =

β G(γ)B(γ, x)dγ,

G(γ) =

a

(4.1)

α

or a series g(x) =

g(ξ)M(γ, ξ)dξ

∞ k=0

β Gk B(γk , x),

Gk =

g(ξ)M(γk , ξ)dξ α

(4.2)


9

and the function B(γ, x) is a solution of a second-order linear equation with respect to x that satisfies the condition of the theorem of Fuchs [12], namely, L − γ 2 B(γ, x) = 0, Lγ B = r(x) s(x)B + t(x)B, a ≤ x ≤ b. (4.3) Here s(x) > 0 for x ∈ (a, b), and r(x) is of definite sign for x ∈ (a, b). Suppose also that the functions B and B are bounded as x → b, and α1 B + α2 B = 0 at x = a. Moreover, the numbers γk make up the countable set of zeros of some transcendental equation, and a ≤ γk < γk+1 ≤ b. We consider the dual integral equation (the dual series-equation) b Q(γ)ρ(γ)K(λγ)B(γ, x)dh(γ) = f (x),

c≤x≤d

a

(4.4)

b Q(γ)B(γ, ξ)dh(ξ) = 0,

α ≤ x ≤ c, d < x ≤ β,

a

where for (4.1) h(γ) ≡ γ, and for (4.2) ∞

h(γ) ≡

1 [1 + sgn(γ − γk )]. 2 k=0

Here the function ρ(γ) is such that for K(λγ) ≡ 1 the solution of (4.4) is known. Let K(γ) denote the properties (3.2), (3.3). We have the Theorem [13]. Theorem 4.1. If K(γ) has the properties (3.2)–(3.3), then it admits approximation by expressions of the form Σ K(λγ) = KN (λγ) + K∞ (λγ).

(4.5)

According to (4.1),

β

Q(γ) =

g(ξ)M(γ, ξ)dξ

forρ(γ) = 1.

(4.6)

α

Substituting (4.6) into (4.4), we have b β g(ξ)ρ(γ)K(λγ)M(γ, ξ)B(γ, x)dξdh(γ) = f (x), a

c ≤ x ≤ d.

(4.7)

α

Below, the integral operator corresponding to a function K(γ) in the class X will also be denoted by X. Using (4.5), we rewrite (4.7) in operator form: ΠN q + Σ∞ q = f.

(4.8)

10


Definition 4.2. Condition A will be said to hold for equation (4.4) if when K(γ) ∈ ΠN one can construct a closed solution for it by following [14]. We denote it by q = Π−1 N f,

x ∈ (c, d).

(4.9)

In other words, Condition A means that for functions f (x) in some class W(c, d) of functions there exists a function q(x) in some class V(c, d) of functions such that (4.9) holds. It follows from the representation (4.9) that ||q||V(c,d) ≤ m(ΠN )||f ||W(c,d) ,

m(ΠN ) = const .

Below, we let m(X) be some constant depending on the concrete form of the function belonging to X. In [6] conditions are obtained under which if λ → 0, then the operator Π−1 N ΣM of equation (4.4) is a contraction operator. 4.1. The proof of solution (4.9) accuracy for the small values of parameter λ We will show that under certain conditions (4.9) is an asymptotically exact solution of Eq. (4.8) in the limit λ → 0. Let us first examine the question of the existence and uniqueness of a solution to the dual equations (4.4) for K(γ) of class SN,M in this case Eqs. (4.4) can be recast as ΠN q + ΣM q = f.

(4.10)

Let us find conditions under which the operator Π−1 N ΣM in (4.4) is contraction operator. For this we will use the following assertions. Lemma 4.3. Consider a bilinear form of the type b γρ(γ)M(γ, ξ)B(γ, x) α ˜ ia (ξ, x) ≡ dh(γ). γ 2 + a2 a ˜ ia (ξ, x) If γρ(γ) = r−1 (γ), M(γ, x) = B(γ, x), and a is a real number, then α admits the representation B− (ia, ξ)B+ (ia, x), ξ < x, α ˜ ia (ξ, x) = B+ (ia, ξ)B− (ia, x), x < ξ, where B− (ia, x) and B+ (ia, x) are linearly dependent solutions of (4.3) such that B− (ia, ξ) → 0 and B+ (ia, ξ) → ∞ as a → ∞. The assertion of Lemma 4.3 follows from Lemma 28.1 of [15] by the substitution γr = ia in the latter. Without loss of generality we set M = 1 in (4.10). Lemma 4.4. If condition A and the hypotheses of Lemma 4.3 are satisfied for (4.4), then the operator Σ1 q in (4.10) admits a series representation (Σ1 q corresponds to K1Σ (λγ)) ∞ Σ1 q = βk B(γk , x) (4.11) k=0


11

and the coefficients βk of the series expansion have the form ⎛ d d −1 Cr λ ⎝C(a) q(ξ)B(γk , ξ)dξ − s(c)Wca (B+ , B) q(ξ)B− (a, ξ)dξ βk (a) = 2 γk − a2 c c ⎞ d + s(d)Wda (B− , B) q(ξ)B+ (a, ξ)dξ ⎠ , a = iD1 λ−1 . (4.12) c

Here γ0 , γ1 , . . . , γn , . . . the set of all eigenvalues of problem (4.3) with the respective boundary conditions, B(γk , x) are the corresponding normalized eigenfunctions, C(a) is a bounded constant which is fixed for each (4.3), and which is related with the Wronskian W (B+ , B− ) of the functions B+ (a, x) and B− (a, x) by the relation W [B+ (a, x), B− (a, x)] = C(a)s−1 (x)

(4.13)

and we used the notation Wba (A, B) = A(a, b)B (γk , b) − B(γk , b)A (a, b).

(4.14)

Proof. To prove Lemma 4.4, we write the representation of the expansion coefficients βk −1

d

βk (a) = cr λ

d q(ξ)Ak (a, ξ)dξ, Ak (a, ξ) =

c

α ˜ a (ξ, x)B(γk , x)r−1 (x)dx.

c

(4.15) Using Lemma 4.3 and the following well-known property of solutions of a second-order differential equation: d c

d B(a, x)B(ib, x) s(x) dx = 2 (B (a, x)B(ib, x) − B(a, x)B (ib, x)) r(x) a + b2 c

where B(a, x) and B(ib, x) are two arbitrary solutions of (4.3) corresponding to the values γ = a and γ = ib, we recast expression (4.15) as ⎧ x)B ⎪ + (a, ξ (γk , x) ⎪ s(x)B− (a, ξ) [B ⎪ ⎨ − B(γk , x)B+ (a, x) c , ξ < x 1 Ak (a, ξ) = 2 (4.16) s(x)B 2 + (a, ξ) [B− (a, x)B ⎪ (γk , x) γk − a ⎪ ⎪ ⎩ − B(γk , x)B− (a, x) d , x < ξ ξ

The assertion of Lemma 4.4 follows from (4.16) using (4.13) and (4.14).

We now consider (4.3). Setting y(x) = B(x) s(x), we obtain for y(x) an equation of the form y − γ 2 q(x)y = 0, where q(x) = p(x) − R(x)γ 0.25(s s−1 )2 .

−2

q(x) = p(x) − R(x)γ −2 , −1

, p(x) = (rs)

(4.17) −1

, R(x) = t(rs)

− s (2s)−1 +

12

S. Aizikovich, V. Alexandrov and I. Trubchik We have the:

Lemma 4.5. Under the assumptions of Lemma 4.4, the operator Π−1 N ΣM is a contraction operator in the space V (c, d) provided: 1. q (x) is continuous for x ∈ [a, b]; 2. q(x) ≥ 0 for x ∈ [a, b] for 0 < λ < λ∗ , where λ∗ is a fixed value of λ. Proof. To prove this lemma, let us estimate with respect to λ the coefficients βk in (4.12). We use the notation F (a, c) = B− (a, ξ)Wca (B+ , B),

Φ(a, d) = B+ (a, ξ)Wda (B− , B).

According to Theorem 2 of [16] if condition 1) and 2) are satisfied, then (4.17) has a solution of the form y1,2 (x, γ) = q −1/4 (x)E± (x0 , x) 1 + γ −1 ε1,2 (x, γ) , ⎧ ⎫ x

⎨ ⎬ (4.18) q(t)dt . E± (x0 , x) = exp ±γ ⎩ ⎭ x0

For the functions ε1,2 , the following bounds hold: |εj (x, γ)| ≤ c,

x ∈ [a, b], γ ≥ γ0 > 0, j = 1, 2

where the constant c does not depend on x, γ. The asymptotic relations (4.18) can be differentiated, i.e., (x, γ) = ±q 1/4 (x)E± (x0 , x) 1 + γ −1 ε˜1,2 (x, γ) , y1,2

(4.19)

(4.20)

where the functions ε˜1,2 , are subject to estimates of the form (4.19). Using (4.18) and (4.20) and taking into account that in (4.11) γ = D1 λ−1 , we have $−1/2 #

E− (c, ξ) (4.21) F (γ, c) = s(ξ)s(c) q(ξ)q(c) % & × B (γk , c) 1 + γ −1 ε1 (ξ, γ) − B(γk , c)γ 1/2 q(c) 1 + γ −1 ε2 (ξ, γ) . Since c < ξ, the behavior of F (γ, c) determines the factor of the form E− (c, ξ), and from (4.21). It follows that there is a γ0 such that for γ ≥ γ0 > 0 the function F (γ, c) tends to zero. An analogous estimate also holds for Φ(γ, c), taking into account that d > ξ. Since the coefficients of the expansion (4.11) have the form (4.12), and since ˜ (λ ˜ = D1 γ −1 ) lie the function B(γk , x) are orthonormal, we obtain for 0 < λ < λ, 0 bound ∞ Σ1 q V (c,d) ≤ ak ≤ λM ∗ , λ → 0 (0 ≤ λ ≤ λ1 ) k=0

where the constant M ∗ does not depend on λ. This proves the assertion of the lemma.


13

Based on Lemma 4.5, we apply the Banach contraction principle to the equation −1 q + Π−1 N ΣM q = ΠN f

to obtain the following result. Theorem 4.6. Under the assumptions of Lemma 4.5, if K(γ) belongs to the class SN,M then equation (4.4) are uniquely solvable in the space V (c, d) and the following bound holds: ||q(x)||V (c,d) ≤ m((ΠN , ΣM ) ||f ||W (c,d) . Furthermore, we have: Theorem 4.7. Suppose conditions 1) and 2) of Lemma 4.5 and condition A are satisfied, and K(γ) belongs to the class ΠN , with γρ(γ) = r−1 (γ) then for 0 < λ < λ∗ , where λ∗ is a fixed value of λ, equation (4.2) is uniquely solvable in the space V (c, d), and the following bound holds: ||q(x)||V (c,d) ≤ m((ΠN , Σ∞ ) ||f ||W (c,d) . Theorem 4.7 is a consequence of Theorems 4.1 and 4.6 and is proved with the aid of the well-known procedure of perturbation theory, based on the method of successive approximations, in the same way is in [15]. 4.2. The proof of solution (4.9) accuracy for the large values of parameter λ We investigate conditions under which (4.9) is an asymptotically exact solution of (4.4) as λ → ∞. For this we follow the scheme presented earlier and determine conditions under which the operator Π−1 N ΣM of equation (4.4) is a contraction operator. Everywhere below we assume that the solutions of (4.3) satisfy the symmetry condition B(γ, x) = B(x, γ). (4.22) According to (4.22), the behavior of B(γ, x) as γ → 0 is determined by the behavior of the corresponding solution of (4.3) as x → 0. We reduce equation (4.3) to selfadjoint form; to do this we multiply it by the function r−1 (x). From (4.3) we obtain

Lγ B(γ, x) ]−1− Q(x)B, = [s(x)B 2 Q(x) = t(x) − γ r (x).

s(x) > 0, a ≤ x ≤ b,

(4.23)

Assume that the coefficients s(x) and Q(x) of (4.23) are analytic in the disk |x| < R. Then every solution B(x) of (4.23) is analytic in this disk, i.e., can be expanded in a power series convergent in the disk |x| < R [7]. Lemma 4.8. The operator Π−1 N ΣM of equation (4.4) is a contraction operator acting in the space V (c, d) if the coefficients s(x) and Q(x) of (4.23) are analytic in the disk |x| < R for λ > λa , where λa is some fixed value of λ, and the symmetry condition (4.22) holds.

14


Proof. To prove the lemma we get an estimate with respect to λ for the coefficients in the eigenfunction expansion in problem (4.3) under the corresponding boundary conditions (the coefficients βk in the expression (4.23) of [6]). It follows from the condition of the lemma and the symmetry condition (4.22) that there is a λa such that for λ > λa the solutions B± (a, x) can be represented as a power series in λ−1 that converges in the disk |λ| > λa . This gives us that Σ1 q V (c,d) ≤

∞

ak < λ−1 M a , λ → ∞ (λ > λa )

k=0 a

where the constant M does not depend on λ. Thus, λ can be chosen so that Π−1 N ΣM is a contraction operator under the condition of the given lemma (λa = M a ). We consider separately the case when x = 0 is a regular singular point for (4.23), i.e., s(x) = xϕ(x), ϕ(0) = 0, (4.24) where ϕ(x) > 0 is a function continuous on [a, b]. Note that a function s(x) of the form (4.24) satisfies the conditions of the Fuchs theorem. Lemma 4.9. ([17], Russian p. 628). Let B+ (x) and B− (x) be two linearly independent solutions of (4.23) with coefficient s(x) satisfying condition (4.24). In this case if B+ = 0 then B− (x) has a logarithmic singularity at x = 0. If B+ (x) has an nth-order zero at x = 0 (n > 0), then B− (x) has a pole of order n at x = 0. Lemma 4.10. Suppose that the coefficient s(x) of equation (4.23) has the form (4.24), condition (4.22) holds, and, moreover, s(c)B(γk , c) = s(d)B(γk , d). In this case the operator Π−1 N ΣM of equation (4.4) is a contraction operator acting in the space V (c, d) for λ > λa , where λa is some fixed value of λ. Proof. We estimate the coefficients βk in (2.2) of [6]. It follows from Lemma 4.9 and condition (4.22) that there is a λa such that for λ > λa the condition B+ (0) = 0 implies that Σ1 q V (c,d) ≤

∞

|ak | ≤ M1 λ−1 ln λ, λ → ∞ (λ > λa )

(4.25)

k=0

and if B+ (x) has a zero of order n at x → 0, then M1 λ−1 ln λ is replaced in (4.25) by M2 λ−1 . Applying the contraction mapping principle to the equation −1 q + Π−1 N ΣM q = ΠN f

we get on the basis of Lemmas 4.8 and 4.10 a proof of the existence and uniqueness of a solution of (4.4) under the restrictions imposed.


15

Thus, we have proved Theorem 4.11. Equation (4.4) is uniquely solvable in the space V (c, d) for K(γ) of class SN,M under the conditions of Lemma 4.8 or 4.10, and q(x) V (c,d) ≤ m(ΠN , ΣM ) f W (c,d) .

(4.26)

Moreover, we have Theorem 4.12. Equation (4.4) is uniquely solvable in the space V (c, d) for K(γ) having the properties (3.2)–(3.3) when γρ(γ) = r−1 (γ), Condition A holds for γ > γ a , and the conditions of Lemma 4.8 or 4.10 hold for λ < λa (λa being some fixed value of λ), and the estimate (2.5) holds with ΣM replaced by Σ∞ . Theorem 4.12 follows from Theorems 4.1 and 4.11, and can be proved with the help of a device known in perturbation theory and based on the method of successive approximations, just as in [18]. Example. Examples of representations of the form (2.2) in [6]. 1. t(x) = 0, r(x) = s(x) = const in (4.3), and B(α, ξ) = cos αξ,

B− (iD, ξ) = 0, 5πD−1 exp(−Dξ),

B+ (iD, x) = cosh Dx, βk (iDλ−1 ) =

4πcλ−1 (kπ)2 + D2 λ2 1 × q(ξ) cos kπξ − exp(−Dλ−1 ) cos kπ cosh Dλ−1 ξ dξ. 0 (0)+

Here the conditions of Lemma 4.8 hold. The space V (c, d) ≡ C1/2 (−l, 1), (0)+

where C1/2 (−l, 1) is the space of even functions continuous with the weight √ 1 − x, equipped with the norm

f C(0)+ ,(−l,1) = max F (x) 1 − x2 , 1/2

x∈[−1,1]

and W (c, d) is the space of functions having on [−1, 1] derivatives of first order satisfying a Holder condition with exponent 1/2 + ε, equipped with the usual norm [18]. 2. r(x) = x−1 , s(x) = x, t(x) = −n2 x−2 in (4.3), and B(α, ξ) = Jn (αξ),

B− (iD, ξ) = Kn (Dξ),

B +( iD, x) = In (Dx), n = 0, 1;

16

S. Aizikovich, V. Alexandrov and I. Trubchik where Jn (x) is the corresponding Bessel function, and In (x) and Kn (x) are the modified Bessel functions. We have that 2cλ−1 iD βk = 2 λ Jn+1 (μk )(μ2k + D2 λ−2 ) ⎤ ⎡ 1 D D μk J1−n (μk )In ρ dρ⎦ , × ⎣ q(ρ)ρ Jn (μk ρ) − Kn λ λ 0

(n = 0, 1) obtains.

References [1] V.S. Nikishin The static contact problems for multylayered foundations. The Contact Interaction Mechanics M: Phizmatlit, 2001. 199–213.(in Russian) [2] V.A. Babeshko, E.V. Glushkov, N.V. Glushkova, The methods of the construction of the Green matrix for the stratified elastic half-space. Numerical Mathematics and Math. Physics Journal 27 (1987), no. 1, 93–101. [3] S.M. Aizikovich, V.M. Alexandrov, J.J. Kalker, L.I. Krenev, I.S. Trubchik, Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties. Int. J. of Solids and Structures 39 (2002), no. 10, 2745–2772. [4] S.M. Aizikovich, V.M. Alexandrov, Axisymmetrical Problem about Indentation of Round Punch into Elastic Inhomogeneous with Depth Half-Space. Izv. Akad. Nank SSSR. Mech. Tverd. Tela 2 (1984), 73–82. [5] S.M. Aizikovich, I.S. Trubchik, and E.V. Shklyarova, Penetration of a die into a vertically inhomogeneous strip. Izv. AN SSSR. Mekh. Tv. Tela 25 (1991), no. 1, 61–71. [6] I.I. Vorovich, V.M. Alexandrov, and V.A. Babeshko, Nonclassical mixed problems of elasticity theory. M: Nauka, 1974. (in Russian) [7] S.M. Aizikovich, V.M. Alexandrov, Properties of compliance functions corresponding to layered continuously inhomogeneous half-spaces. Dokl. Akad. Nauk SSSR 266 (1982), no. 1, 40–43. [8] A.K. Privarnikov, Spatial Deformation of a Multilayered Foundation. Stability and Strength of Structure Elements. Dnepropetrovsk University. Dnepropetrovsk, 1973, 27–45. [9] N.I. Akhiezer,The Lecture of Theory of Approximation. M: Nauka, 1965. [10] V.A. Babeshko, Asymptotic Properties of Solutions of Certain Two-Dimensional Integral Equations. Dokl. Akad. Nauk SSSR 206 (1972), no. 5, 1074–1077. English transl. in Soviet Phys. Dokl. 17. [11] V.L. Goncharov, Theory of interpolation and function approximation. M: ONTIGTTI, 1934 [12] A.I. Lurje The elasticity theory. M: Nauka, 1970. (in Russian)


17

[13] R.E. Gibson, P.T. Brown, and K.R.F. Andrews, Some results concerning displacements in a nonhomogeneous elastic layer. Z. angew. Math. und Phys. 22 (1971), no. 5, 855–864 [14] V.M. Alexandrov, The solution of a class of dual equations. Dokl. Akad. Nauk SSSR 307 (1973), no. 2, 55–58. [15] I.I. Vorovich, and V.A. Babeshko, Dynamical mixed problems of elasticity theory for non-classical domains. M: Nauka, 1979. (in Russian) [16] M.V. Fedoruk,Ordinary differential equations. M: Nauka, 1985. (in Russian) [17] A. Zigmund, Trigonometric Series. Cambridge, 1988. [18] S.M. Aizikovich, Asymptotic solutions of contact problems of elasticity theory for depthwise inhomogeneous media. Prikl. Math. Mekh. 46 (1982), no. 1, 148–158. Sergey Aizikovich Gagarina sqr.,1 P.O. Box 4845 344090 Rostov-on-Don, Russia e-mail: [email protected] Victor Alexandrov Institute on Mechanics Problems at RAS Moscow, Russia e-mail: [email protected] Irina Trubchik Institute of Mechanics and Applied Math. at SFU Rostov-on-Don, Russia e-mail: [email protected]



Krein Systems D. Alpay, I. Gohberg, M.A. Kaashoek, L. Lerer and A. Sakhnovich In memory of Mark Grigorievich Krein, with appreciation of his many great discoveries, on the occasion of his Centennial.

Abstract. In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix-valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role. Mathematics Subject Classification (2000). Primary: 34A55, 49N45, 70G30; Secondary: 93B15, 47B35.

1. Introduction The following result is due to M.G. Krein, see [14]: Theorem 1.1. Let T > 0, and let k be a scalar continuous and hermitian function on the interval [−T, T] such that for each 0 < τ ≤ T the corresponding convolution integral operator Tτ on L1 [0, τ ], τ (Tτ f )(t) = f (t) − k(t − s)f (s) ds, 0 ≤ t ≤ τ, (1.1) 0

is invertible. Let γτ (t, s) denote the resolvent kernel τ γτ (t, s) − k(t − v)γτ (v, s)dv = k(t − s),

0 ≤ t, s ≤ τ.

(1.2)

0

Daniel Alpay wishes to thank the Earl Katz family for endowing the chair which supported his research. The work of Alexander Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant no. Y330.

20

D. Alpay, I. Gohberg, M. Kaashoek, L. Lerer and A. Sakhnovich

Consider the entire function

τ −iλx 1+ e γτ (x, 0)dx , P(τ, λ) = e 0 τ P∗ (τ, λ) = 1 + eiλx γτ (τ − x, τ )dx. iλτ

(1.3) (1.4)

0

Then with a(τ ) = γτ (τ, 0) and for λ ∈ C it holds that ⎧ ∂ ⎪ ⎪ P(τ, λ) = iλP(τ, λ) + P∗ (τ, λ)a(τ ), 0 ≤ τ ≤ T, ⎨ ∂τ (1.5) ⎪ ⎪ ∂ P (τ, λ) = P(τ, λ)a(τ )∗ . ⎩ ∗ ∂τ Putting Y (τ, λ) = P(τ, λ) P∗ (τ, λ) , the system (1.5) can be rewritten as ∂ 0 a(τ ) Ir 0 + . (1.6) Y (τ, λ) = Y (τ, λ) iλ a(τ )∗ 0 0 0 ∂τ Here τ ∈ [0, T]. We call (1.6) a Krein system when, as in (1.5), the function a is given by a(τ ) = γτ (τ, 0), where γτ (t, s) is the resolvent kernel corresponding to some k on [−T, T] with the properties described in the Theorem 1.1. In that case, following Krein, the function k is called an accelerant for (1.6), and we shall refer to a as the potential associated with the accelerant k. The functions P(τ, ·), P∗ (τ, ·) are called Krein orthogonal functions at τ associated to the weight δ − k, where δ is the delta function. In this paper we prove the analogue of Theorem 1.1 for systems with accelerants that are allowed to have a jump discontinuity at the origin. We also present explicit formulas for determining the unique accelerant k from the given potential a. As for continuous accelerants in [14], the results are proved not only for scalar functions but also for the matrix-valued case, when in (1.5) the functions P, P∗ and a are Cr×r -valued. The result expressing the accelerant in terms of the potential referred to in the previous paragraph is based on a recent theorem involving a certain analog R(B, D) of the resultant operator for a class of entire matrix functions B and D. The resultant R(B, D) is defined as follows (see Section 3 for more details). Let B and D be of the form 0 τ iλu B(λ) = Ir + e b(u)du and D(λ) = Ir + eiλu d(u)du, −τ

where the functions b resultant of B and D

0

and d belong respectively to Lr×r [−τ, 0] and Lr×r [0, τ ]. 1 1 [−τ, τ ] by: is the operator defined on the space Lr×r 1 ⎧ τ ⎪ ⎪ ⎪ ⎨q(u) +

(R(B, D)q)(u) =

⎪ ⎪ ⎪ ⎩q(u) +

−τ τ −τ

d(u − s)q(u)du,

0 ≤ u ≤ τ,

b(u − s)q(u)du,

−τ ≤ u < 0.

The

Krein Systems

21

Let us now state our main results. Theorem 1.2. Let k be a r × r-matrix-valued accelerant on [−T, T], with possibly a jump discontinuity at the origin, and let γτ (t, s) be the corresponding resolvent kernel as in (1.2). Put τ iλτ −iλx Ir + P(τ, λ) = e e γτ (x, 0)dx (1.7) 0 τ P∗ (τ, λ) = Ir + eiλx γτ (τ − x, τ )dx. (1.8) 0

Then a(τ ) = γτ (0, τ ), with 0 < τ ≤ T, extends to a continuous function on [0, T] and Y (τ, λ) = P(τ, λ) P∗ (τ, λ) , satisfies the Krein system (1.6) with potential a. For our second main result we need the matrizant of (1.6). By definition, this is the unique C2r×2r -valued solution U (τ, λ) of (1.6) satisfying the initial condition U (0, λ) ≡ I2r . Theorem 1.3. Let k be a r × r-matrix-valued accelerant on [−T, T], with possibly a jump discontinuity at the origin, and let a be the corresponding potential. Then k is uniquely determined by a, and k can be obtained from a in the following way. Let U (τ, λ) be the matrizant of (1.6), and put I 0 F (λ) = eiλT Ir Ir U (T, −λ) r , G(λ) = Ir Ir U (T, −λ) . Ir 0 Then F and G are entire r × r matrix functions of the form T 0 f (x)eiλx dx, G(λ) = Ir + g(x)eiλx dx, F (λ) = Ir + −T

0

where f and g are continuous C -valued functions on [0, T] and [−T, 0], respectively. Moreover, the resultant operator R(F , G ) is invertible, and the function k is given by the formula k = [R(F , G )]−1 q. (1.9) ∗ ∗ ∗ ¯ and G (λ) = G(λ) ¯ , where the superscript means taking Here F (λ) = F (λ) adjoints. Finally, q is the function on the interval [−T, T] given by + f (−x)∗ , −T ≤ x < 0, q(x) = g(−x)∗ , 0 ≤ x ≤ T. r×r

To prove Theorem 1.2 we use in an essential way the results of [12]. The proof of Theorem 1.3 is based on recent results of [7] on the continuous analog of the resultant. In each of the two theorems above our starting point is a given accelerant. In a next paper we plan to study the inverse situation, which includes, in particular,

22


the question whether or not a continuous potential is always generated by an accelerant. Let us illustrate Theorem 1.2 with an example. Take k to be + i, if t ∈ [0, T], k(t) = (1.10) −i, if t ∈ [−T, 0]. Clearly, k is continuous with a jump discontinuity at zero, and k is hermitian. Note that this function is of the form + iCe−itA (I − P )B, t ∈ [0, T], k(t) = (1.11) −iCe−itA P B, t ∈ [−T, 0], 0 0 0 0 A= , P = , and C = B ∗ = 1 1 . 0 0 0 1 The formulas from [4] allow us to show that for this k the integral operator Tτ in (1.1) is invertible for τ < π2 . Hence k is an accelerant on [−T, T] whenever T < π2 . Furthermore, again using the formulas from [4], one computes that for each τ < π2 the resolvent kernel associated to k, that is, the solution γτ (t, s) of (1.2), is given by ⎧ 2i(t−s) ie ⎪ ⎪ , 0 ≤ s < t ≤ τ, ⎨ 1 + e2iτ γτ (t, s) = 2i(t−s) ⎪ ⎪ ⎩ −ie , 0 ≤ t < s ≤ τ. 1 + e−2iτ Direct computations show then that the functions P and P∗ defined by the formulas (1.5) are equal to with

e2iτ − eiλτ 2 , 2iτ 1+e 2−λ 2 e2iτ − eiλτ P∗ (τ, λ) = 1 + , 1 + e2iτ 2−λ and that these functions satisfy the system (1.5) with P(τ, λ) = eiλτ +

2i , τ ∈ [0, T]. (1.12) 1 + e−2iτ Other examples will be given in the final two sections of the paper. We now give the outline of the paper. The rest of the paper consists of five sections. In Section 2 we show that a Krein system can be associated to accelerants with jump discontinuities and prove Theorem 1.2. In Section 3 we review the notion of continuous analogue of the resultant and state the results from [7] used in this paper. The proof of Theorem 1.3 is given in Section 4. The last two sections present examples. In Section 5 we consider the case of accelerants k of the form (1.11), where A, B and C are matrices of appropriate sizes and P is a projection commuting with A. This includes in particular the case when the Fourier transform a(τ ) =

Krein Systems

23

of k (considered as a function on R) is a rational matrix-valued function vanishing at infinity. Such functions k have in general a jump discontinuity at the origin. In Section 6 a class of continuous accelerants is elaborated.

2. Krein system for accelerants with jump discontinuity and proof of Theorem 1.2 In the proof of Theorem 1.1 an important role is played by the equations ∂ γτ (t, s) = γτ (t, τ )γτ (τ, s), ∂τ

0 ≤ s, t ≤ τ,

(2.1)

∂ γτ (τ − t, τ − s) = γτ (τ − t, 0)γτ (0, τ − s), 0 ≤ t, s ≤ τ. (2.2) ∂τ Equation (2.1) is called the Krein-Sobolev identity. The second equation is obtained by replacing in equation (1.2) the function k(t) by k(−t). The corresponding resolvent kernel is equal to γτ (τ − t, τ − s), as can be seen by a change of variables; see the discussion [5, p. 450] and in particular equation (3.5) there. The above equations have been used by M.G. Krein in [15] to deduce his system (1.5) in the case of a continuous accelerant. It is known [13, Section 7.3, p. 187] that continuity of the accelerant is not necessary to insure that the Krein-Sobolev identity holds. In fact, when k has a jump discontinuity at the origin appropriate generalizations of (2.1)–(2.2) have been established in [12]. Before presenting the proof of Theorem 1.2, we first review the necessary results from [12]. In what follows k is a r × r accelerant on [−T, T] with a possible jump discontinuity at the origin and γτ (t, s) is the corresponding resolvent kernel as in (1.2). From [12] we know that the function (t, s, τ ) → γτ (t, s) is continuous on the domain 0 ≤ s < t ≤ T, 0 < τ ≤ T and on the domain 0 ≤ t < s ≤ T, 0 < τ ≤ T. Moreover, (t, s, τ ) → γτ (t, s) admits continuous extensions on the closures of these domains. In particular, a(τ ) = γτ (τ, 0) is continuous on the left open interval (0, T] and has a continuous extension to the closed interval [0, T]. Next, we consider the modifications of equations (2.1)–(2.2). Using the fact that k has a jump discontinuity at the origin, we let k+ be the function equal to k for t = 0 k+ (0) = lim k(h). h→0 h>0

Similarly, let k− be the function equal to k for t = 0 and defined at the origin by k− (0) = lim k(h). h→0 h 0: τ ∂ iλτ ∂ P(τ, λ) = iλP(τ, λ) + e e−iλx γτ (x, 0) dx ∂τ + ∂τ + 0 τ ∂ = iλP(τ, λ) + γτ (τ, 0) + eiλ(τ −x) + γτ (x, 0) dx ∂τ 0 τ = iλP(τ, λ) + γτ (τ, 0) + eiλ(τ −x) γτu (x, τ )γτl (τ, 0) dx 0 τ iλ(τ −x) = iλP(τ, λ) + In + e γτ (x, τ ) dx γτ (τ, 0) 0 τ = iλP(τ, λ) + In + eiλx γτ (τ − x, τ ) dx γτ (τ, 0) 0

= iλP(τ, λ) + P∗ (τ, λ)γτ (τ, 0).

Krein Systems

25

Here we removed the superscripts u and l using (2.3) and using the fact that the value of an integral does not depend on the value of the integrand at one point. Using now (2.10) we obtain in a similar way that ∂ P(τ, λ) = iλP(τ, λ) + P∗ (τ, λ)γτ (τ, 0). ∂τ − ∂ It follows that ∂τ P(τ, λ) exists and that the first equality in (1.5) holds. Analogously, using the (2.8) and (2.10), we have τ ∂ ∂ P∗ (τ, λ) = eiλx γτ (τ − x, τ ) dx ∂τ ± ∂τ ± 0 τ ∂ = eiλτ γτ (0, τ ) + eiλx ± γτ (τ − x, τ ) dx ∂τ 0 τ iλτ = e γτ (0, τ ) + eiλx γτ (τ − x, 0)γτ (0, τ ) dx 0 τ = eiλτ Ir + eiλ(x−τ ) γτ (τ − x, 0) dx γτ (0, τ ), 0 τ eiλx γτ (x, 0) dx γτ (0, τ ) = eiλτ Ir + 0

= P(τ, λ)γτ (0, τ ). ∗ Since k is hermitian, we have γτ (0, τ ) = γτ (τ, 0) . Thus P and P∗ satisfy (1.5), and hence Y (τ, λ) = P(τ, λ) P∗ (τ, λ) satisfies (1.6).

3. Intermezzo: The continuous analogue of the resultant We review here the results of [7] needed in the proof of Theorem 1.3. The definition of the resultant operator R(B, D) has already been given in the introduction. Consider an entire matrix function of the form τ L(λ) = Ir + eiλx (x)dx, ∈ Lr×r [0, τ ]. (3.1) 1 0

With a slight abuse of terminology, following [7], we call L(λ) a Krein orthogonal matrix function if there exists a hermitian Cr×r -valued function k ∈ Lr×r [−τ, τ ] 1 such that τ

(t) −

k(t − u)(u) du = k(t),

0 ≤ t ≤ τ.

0

In that case we refer to δ − k as the associate weight. The following result is proved in [7, Theorem 5.6]. Theorem 3.1. Let L be a Cr×r -valued entire function of the form (3.1). Then there exists a hermitian matrix function k ∈ Lr×r [−τ, τ ] such that L is the Krein 1

26


orthogonal matrix function with weight δ − k if and only if there exists a matrix function M of the form τ M (λ) = Ir + eiλu m(u)du, m ∈ Lr×r [0, τ ], (3.2) 1 0

such that the following two conditions are satisfied: L(λ)L (λ) = M (λ)M (λ),

λ ∈ C,

Ker L (λ) ∩ Ker M (λ) = {0} ,

(3.3)

λ ∈ C.

(3.4)

Furthermore, when these conditions hold, the function k is given by the formula + (−u)∗ , −τ ≤ u ≤ 0, −1 (3.5) q, q(u) = k = R(L , M ) m(u), 0 ≤ u ≤ τ. In [7] the above theorem is derived as a corollary of the following somewhat more general theorem ([7, Theorem 5.5]). Theorem 3.2. Given , m ∈ Lr×r [0, τ ], put 1 τ iλu L(λ) = Ir + e (u)du, M (λ) = Ir + 0

τ

eiλu m(u)du.

0

Then there is a hermitian matrix function k ∈ Lr×r [−τ, τ ] such that 1 τ (t) − k(t − u)(u) du = k(t), 0 ≤ t ≤ τ,

(3.6)

0

τ

m(u)k(t − u) du = k(t),

m(t) −

0 ≤ t ≤ τ,

(3.7)

0

if and only if the two conditions (3.3) and (3.4) are satisfied, and in that case the function k is uniquely determined by (3.5). In general, a Krein orthogonal matrix function L may have many different weights. This is reflected by the fact that given L as in Theorem 3.1 there may be many different functions M of the form (3.2) satisfying (3.3) and (3.4). However, as soon as M is fixed, then the weight is uniquely determined by (3.5) (as we see from Theorem 3.2). Remark. If in (3.5) the functions and m are continuous on the interval [0, τ ], then the function q in the right-hand side of (3.5) is a continuous function on [−τ, τ ] with a possible jump discontinuity at zero. This implies that the function k defined by (3.5) is also continuous on [−τ, τ ] with a possible jump discontinuity at zero.

Krein Systems

27

4. Proof of Theorem 1.3 Throughout this section k is a r × r-matrix-valued accelerant on [−T, T], with possibly a jump discontinuity at the origin, and we consider the Krein system (1.6) with the potential a defined by k. Furthermore U (τ, λ) will be the matrizant of (1.6). Our aim is to prove Theorem 1.3. As in Theorem 1.3, put Ir 0 iλT Ir Ir U (T, −λ) F (λ) = e , G(λ) = Ir Ir U (T, −λ) . Ir 0 First let us show that F (λ) = eiλT P(T, −λ)

and G(λ) = P∗ (T, −λ),

(4.1)

where P(T, λ) and P∗ (T, λ) are defined by (1.7) and (1.8) with τ = T. To obtain (4.1) note that for each λ ∈ C the two r × 2r matrix functions

Ir

Ir U (τ, λ)

and

P(τ, λ)

P∗ (τ, λ)

satisfy thelinear differential equation (1.6), and at τ = 0 both functions are equal to Ir Ir . Thus both have the same initial condition at τ = 0. It follows that these two functions coincide on 0 ≤ τ ≤ T. For τ = T this yields the identities in (4.1). Using (4.1), we see from the formulas for P and P∗ in (1.7) and (1.8) that F (λ) = Ir +

T

f (x)e

iλx

dx,

G(λ) = Ir +

0

0

g(x)eiλx dx,

(4.2)

−T

with f (x) = γT (x, 0) on 0 ≤ x ≤ T and g(x) = γT (T + x, T) on the interval −T ≤ x ≤ 0. In particular, the functions f and g are continuous on their respective domains as desired. It remains to prove (1.9). To do this we first derive the following lemma. Lemma 4.1. The functions P and P∗ given by (1.7) and (1.8), respectively, satisfy the identity P(τ, λ)P (τ, λ) = P∗ (τ, λ)P∗ (τ, λ)

(0 ≤ τ ≤ T, λ ∈ C).

(4.3)

Furthermore, for each 0 ≤ τ ≤ T the left-hand side in the above identity is a right canonical factorization (that is, for each 0 ≤ τ ≤ T the function det P(τ, λ) has no zero in the closed lower half-plane) while the right side is a left canonical factorization (that is, for each 0 ≤ τ ≤ T the function det P∗ (τ, λ) has no zero in the closed upper half-plane).

28


Proof. Fix 0 ≤ τ ≤ T. Recall that the integral operator Tτ defined by (1.1) is selfadjoint and invertible. Let aτ , bτ , bτ , dτ be the L1 -functions defined by τ aτ (t) − k(t − u)aτ (u) du = k(t), 0 ≤ t ≤ τ, 0 0

bτ (t) −

−τ 0

cτ (t) −

−τ τ

bτ (u)k(t − u) du = k(t),

−τ ≤ t ≤ 0,

k(t − u)cτ (u) du = k(t),

−τ ≤ t ≤ 0,

dτ (u)k(t − u) du = k(t),

0 ≤ t ≤ τ,

dτ (t) −

0

and put

Bτ (λ) = I +

eiλs aτ (s) ds,

Cτ (λ) = I +

τ

Aτ (λ) = I + 0 0

−τ

0 −τ τ

Dτ (λ) = I +

eiλs cτ (s) ds,

eiλs bτ (s) ds, eiλs dτ (s) ds.

0

In terms of the resolvent kernel γτ (t, s) associated with k we have (0 ≤ x ≤ τ );

aτ (x) = γτ (x, 0),

bτ (−x) = γτ (0, x)

cτ (x) = γτ (τ + x, τ )

dτ (−x) = γτ (τ, τ + x)

(−τ ≤ x ≤ 0).

Note that in this terminology, the functions P and P∗ given by (1.7) and (1.8) are equal to P(τ, λ) = eiλτ Aτ (−λ),

P∗ (τ, λ) = Cτ (−λ).

(4.4)

From Theorem 5.3 in [7] we know that Aτ (λ)Bτ (λ) = Cτ (λ)Dτ (λ),

Ker Bτ (λ) ∩ Ker Dτ (λ) = {0}.

(4.5)

Next recall that k is hermitian. This implies that bτ (−x) = aτ (x)∗ ,

cτ (−x) = dτ (x)∗

(0 ≤ x ≤ τ ),

and hence Aτ (λ) = Bτ (λ) and Dτ (λ) = Cτ (λ). In particular, (4.5) reduces to Aτ (λ)Aτ (λ) = Dτ (λ)Dτ (λ),

Ker Aτ (λ) ∩ Ker Dτ (λ) = {0}.

(4.6)

Finally, since for each 0 ≤ τ ≤ T the operator Tτ in (1.1) is selfadjoint and invertible, it follows that Tτ is strictly positive for each 0 ≤ τ ≤ T. Then we know (using the theory of Krein orthogonal functions; see Theorem 8.1.1 in [6]) that the function det Aτ (λ) has no zero in the closed upper half-plane, and the function det Dτ (λ) has no zero in the closed lower half-plane. Thus Aτ (λ)Aτ (λ) is a left canonical factorization and Dτ (λ)Dτ (λ) is a right canonical factorization. Using (4.4) the above remarks provide the proof of the lemma.

Krein Systems

29

We are now ready to prove (1.9). From (4.1) and (4.3) it follows that F (λ)F (λ) = G(λ)G (λ).

(4.7)

Moreover the left-hand side of this identity is a left canonical factorization and the right-hand side is a right canonical factorization. In particular, Ker F ∩ Ker G = {0}. This allows us to apply Theorem 3.2 with τ = T, (u) = f (u) and m(u) = g(−u)∗ , where the functions f and g are as in (4.2). In other words, we apply L = F [−T, T] such and M = G . It follows that there exists a unique hermitian k˜ ∈ Lr×r 1 that T ˜ − s)f (s) ds = k(t), ˜ f (t) − k(t 0 ≤ t ≤ T, (4.8) g(t) −

0 0

−T

˜ − s)g(s) ds = k(t), ˜ k(t

Moreover, k˜ is given by the formula k˜ = [R(F , G )]−1 q

with q(x) =

+

−T ≤ t ≤ 0.

f (−x)∗ , g(−x)∗ ,

(4.9)

−T ≤ x ≤ 0, 0 ≤ x ≤ T.

Since f (x) = γT (x, 0) on 0 ≤ x ≤ T and g(x) = γT (T + x, T) on the interval −T ≤ x ≤ 0, we know from the proof of Lemma 4.1 that (4.8) and (4.9) also hold with k˜ being replaced by the original accelerant k. But then, by the uniqueness statement in Theorem 3.2, the functions k˜ and k coincide. Thus (1.9) holds, which completes the proof of Theorem 1.3. Remark. In the proof of Lemma 4.1 we used in an essential way the accelerant and its properties. However, this is not necessary. It is possible to give a proof of Lemma 4.1 without any reference to the accelerant. In fact, such a proof can be obtained by using the properties of a canonical differential systems of Dirac ¯ ∗ is a solution of a canonical differential type. To see this note that e−iτ λ Y (τ, −2λ) system of Dirac type with potential v(τ ) = −ia(τ ) whenever Y (τ, λ) is a solution of (1.6). We will come back to this in a later paper.

5. An example with jump discontinuity: the rational case In this section we consider the case where the accelerant is of the form + iCe−itA (I − P )B, t > 0, k(t) = −iCe−itA P B, t < 0.

(5.1)

In this expression, A, B and C are matrices of appropriate sizes and P is a projection commuting with A. Motivation for such a form originates with linear system theory. Indeed, let W be a rational Cp×q -valued function, analytic at infinity. Then, as is well known, W admits a realization of the form W (λ) = D + C(λIN − A)−1 B,

30


where D = W (∞) and (A, B, C) ∈ CN ×N × CN ×q × Cp×N . Assume furthermore that A has no real eigenvalues. Then, the function W belongs to the Wiener algebra, and eiλt k(t)dt,

W (λ) = D + R

where k is of the form (5.1) with P being the Riesz projection corresponding to the eigenvalues of A in the upper half-plane. Note that, in general, functions k of the form (5.1) need not have summable entries. In this section we first take × a −bb∗ A= , 0 a×∗

B=

b , c∗

C = −c

−b∗ ,

(5.2)

where (a, b, c) ∈ Cn×n × Cn×k × Ck×n , and throughout it is assumed that the spectra of a and a× = a − bc are both in the open upper half-plane. For P we take the Riesz projection of A corresponding to the eigenvalues in the upper half-plane. In other words P is given by I iΩ , (5.3) P = 0 0 where Ω is the unique solution of the Lyapunov equation i(Ωa×∗ − a× Ω) = bb∗ . With A, B, C and P as in (5.2) and (5.3), the function W (λ) = Ir + eiλt k(t)dt

(5.4)

(5.5)

R

is positive definite on the real line. Conversely, any rational r×r matrix function W which is positive definite on the real line and analytic at infinity with W (∞) = Ir can be represented in this way (see [1]). Proposition 5.1. When k is of the form (5.1) with A, B, C and P being given by (5.2) and (5.3), then k is an accelerant on each interval [−T, T]. Moreover, in this case the corresponding potential is given by ∗ ∗ a(τ ) = i (In + Ω(Y − e−iτ a Y aiτ a ))−1 (b + iΩc∗ ) , (5.6) where Ω is given by (5.4), and where Y is the solution of the Lyapunov equation i(Y a − a∗ Y ) = −c∗ c.

(5.7)

Proof. The fact that the function W in (5.5) is positive definite on the real line implies that for each τ the integral operator Tτ in (1.1) is strictly positive. Hence k is an accelerant on each interval [−T, T]. Using Theorem 4.1 in [4] one computes that in this setting γτ (0, τ ) = −iC(P e−iτ (A−BC)Im P )−1 P B.

Krein Systems

31

Since A, B and C are given by (5.2), we have a 0 A − BC = ∗ . c c a∗ It then follows, as computed in [1, p.15], that ∗

γτ (0, τ ) = −i(In + Ω(Y − e−iτ a Y aiτ a ))−1 (b + iΩc∗ ), where Ω and Y are given by (5.4) and (5.7), respectively. Since the potential is given by a(τ ) = γτ (τ, 0) and γτ (τ, 0) = γτ (0, τ )∗ , we see that a is given by (5.6). Next we assume that the matrices A, B, and C in (5.1) are given by ∗ √ γ2 √ β γ2 γ2∗ A=2 , B= 2 , C = 2 γ1∗ γ2∗ , 0 β γ1

(5.8)

where β is a square matrix of order n, and γ1 and γ2 are matrices of sizes n × r. Furthermore, we assume that β ∗ − β = iγ2 γ2∗ . A triple of matrices β, γ1 and γ2 with these properties will be called admissible. For the matrix P in (5.1) we take I −iIn . (5.9) P = n 0 0 The fact that the triple of matrices β, γ1 and γ2 is assumed to be admissible implies that with A, B, C and P as in (5.8) and (5.9), the function (5.5) is positive semidefinite on the real line. Conversely, any rational r × r matrix function W which is positive semi-definite on the real line and analytic at infinity with W (∞) = Ir can be represented in this way (see [9], also [10]). For information about the connection between the matrices A, B, C and P in (5.2) and (5.3) and those in (5.8) and (5.9), we refer to the introduction of [3]. Proposition 5.2. Let β, γ1 and γ2 be an admissible triple, and put k(t) = −2(γ1 + iγ2 )∗ e−2itβ γ1 ,

k(−t) = k(t)∗

(t > 0).

(5.10)

Then k is an accelerant on each interval [−T, T], and the corresponding potential is given by ∗

a(τ ) = −2(γ1 + iγ2 )∗ e−iτ α Σ(τ )−1 e−iτ α γ1 , where

Σ(t) = In +

t

Λ(s)Λ(s)∗ ds,

Λ(t) = e−itα γ1

α = β − γ1 γ2∗ ,

(5.11)

−eitα (γ1 + iγ2 ) .

(5.12)

0

Proof. Let A, B, C and P be given by (5.8) and (5.9). Put iIn I . S= n 0 In

32


Then S is invertible, and one computes that ∗ √ 2β 0 −i(γ1 + iγ2 ) S −1 , B = 2S A=S , 0 2β γ1 √ 0 −1 I S . C = 2 γ1∗ i(γ1 + iγ2 )∗ S −1 , P = S n 0 0 It follows that iCe

−itA

0 −i(γ1 + iγ2 ) 0 i(γ1 + iγ2 ) γ1 0 e−2itβ

(I − P )B = 2i γ1∗

∗

= −2(γ1 + iγ2 )∗ e−2itβ γ1 . Analogously −iCe

−itA

P B = −2i γ1∗

e−2itβ ∗ i(γ1 + iγ2 ) 0

0 0

∗

−i(γ1 + iγ2 ) γ1

∗

= −2γ1∗e−2itβ (γ1 + iγ2 ). It follows that k given by (5.10) can be written in the form (5.1) with A, B, C, and P as in (5.8) and (5.9). Next, we consider A× = A − BC. We have ∗ 0 α , where α = β − γ1 γ2∗ . A× = 2 −γ1 γ1∗ α The proof of Proposition 4.1 in [9] shows that ×

P e−itA |Im

∗

P

= e−itα Σ(t)e−itα ,

t ≥ 0.

Since Σ(t) is positive definite, the matrix Σ(t) is invertible. Hence the map × P e−itA |Im P , viewed as an operator acting on Im P , is invertible. By Theorem 4.3 in [4] this implies that for our k the integral operator Tτ given by (1.1) is invertible for each τ , and γτ (0, τ ) = −iC(P e−iτ (A−BC) Im P )−1 P B ∗

= −2γ1∗ eiτ α Σ(τ )−1 eiτ α (γ1 + iγ2 ). Here we used that PB =

√

2

−i(γ1 + iγ2 ) , 0

C|Im

P

=

√ ∗ 2γ1 .

Since the potential is given by a(τ ) = γτ (τ, 0) and γτ (τ, 0) = γτ (0, τ )∗ , we see that a is given by (5.11). Remark. Note that the two propositions in this section do not cover the example presented in the introduction. Indeed, when k is given by (1.10), then k is not an accelerant for [−π/2, π/2].

Krein Systems

33

6. Another class of potentials We now consider the case where the Cr×r -valued accelerant k admits a representation of the form k(t) = CetA B,

t ∈ [−T, T],

(6.1)

where A, B and C are matrices of appropriate sizes. We assume that there exists a hermitian matrix H such that HA + A∗ H = 0

and C = B ∗ H.

(6.2)

The latter implies that k(t)∗ = k(−t) on [−T, T], and hence k is a hermitian kernel. Under certain minimality conditions the converse statement is also true. More precisely, if k given by (6.1) with the pair (A, B) being controllable and the pair (C, A) being observable, then k(t)∗ = k(−t) implies that there exists a unique invertible hermitian matrix H such that (6.2) holds. Let τ ∈ (0, T]. As proved in [8], equation (1.2) has a unique solution if and only if the matrix τ Mτ = I − e−sA BCesA ds (6.3) 0

is invertible. When this is the case, we have: γτ (t, s) = CetA Mτ−1 e−sA B,

s, t ∈ [−τ, τ ].

(6.4)

Proposition 6.1. Assume k is given by (6.1), and let H be a hermitian matrix H such that (6.2) holds. Then k is an accelerant if and only if the matrix Mτ in (6.3) is non-singular for 0 ≤ τ ≤ T. In that case the corresponding potential is given by a(t) = CetA Mt−1 B,

0 < t ≤ T,

(6.5)

and the functions , P(τ, λ) = eiλτ Ir + C(A − iλIr )−1 eτ A − eiλτ Ir Mτ−1 B, , P∗ (τ, λ) = Ir + C(A − iλIr )−1 eτ A − eiλτ Ir Mτ−1 e−τ A B, are the associate Krein orthogonal matrix functions. Proof. Since k is hermitian, the operator Tτ will be strictly positive if and only Tτ is invertible. The latter happens if and only if Mτ is non singular. Thus k is an accelerant if and only if Mτ is non-singular for 0 ≤ τ ≤ T. Assume k to be an accelerant. Then the potential is given by a(t) = γt (t, 0) on (0, T]. Using (6.4), this yields (6.5). Furthermore, the associate Krein orthogonal

34


function P for k can be computed as follows: τ iλτ −iλx Ir + P(τ, λ) = e e γτ (x, 0)dx 0 τ iλτ −iλx xA −1 Ir + e Ce Mτ Bdx =e 0 τ iλτ −iλx xA −1 Ir + C e e dx Mτ B =e 0 % & = eiλτ Ir + C(A − iλI)−1 eτ (A−iλ) − Ir Mτ−1 B , = eiλτ Ir + C(A − λI)−1 eτ A − eiλτ Ir Mτ−1 B. Analogously,

τ

P∗ (τ, λ) = Ir +

eiλx γτ (τ − x, τ )dx 0

τ

eiλx Ce(τ −x)AMτ−1 e−τ A Bdx τ = Ir + Ceτ A eiλx e−xA dx Mτ−1 e−τ A B 0 τA = Ir + Ce (iλ − A)−1 e(iλ−A)τ − (iλ − A)−1 Mτ−1 e−τ A B = Ir + C(A − iλIr )−1 eτ A − eiλτ Ir Mτ−1 e−τ A B.

= Ir +

0

This completes the proof.

Corollary 6.2. Assume k is given by (6.1), and assume that (6.2) holds with H = −I. Then k is an accelerant. In particular, if rj > 0 and βj ∈ R for j = 1, . . . , n, then the function n k(t) = − rn eiβν t (6.6) ν=1

is an accelerant for each each interval [−T, T]. Proof. From H = −I, we see that the matrix Mτ in (6.3) can be rewritten as τ sA ∗ sA Ce Ce ds. Mτ = I + 0

It follows that Mτ is positive definite and hence non-singular for each τ ≥ 0. Thus k is an accelerant by Proposition 6.5 above. Next, consider the function k in (6.6). Since rj > 0 and βj ∈ R for each j = 1, . . . , n, we can represent k as in (6.1) by taking √ √ √ r1 r2 · · · rn , B = −C ∗ . A = diag (iβ1 , iβ2 , . . . , iβn ), C = But then (6.2) holds with H = I. By the result of the first paragraph, this shows that k is an accelerant on [−T, T] for each T > 0.

Krein Systems

35

From (6.3) it follows that d Mτ = −e−τ A BCeτ A . (6.7) dτ This together with the explicit formula (6.4) allows one to give a direct proof of the Krein-Sobolev equation (2.1) and of equation (2.2) for accelerants as in (6.1). The class of accelerants considered in this section includes the restrictions of polynomials to [−T, T]. On the other hand, when considered for t on the whole real line, k is never integrable (except for the trivial case k = 0). Thus this class of accelerants has a zero intersection with the accelerants considered in the first part of the previous section. Nevertheless the class of potentials corresponding to the accelerants considered in this section shares a number of common properties with the strictly pseudo-exponential potentials. For instance, using (6.7), we have Mτ =

a(0) = CB a (0) = CAB + (CB)2 .. . and there exist non commutative polynomials f0 , f1 , . . . such that CA B = f (v(0), . . . , v ( ) (0)),

= 0, 1, . . .

Thus, and as for strictly pseudo-exponential potentials (see [2]), one can in principle recover the potential from the values of its first derivatives at the origin (cf., [11], where such results are proved for pseudo-exponential potentials).

References [1] D. Alpay and I. Gohberg, Inverse spectral problem for differential operators with rational scattering matrix functions. Journal of differential equations 118 (1995), 1–19. [2] D. Alpay and I. Gohberg, Potentials associated to rational weights. New results in operator theory and its applications, Oper. Theory Adv. Appl. 98 (1997), 23–40. [3] D. Alpay, I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential. Math. Nachr. 215 (2000), 5–13. [4] H. Bart, I. Gohberg, and M.A. Kaashoek, Convolution equations and linear systems. Integral Equations Operator Theory 5 (1982), 283–340. [5] H. Dym, On reproducing kernels and the continuous covariance extension problem. Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math. 122 (1990), 427–482. [6] R. Ellis and I. Gohberg, Orthogonal systems and convolution operators. Oper. Theory Adv. Appl. 140, Birkh¨ auser Verlag, Basel, 2003. [7] I. Gohberg, M.A Kaashoek, and L. Lerer, The continuous analogue of the resultant and related convolution operators. The extended field of operator theory, Oper. Theory Adv. Appl. 171 (2007), 107–127.

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[8] I. Gohberg, M.A. Kaashoek, and F. van Schagen, On inversion of convolution integral operators on a finite interval. Operator theoretical methods and applications to mathematical physics, Oper. Theory Adv. Appl. 147 (2004), 277–285. [9] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications. Math. Nach. 194 (1998), 93– 125. [10] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential. Asymptotic Analysis 29 (2002), 1–38. [11] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Taylor coefficients of a pseudoexponential potential and the reflection coefficient of the corresponding canonical system. Math. Nach. 12/13 (2005), 1579–1590. [12] I. Gohberg and I. Koltracht, Numerical solution of integral equations, fast algorithms and Krein–Sobolev equations. Numer. math. 47 (1985), 237–288. [13] I. Gohberg and M.G. Kre˘ın, Theory and applications of Volterra operators in Hilbert spaces. Vol. 24 of Translations of mathematical monographs. American Mathematical Society, Rhode Island, 1970. [14] M.G. Krein, On the theory of accelerants and S-matrices of canonical differential systems. Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 1167–1170. [15] M.G. Kre˘ın, Continuous analogues of propositions for polynomials orthogonal on the unit circle. Dokl. Akad. Nauk. SSSR 105 (1955), 637–640. D. Alpay Department of Mathematics, Ben–Gurion University of the Negev 84105 Beer-Sheva, Israel e-mail: [email protected] I. Gohberg School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel–Aviv University, 69989 Tel–Aviv, Ramat–Aviv, Israel e-mail: [email protected] M.A. Kaashoek Afdeling Wiskunde, Faculteit der Exacte Wetenschappen Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: [email protected] L. Lerer Department of Mathematics, Technion, Israel Institute of Technology 32000 Haifa, Israel e-mail: [email protected] A. Sakhnovich Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien 15 Nordbergstrasse, A-1090 Wien, Austria e-mail: al [email protected]


Spectral Theory of the Infinite Block Jacobi Type Normal Matrices, Orthogonal Polynomials on a Complex Domain, and the Complex Moment Problem Yu.M. Berezansky To the memory of M.G. Krein

Abstract. In this survey, we describe new results in the spectral theory of normal block Jacobi matrices and the corresponding questions concerning the complex moment problem and orthogonal polynomials. Mathematics Subject Classification (2000). Primary 44A60, 47A57, 47A70. Keywords. Block Jacobi matrix, generalized eigenvector, orthogonal polynomials, direct and inverse spectral problems, moment problem.

1. Introduction In this talk, we propose an analog of the Jacobi matrix related to the complex moment problem and to a system of polynomials orthogonal with respect to some probability measure on the complex plane. Such a matrix has a block threediagonal structure and gives rise to a normal operator acting on a space of 2 type. Roughly speaking, these results are a generalization of the classical theory of Jacobi Hermitian matrices to the case of normal operators. They are deeply connected with some works by M.G. Krein ([33, 34], 1948–1949) devoted to the spectral approach to proving the integral representation for positive definite kernels and Jacobi matrices with operator-valued elements. The results of the talk are devoted to the complex moment problem and are connected with a large number of works starting from the year 1957: Y. Kilpi [32], N.I. Akhiezer [1], A. Atzmon [2], C. Berg, J.P.R. Christensen, P. Ressel [21], T.M. Bisgaard [22], J. Stochel, F.H. Szafraniec [39], and other mathematicians cited in the above-mentioned articles.

38

Yu.M. Berezansky

The results devoted to unitary block Jacobi matrices are connected with new works on orthogonal polynomials on the unit circle, in particular, with works by M.J. Cantero, L. Moral, L. Vel´ azques ([23], 2003), B. Simon ([38], 2005), and L.B. Golinskii ([30], 2006). Some results of this talk were obtained together with M.E. Dudkin.

2. Classical Jacobi matrices and moment problem, orthogonal polynomials on the axis 10 . At first, I remind the corresponding classical situation. In the classical theory, investigated is, in the space 2 of sequences f = (fn )∞ n=0 , fn ∈ C, the Hermitian Jacobi matrix ⎡ ⎤ b 0 a0 0 0 0 . . . ⎢a0 b1 a1 0 0 . . .⎥ ⎢ ⎥ J = ⎢ 0 a1 b2 a2 0 . . .⎥ , bn ∈ R, an > 0, n ∈ N0 = {0, 1, 2, . . .}. (2.1) ⎣ ⎦ .. .. .. .. .. . . . . . On finite sequences f ∈ f in , this matrix generates an operator on 2 which is Hermitian with equal defect numbers and therefore has a selfadjoint extension on ∞ 0 1 1 2 . Under some conditions on J (for example, an = ∞), the closure J of J is n=0

selfadjoint. The direct spectral problem, i.e., the eigenfunction expansion for J1 (or for some selfadjoint extension of J) is constructed in the following way (for simplicity, we will assume that J1 is selfadjoint). For any λ ∈ R, we introduce the sequence of polynomials ∞ P (λ) = Pn (λ) n=0 ∈ = C∞ as a solution of the equation JP (λ) = λP (λ),

P0 (λ) = 1,

i.e., ∀n ∈ N0

an−1 Pn−1 (λ) + bn Pn (λ) + an Pn+1 (λ) = λPn (λ), P−1 (λ) = 0,

(2.2)

P0 (λ) = 1.

The solution of this recurrence exists: it is necessary to go step by step, starting from P0 (λ); such a procedure is possible because all an > 0. The sequence of the polynomials P (λ) is a generalized eigenvector for J1 with eigenvalue λ (we use some quasinuclear rigging of the space H = 2 : H − ⊃ H0 ⊃ H+ ,

P (λ) ∈ H− ).

(2.3)

The corresponding Fourier transform F =2 is: 2 ⊃ f in f =

(fn )∞ n=0

→ f2(λ) =

∞ n=0

fn Pn (λ) ∈ L2 (R, dρ(λ)) =: L2 .

(2.4)

Spectral Theory of Block Jacobi Type Matrices

39

This mapping is a unitary operator (after closure) between 2 and L2 . The image of J1 is the operator of multiplication by λ on the space L2 . The polynomials Pn (λ) are orthonormal w.r.t. the spectral measure dρ(λ) : Pj (λ)Pk (λ)dρ(λ) = δj,k , j, k ∈ N0 . (2.5) R

Note that (2.5) is a consequence of the Parseval equality which holds for the mapping (2.4): ∀f, g ∈ f in (f, g) 2 = f2(λ)2 g (λ)dρ(λ). (2.6) R

20 . The inverse spectral problem is stated in this classical case as follows. Suppose that we have a Borel probability measure dρ(λ) on R for which all moments sn , sn = λn dρ(λ), n ∈ N0 , (2.7) R

exist (and the support of dρ(λ) contains an infinite set on a finite interval). The question is: is it possible to recover the corresponding Jacobi matrix J in such a manner that the initial measure dρ(λ) is equal to the spectral measure 1 What is the way for such a reconstruction? for J? The answer is simple: it is necessary to take the sequence of functions 1, λ, λ2 , . . . ∈ L2

(2.8)

(which are linearly independent) and apply the classical procedure of orthogonalization (by Schmidt) to it. As a result, we get the sequence of orthonormal polynomials P0 (λ) = 1, P1 (λ), P2 (λ), . . . . (2.9) Then the matrix J is reconstructed by these formulas: ∀n ∈ N0 2 an = λPn (λ)Pn+1 (λ)dρ(λ), bn = λ Pn (λ) dρ(λ). (2.10) R

R

30 . The classical moment problem. The question is: if we have a sequence s = (sm )∞ m=0 , sm ∈ R, when does a finite Borel measure dρ(λ) exist such that sm = λm dρ(λ), m ∈ N0 , (i.e., (2.7))? (2.11) R

The answer is: iff ∀f = (fj )∞ j=0 ∈ f in ∞

sj+m fj fm ≥ 0.

(2.12)

j,m=0

It is known that this result is deeply connected with the spectral theory of Jacobi matrices (see below, the second part of talk).

40

Yu.M. Berezansky

Now, we will explain in what manner it is possible to get the representation (2.11) if the condition (2.12) is fulfilled. In accordance with (2.12), we introduce the (quasi) scalar product (f, g)S =

∞

f, g ∈ f in ,

sj+m fj gm ,

(2.13)

j,m=0

and construct, in the usual way, the corresponding Hilbert space S. The shift operator ∀f ∈ f in (T f )j = fj−1 , j ∈ N0 (f−1 = 0) is (as is easy to understand) Hermitian on S with equal defect indexes. Therefore, this operator has a selfadjoint extension T1 in S. Further, we construct the generalized eigenvector expansion of this operator on the space S. For this purpose, it is necessary to introduce a quasinuclear rigging of the space S of type (2.3). A simple calculation shows that now the generalized eigenvector P (λ), λ ∈ R, has the form P (λ) = (1, λ, λ2 , . . .) ∈ ,

λ ∈ R.

(2.14)

The Fourier transform is: 2 S ⊃ f in (fj )∞ j=0 = f → f (λ) =

∞

fj λj ∈ L2 (R, dρ(λ)) =: L2 .

(2.15)

j=0

Here dρ(λ) is the spectral measure of the operator T1. The Parseval equality has the form (2.6), as before. From (2.15) we conclude that for the vector δn = (0, . . . , 0, 1, 0, 0, . . .), the n

Fourier transform δ2n = λn . Therefore, the Parseval equality (2.6) gives the required representation (2.11): sm = (δm , δ0 )S = (δ2m , δ20 )L2 = λm dρ(λ), m ∈ N0 . R

3. The complex moment problem 10 . To change from the classical Jacobi matrices to the normal Jacobi type block matrices, we begin with the corresponding moment problem. In the simplest case, the problem is stated as follows. Instead of the sequence s = (sm )∞ m=0 , sm ∈ R, we have the sequence s = (sm,n )∞ m,n=0 ,

sm,n ∈ C.

The question is: under what conditions does a finite Borel measure dρ(z) on C exist such that sm,n = z m z n dρ(z), m, n ∈ N0 ? (3.1) C


41

In this case, some conditions of positiveness of type (2.12) play an essential role as well. So, as in the case of the classical moment problem, we use the sequence f = (fj,k )∞ j,k=0 ,

fj,k ∈ C;

= (C2 )∞ is the set of all such sequences, and f in denotes the set of finite sequences of . The condition of positiveness, analogous to (2.12), is the following: ∞

sj+n,k+m fj,k f m,n ≥ 0,

f ∈ f in

(3.2)

j,k,m,n=0

(note that the disposition of the indexes of s, f and f in this sum is essential). As for the classical moment problem, we introduce the scalar product connected with (3.2): ∞ (f, g)S = sj+n,k+m fj,k gm,n , f, g ∈ f in , (3.3) j,k,m,n=0

and construct the corresponding Hilbert space S. Now, we consider two operators T and T + on the space S, acting on f ∈ f in according to the rules (T f )j,k = fj,k−1 ,

(T + f )j,k = fj−1,k ,

j, k ∈ N0 , f−1,k = fj,−1 = 0.

(3.4)

It is clear that T is formally normal: algebraically, T T + = T + T. Under some additional conditions on the growth of sm,n as m, n → ∞, we can assert that the closure T1 is a normal operator and can apply, to our situation, a scheme similar to that in Section 2.30 , but now involving a normal operator instead of a selfadjoint operator. This way gives the following result. Theorem 3.1. Consider the sequence s = (sm,n )∞ m,n=0 , sm,n ∈ C. If on C a finite Borel measure dρ(λ) exists such that (3.1) sm,n = z m z n dρ(z), m, n ∈ N0 , C

then the condition of positiveness (3.2) is fulfilled. Conversely, if for s the condition (3.2) is fulfilled and ∞ 1 = ∞, (3.5) √ 2p s 2p,2p p=1 then the representation (3.1) holds. a) It should be emphasized that the condition (3.5) provides the normality (not only formal normality) of the operator T. For a formally normal operator, it is impossible to assert in general that it can be extended to a normal operator

42

Yu.M. Berezansky

(unlike the case of Hermitian operators with equal defect numbers and selfadjoint operators). Therefore, the condition of type (3.5) is necessary to assume. b) The proof of Theorem 3.1 is analogous to that for the classical moment problem, but now it is necessary to use the generalized eigenfunction expansion for a normal operator (instead of a selfadjoint one). Instead of (2.14), we now have the following generalized eigenvector: P (z) = (z m z n )∞ m,n=0 ∈ ,

z ∈ C is eigenvalue.

(3.6)

The corresponding Fourier transform of type (2.15) is: 2 S ⊃ f in (fj,k )∞ j,k=0 = f → f (z) =

∞

fj,k z j z k ∈ L2 (C, dρ(z)) =: L2 .

(3.7)

j,k=0

c) In the classical situation, the sequence 1,λ,λ2 , . . .∈ L2 = L2 (R, dρ(λ)) 3 (2.8) n is connected with the moments (2.7), sn = λ dρ(λ), n ∈ N0 . The orthogonalizaR

tion of the functions (2.8) gives the orthonormal polynomials Pn (λ), connected with the Jacobi matrix (2.1). But now instead of the sequence (2.8), we have the double sequence (z j z k )∞ z ∈ C, z j z k ∈ L2 . (3.8) j,k=0 , The question is: what is a convenient way of introducing the linear order into (3.8) in order to apply the orthogonalization procedure and get the analog of Pn (λ) for “Jacobi” normal matrices? Answer. The order is this: 0 0

z z zn ↓

z 1 z¯0 z 2 z¯0

–

0 1

z z¯

z 1 z¯1

–

zn z 0 z¯2

→ – (3.9)

z j z¯k

4. Block Jacobi type normal matrices and their spectral theory The natural problem arises: in what way is it possible to develop the classical theory for the complex moment problem? What are the corresponding analogs of Jacobi matrices and orthogonal polynomials in the complex plane (or on some set of C, for example, on the unit circle T ⊂ C)? The previous account gives the following picture. 10 . Direct spectral problem. Instead of the space 2 = C ⊕ C ⊕ · · · , it is necessary to take the space l2 = H0 ⊕ H1 ⊕ H2 ⊕ · · · , Hn = Cn+1 (C1 = C)

(4.1)

Spectral Theory of Block Jacobi Type Matrices and, instead of the acting on the space ⎡ b 0 c0 ⎢a0 b1 ⎢ J = ⎢ 0 a1 ⎣ .. .. . .

43

Jacobi matrix (2.1), the following Jacobi type block matrix l2 (4.1), first on finite vectors lf in ⊂ l2 : ⎤ here, a , b , c are operators n n n 0 0 0 ... (finite-dimensional matrices): c1 0 0 . . .⎥ ⎥ ; an : Hn → Hn+1 , (4.2) b2 c2 0 . . .⎥ ⎦ b :H →H , .. .. .. n n n . . . cn : Hn+1 → Hn .

> 0 in (2.1) now have the form: ∀n ∈ N0 ⎞ an; 0,1 . . . an;0,n an; 1,1 . . . an;1,n ⎟ ⎟ .. .. ⎟ , .. . . . ⎟ ⎟ 0 . . . an; n,n ⎠ 0 ... 0 ⎞ cn; 0,1 0 0 ... 0 ⎟ cn; 1,1 cn; 1,2 0 ... 0 ⎟ ⎟; .. .. .. .. ⎠ . . . . cn; n,0 cn; n,1 cn; n,2 cn; n,3 . . . cn; n,n+1 an;0,0 > 0, an;1,1 > 0, . . . , an;n,n > 0; cn;0,1 > 0, cn;1,2 > 0, . . . , cn;n,n+1 > 0.

The essential conditions an ⎛ an; 0,0 ⎜ 0 ⎜ ⎜ an = ⎜ ... ⎜ ⎝ 0 0 ⎛ cn; 0,0 ⎜ cn; 1,0 ⎜ cn = ⎜ . ⎝ ..

(4.3)

Under some simple conditions on an , bn , and cn , the matrix J is formally normal: JJ + = J + J (J + is the adjoint matrix to J). If an , bn , and cn are uniformly bounded operators, then the closure J1 is a bounded normal operator on l2 (for simplicity, we will speak here only about this case). 1 The corresponding generalized eigenLet z ∈ C belong to the spectrum of J. vector has the form ∞ P (z) = (Pn (z))n=0 ; (4.4) here, Pn (z) ∈ Hn is a vector-valued polynomial w.r.t. z, z of degree n (i.e., its coordinates are some linear combinations of z j z k , j + k ≤ n). This eigenvector P (z) is a solution of two equations of (2.2) type: JP (z) = zP (z),

J + P (z) = zP (z).

(4.5)

The corresponding Fourier transform2 has the form: ∞ 2 l2 ⊃ lf in f = (fn )∞ (fn , Pn (z))H ∈ L2 (C, dρ(z)) =: L2 , (4.6) n=0 → f (z) = n=0

where dρ(z) is a spectral measure of J1 with compact support. So, we have the following result. Theorem 4.1. On the space l2 (4.1), consider the bounded normal operator J1 which is generated by the block Jacobi matrix (4.2) satisfying the conditions (4.3).

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Yu.M. Berezansky

The corresponding generalized eigenvectors of the form (4.4) are solutions of the equations (4.5) and give rise to the Fourier transform2 (4.6). This transform is a unitary operator between l2 and L2 constructed by spectral measure dρ(z) with compact support. The polynomials Pn (z) generate an orthonormal basis in the space L2 . It is necessary to remark that every bounded normal operator in Hilbert space for which one cyclic vector exists is unitarily equivalent to the operator J1 which is generated in the space (4.1) by the matrix (4.2) satisfying the conditions (4.3). 20 . Inverse spectral problem. Suppose we have the probability Borel measure dρ(z) on C with compact support. As before, the question is: is it possible to recover the corresponding block Jacobi normal matrix (4.2), (4.3) in such a manner that dρ(z) is a spectral 1 What is the way for such a reconstruction? measure for J? As before, it is necessary to take the sequence (3.8) of the functions z j z k and apply the Schmidt orthogonalization procedure to it in L2 . As a result, we get the polynomials Pn (z) of type (4.4). It is necessary to take such a measure dρ(z) that the functions z j z k are linearly independent (for example, the support of dρ(z) contains some open set). The linear order for the sequence (3.8) should be taken as that in the picture (3.9). So, we have the following order: z 0 z 0 ; z 1 z 0 , z 0 z 1 ; z 2 z 0 , z 1 z 1 , z 0 z 2 ; . . . ; z n z 0 , z n−1 z 1 , . . . , z 0 z n ; . . . .

(4.7)

After the orthogonalization, we get the following table: P0;0 (z) ≡ 1; P1;0 (z), P2;0 (z), . . . P1;1 (z); P2,1 (z), . . . P2;2 (z); . . .

Pn;0 (z), Pn;1 (z), Pn;2 (z), .. .

... ... ...

Pn;n (z); . . . Now, we can construct the “generalized eigenvector” (4.4) by setting Pn (z) = (Pn;0 (z), Pn;1 (z), . . . , Pn;n (z)). In this way, it is possible to prove the following result. Theorem 4.2. Let dρ(z) be a probability Borel measure with compact support; assume that the functions (4.7) are linearly independent. Then this measure is the spectral measure for a normal bounded operator J1 which is generated on the space l2 (4.1) by the block Jacobi type matrix (4.2) with satisfying the conditions (4.3).

Spectral Theory of Block Jacobi Type Matrices Its elements are: an;α,β = zPn;β (z)Pn+1;α (z)dρ(z),

45

α = 0, . . . , n + 1, β = 0, . . . , n;

C bn;α,β =

zPn;β (z)Pn;α (z)dρ(z),

α, β = 0, . . . , n;

(4.8)

C

cn;α,β =

zPn+1;β (z)Pn+1;α (z)dρ(z),

α = 0, . . . , n, β = 0, . . . , n + 1.

C

If we start from the spectral measure dρ(z) of the bounded normal operator 1 J generated by the matrix J (4.2)–(4.3), then the formulae (4.8) give the elements of this matrix.

5. Block Jacobi unitary matrices and orthogonal polynomials on the unit circle 10 . The theory of orthogonal polynomials on the unit circle T ⊂ C has been intensively developed for the last 50–60 years. In 2005, B. Simon published a twovolume book [38] on this topic. But the idea of constructing a Jacobi type block matrix is new for this theory as well. It is necessary to say that in 2003, M.J. Cantero, L. Moral and L. Vel´ asques published article [23] dealing with a deeply connected area. However, they considered the 5-diagonal matrix on the ordinary space 2 and did not use the natural Jacobi type block matrix in the corresponding space. Roughly speaking, the orthogonal polynomials on T and the corresponding trigonometric moment problem are a particular case of the above-mentioned theory. The difference is this: the functions z j z k from (3.8), with z ∈ T, are linearly dependent in the space L2 (T, dρ(z)) for an arbitrary measure dρ(z) on T because ∀n ∈ N0 z j z k = z j+n z k+n . Therefore, instead of all functions z j z k from (3.8), it is necessary to take only these functions: z 0 z 0 = 1; z 1 z 0 = z 1 , z 0 z 1 = z −1 ; z 2 z 0 = z 2 , z 0 z 2 = z −2 ; . . . ; z ∈ T.

(5.1)

The linear order for the orthogonalization is previous: as that in (3.9), (4.7), i.e., (5.1). For the linear independence of the functions (5.1), it is necessary to assume that the support of dρ(z) consists of infinitely many points. In this case, instead of the space (4.1), we take the space l2,u ⊂ l2 : l2,u = H0 ⊕ H1 ⊕ H2 ⊕ · · · , where H0 = C, H1 = H2 = · · · = C2 .

(5.2)

46

Yu.M. Berezansky

The Jacobi type block matrix J of the form (4.2) now acts on the space l2,u and its blocks have the form different from that in (4.3), namely: a a0 = 0;0,0 , b0 = b0;0,0 , c0 = c0;0,0 c0;0,1 , 0 an;0,1 a 0 0 , cn = ; an = n;0,0 cn;1,0 cn;1,1 0 0 n = 1, 2, . . . . a0;0,0 , c0;0,1 , an;0,0 , cn;1,1 > 0, For the unitary case, it is possible to repeat all constructions from Sections 3 and 4, including Theorems 3.1, 4.1, and 4.2.

6. Some applications to the integration of nonlinear difference equations and concluding remarks 10 . At first, we give some additional references. An account of the classical spectral theory of Jacobi matrices, corresponding moment problem, and spectral approach to the representation of positive definite kernels can be found in [1, 4, 37, 43]; see also [19, 8, 9, 12]. The classical theory of orthogonal polynomials, including those on sets (in particular, on the unit circle) in the complex plane can be found in [42, 29, 40, 38, 31]. The book [41] contains results concerning orthogonal polynomials of two real variables x =Re z, y = Imz. The material of Sections 3–5 is an account of the results by Yu.M. Berezansky and M.E. Dudkin which were published with proofs in articles [13, 14, 15]. Note also that the main idea of changing from ordinary infinite matrices in the space 2 to the block Jacobi type matrices and the corresponding spectral theory in the space l2 or l2,u was contained in the talk by Yu.M. Berezansky during the International Conference in Munich, Germany, July 2005 [10]. The article [14] also contains some conditions for the normality of the Jacobi type block matrices. The order of orthogonalization (3.9) is actually not new (for the one in terms of the variables x, y ∈ R, see [3], Ch. 12, [41]). For the case under consideration, it should be taken into account that, e.g., for a bounded operator A to be normal, its parts ReA = 1/2(A + A∗ ) and ImA = 1/2i(A − A∗ ) must be selfadjoint and commuting. Note that books [3] and [41] contain many interesting facts connected with Sections 3–5. Our Theorems 4.1 and 4.2 also provide the answers to some questions formulated in [3], Ch. 12, Subsection 12.3. It is now worth making two remarks on the results of Sections 3–5. At first, it is interesting to find the form of the five-diagonal matrix in ordinary space 2 if we know that the corresponding operator on 2 is unitary with one cyclic vector. Of course, it is generated by a block Jacobi matrix in the space l2,u , but it is necessary to find the formulae for its elements (their representation using the Verblunsky coefficients). This problem is solved in [24, 26].


47

The second remark. In the theory of orthonormal polynomials on the unit circle, there exists an important formula which presents a possibility of finding these polynomials step by step (the Szegö recursion) [42, 38]. Therefore, the following question arises: is it possible to find an analogous formula for orthogonal polynomials on the complex plane? Article [17] contains the solution of this problem: in reality, the Szeg¨ o recursion is equivalent to the two equalities (4.5). They can be rewritten as one recursion which generalizes the Szeg¨ o recursion to general orthogonal polynomials on the complex plane. 20 . Let us change to a short account of the results concerning applications of the theory in Sections 3–5 to the integration of some nonlinear differential-difference equations. Consider the classical Toda lattice on a semi-axis: 1 an (t)(bn+1 (t) − bn (t)), 2 b˙ n (t) = a2n (t) − a2n−1 (t); a−1 (t) = 0;

a˙ n (t) =

(6.1) n ∈ N0 , t ∈ [0, T ].

d , T ≤ ∞. Here an (t) > 0, bn (t) are real continuously differentiable functions, · = dt Let us formulate the Cauchy problem for (6.1): for given initial data an (0), bn (0), n ∈ N0 , it is necessary to find the solution an (t), bn (t), n ∈ N0 , for t > 0; the equality a−1 (t) = 0 is some boundary condition. To find the solution of this problem, the following procedure can be applied. With an (t), bn (t) for every t ∈ [0, T ], we construct the Jacobi matrix J(t) (2.1). This matrix is Hermitian. If we assume the boundedness of this solution, then the 1 is selfadjoint on the space 2 . Denote by dρ(λ; t) its corresponding operator J(t) spectral measure. The change of the solution an (t), bn (t) with t is, of course, very complicated. However, it is possible to prove that the change of dρ(λ; t) is very simple:

dρ(λ; t) = c(t)eλt dρ(λ; 0),

t ∈ [0, T ], λ ∈ R,

(6.2)

where c(t) is a normalizing factor (spectral measure ∀t is a probability measure). As a result, the procedure for finding the solution of the Cauchy problem for (6.1) is as follows: we find the initial spectral measure dρ(λ; 0) of the operator J1(0), then calculate, using (6.2), the spectral measure dρ(λ; t) for t > 0 and, finally, reconstruct the matrix J(t), using the inverse spectral problem. As a result, the elements of J(t) are the solution of our Cauchy problem. Note that the equation 1 is stable w.r.t. t ∈ [0, T ] (see (6.2)). (6.1) is “isospectral”: the spectrum of J(t) Such an approach was proposed in [5, 6, 7]; it is a difference analog of the classical inverse spectral problem method for solving the Cauchy problem for the Korteweg-de Vries differential equation (instead of the Sturm-Liouville equation, we use the spectral theory of Jacobi matrices, which is simpler). This approach was generalized to equations more complicated than (6.1). In particular, the “nonisospectral” equations were investigated for which the change of the spectral measure is more complicated than (6.2). Also, the non-Abelian case

48

Yu.M. Berezansky

where an (t), bn (t) are matrices etc. was investigated. Some of the corresponding and connected results can be found in [25, 16, 28, 20, 27, 35] and in book [43]. Two years ago, L.B. Golinskii published article [30] in which he applied the approach [5, 6, 7], but with the spectral theory of Jacobi matrices changed by the spectral theory of five-diagonals unitary matrices in the space 2 (using the results of works [23, 38]). Such an approach provides a possibility of integrating other equations, different from (6.1), namely, the Schur flows. This point of view is fruitful: in [18, 11], it was shown that it is possible to apply, for integration, the spectral theory of normal (and unitary) block Jacobi matrices, i.e., the results of Sections 3–5. Now, instead of the Toda equation (6.1), we can investigate other differential-difference equations, including a non-Abelian one (for example, the Polyakov-type systems). Using the results of Sections 3–5, it is also possible to investigate the corresponding “nonisospectral” systems. For the case of unitary operators (described in the Section 5), such results were obtained in [36].

References [1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions. Hafner, New York, 1965. (Russian edition: Fizmatgiz, Moscow, 1961.) [2] A. Atzmon, A moment problem for positive measure on the unit disc. Pasif. J. Math. 59 (1975), no. 2, 317–325. [3] H. Bateman and A. Erdélyi, Higher Transcendental Functions. Bessel Functions of Parabolic Cylinder, Orthogonal Polynomials. Fizmatgiz, Moscow, 1966. (English edition: Higher Transcendental Functions. 2, Mc. Graw-Hill Book Company, Inc., New York–Toronto–London, 1953.) [4] Ju.M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators. Amer. Math. Soc., Providence, RI, 1968 (Russian edition: Naukova Dumka, Kiev, 1965). [5] Yu.M. Berezansky, The integration of semi-infinite Toda chain by means of inverse spectral problem. Preprint 84.79 of Inst. of Math., Acad. Sci. Ukr., Kiev, 1984. [6] Yu.M. Berezansky, Integration of nonlinear difference equations by the inverse spectral problem method. Dokl. Acad. Nauk SSSR 281 (1985), no. 1, 16–19. [7] Yu.M. Berezanski, The integration of semi-infinite Toda chain by means of inverse spectral problem. Reports Math. Phys. 24 (1986), no. 1, 21–47. [8] Yu.M. Berezansky, The works of M.G. Krein on eigenfunction expansion for selfadjoint operators and their applications and development. Operator Theory: Advances and Applications 117 (2000), 21–43. [9] Yu.M. Berezansky, Some generalizations of the classical moment problem. Integr. Equ. Oper. Theory 44 (2002), 255–289. [10] Yu.M. Berezansky, The complex moment problem. Intern. Conference on Difference Equations, Special Functions and Applications, July 25–July 30, 2005, Munich; Collection of abstracts, p. 17. [11] Yu.M. Berezansky, Integration of the modified double-infinite Toda lattice with the help of inverse spectral problem. Ukr. Mat. Zh. 60 (2008), no. 4, 453–469.


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[12] Yu.M. Berezansky and M.E. Dudkin, The complex moment problem in the exponential form. Methods Funct. Anal. Topology 10 (2004), no. 4, 1–10. [13] Yu.M. Berezansky and M.E. Dudkin, The direct and inverse spectral problems for the block Jacobi type unitary matrices. Methods Funct. Anal. Topology 11 (2005), no. 4, 327–345. [14] Yu.M. Berezansky and M.E. Dudkin, The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices. Methods Funct. Anal. Topology 12 (2006), no. 1, 1–31. [15] Yu.M. Berezansky and M.E. Dudkin, On the complex moment problem. Math. Nachr. 280 (2007), no. 1–2, 60–73. [16] Yu.M. Berezanskii and M.I. Gekhtman, Inverse problem of the spectral analysis and non-Abelian lattices. Ukr. Math. J. 42 (1990), no. 6, 645–658. [17] Yu.M. Berezanky, I.Ya. Ivasiuk, and O.A. Mokhonko, Recursion relation for orthogonal polynomials on the complex plane. Methods Funct. Anal. Topology 14 (2008), no. 2. [18] Yu.M. Berezansky and A.A. Mokhon’ko, Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices. Funct. Anal. and its Appl. 42 (2008), no. 1, 1–21. [19] Yu.M. Berezansky, Z.G. Sheftel, G.F. Us, Functional Analysis. Vols. 1,2, Birkh¨ auser Verlag, Basel–Boston–Berlin, 1996 (Russian edition: Vyshcha shkola, Kiev, 1990). [20] Yu.M. Berezansky and M. Shmoish, Nonisospectral flows on semi-infinite Jacobi matrices. Nonlinear Math. Phys. 1 (1994), no. 2, 116–146. [21] C. Berg, J.P.R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups. Springer Verlag, Berlin–New York, 1984. [22] T.M. Bisgaard, The two-sided complex moment problem. Ark. Math. 27 (1989), no. 1, 23–28. [23] M.J. Cantero, L. Moral, and L. Vel´ azques, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362 (2003), 29–56. [24] M.J. Cantero, L. Moral, and L. Vel´ azquez, Minimal representations of unitary operators and orthogonal polynomials on the unit circle. Linear Algebra Appl. 408 (2005), 40–65. [25] P. Deift, L.C. Li, and C. Tomei, Toda flows with infinitely many variables. J. Funct. Anal. 64 (1985), 358-402. [26] M.E. Dudkin, The exact inner structure of the block Jacobi type unitary matrices connected with the correspondence direct and inverse spectral problems. Methods Funct. Anal. Topology 14 (2008), no. 2. [27] L. Faybusovich and M. Gekhtman, Elementary Toda orbits and integrable lattices. J. Math. Phys. 41 (2000), no. 5, 2905–2921. [28] M.I. Gekhtman, Integration of non-Abelian Toda type lattices. Funct. Anal. i Prilozh. 24 (1990), no. 3, 72–73. [29] Ya.L. Geronimus, Theory of Orthogonal Polynomials. Gostekhizdat, Moscow-Leningrad, 1950. [30] L.B. Golinskii, Schur flows and orthogonal polynomials on the unit circle. Mat. Sbornik 197 (2006), no. 8, 41–62.

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[31] L. Golinskii, V. Totic, Orthogonal polynomials: from Jacobi to Simon. Spectral theory and Mathematical Physics: a Festschrift in honor of Barry Simon’s 60th birthday, 821–874; Proc. Sympos. Pure Math. 76, Part 2, Amer. Math. Soc., Providence, RI, 2007. ¨ [32] Y. Kilpi, Uber das komplexe Momentenproblem. Ann. Acad. Sci. Fenn., Ser. A. 1., no. 236, 1957. [33] M.G. Krein, On Hermitian operators with directed functionals. Zbirnyk prac’ Inst. Math. Akad. Nauk. Ukrain. RSR (1948), no. 10, 83–106. [34] M.G. Krein, Infinite J-matrices and a matrix moment problem. Dokl. Acad. Nauk SSSR 69 (1949), no. 2, 125–128. [35] O.A. Mokhonko, Some solvable classes of nonlinear nonisospectral difference equations. Ukrain. Mat. J. 57 (2005), no. 3, 427–439. [36] O. Mokhonko, Nonisospectral flows on semi-infinite unitary block Jacobi matrices. Ukr. Mat. Zh. 60 (2008), no. 4, 521–544. [37] B. Simon, The classical moment problem as a self-adjoint finite difference operator. Adv. in Math. 137 (1998), 82–203. [38] B. Simon, Orthogonal polynomials on the unit circle. Part 1: Classical Theory; Part 2: Spectral Theory. AMS Colloquium Series, Amer. Math. Soc., Providence, RI, 2005. [39] J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: a polar decomposition approach. J. Func. Anal. 159 (1998), 432–491. [40] P.K. Suetin, Fundamental properties of polynomials orthogonal on a contour. Uspekhi Math. Nauk 21 (1966), no. 2, 41–88. [41] P.K. Suetin, Orthogonal Polynomials in Two Variables. Nauka, Moscow, 1988. [42] G. Szeg¨ o, Orthogonal Polynomials. Fizmatgiz, Moscow, 1962. (English edition: Amer. Math. Soc., New York, 1959.) [43] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices. AMS Mathematical Surveys and Monographs, Vol. 72, Amer. Math. Soc., Providence, RI, 2000. Yu.M. Berezansky Institute of Mathematics National Academy of Science of Ukraine 3 Tereshchenkivs’ka St. 01601 Kyiv, Ukraine e-mail: [email protected]


On Geometrical and Analytical Aspects in Formulations of Problems of Classic and Non-classic Continuum Mechanics George L. Brovko, Olga A. Ivanova and Alexandra S. Finoshkina Abstract. The aim of this work is to touch some analytical and geometrical aspects in formulations of mathematical problems in classic and non-classic continuum mechanics and to demonstrate the connection of these aspects in generalized theory of stress and strain tensor measures, in finite plasticity, in application of the method of mechanical modeling to building-up models of Cosserat type structures and saturated porous media. Keywords. Objective tensors, generalized theory of strain and stress tensor measures, constitutive relations, finite plasticity, method of mechanical modeling, Cosserat type systems, saturated porous media, heterogeneous media, internal interactions.

1. Introduction Formulations of boundary value problems in continuum mechanics provide for the assignment of a region occupied by a body (in its current or reference configuration), the setting of balance equations and constitutive relations to be satisfied in the region, and the determination of initial and boundary conditions. Such formulations in different classic and non-classic parts of continuum mechanics have their geometrical and analytical features caused by the respective mathematical model of continuum, its motions and interactions. The topology of continuum itself, its inertial characteristics, the geometry of motions and interactions determine the structure of scalar, vector and tensor variables accompanying a process, describing stress and strain state. Constitutive relations specifying the mechanical properties of a body [1–7] analytically express the restrictions on an inner stress state and a motion being proper for the This work was completed with the support of Russian Foundation for Basic Research, project No. 06-01-00565.

52

G.L. Brovko, O.A. Ivanova and A.S. Finoshkina

body in all processes from the considered area. Typical forms of such relations are stress-strain relations and kinematical constraints (e.g., incompressibility, inextensibility, rigidity – in classic continuum mechanics [3], Nowacki pseudocontinuum constraints [8] – in Cosserat models [9] etc.). Constitutive relations essentially determine the mathematical character of the whole problem: the type of the equations system, the space of solutions and right-hand terms, and mathematical properties of the generalized operator of the problem. Such is the boundary value problem of the theory of small elastic-plastic deformations [10] which constitutive equations (stress-strain relations) determine the quasi-elliptical type of the system of equations, lead naturally to Mikhlin’s “energetic” type [11] of Hilbert space for solutions embeddably linked with Sobolev (1) space W2 by Korn’s inequality [12], then, through Sobolev’s embedding theorems, to Lp type of Banach space for volume (p 6/5) and contact (p > 4/3) external forces (right-hand members), and, finally, determine the main properties of the operator (of the generalized formulation of the problem) such as finiteness, coercitivity, strict monotony, and potentiality [13–18]; these properties of the operator remarkably duplicate the similar properties of the constitutive stress-strain relation. Each other mechanical theory gives the similar example. The leading role of constitutive relations turns attention to possibilities of their building-up, comparison and interpretation from the geometrical point of view. In Ilyushin’s general theory of plasticity (theory of elastic-plastic processes) [19], analytical properties of constitutive relations have got a pictorial geometrical interpretation in 5-dimensional vector space (space of images). The classification of plastic processes (and corresponding forms of constitutive relations) by degrees of their complexity was also made by geometrical criteria. Geometrical visuality useful at small strains becomes necessary for study analytical features of constitutive equations in the case of finite strains when continuous variety of stress and strain tensor measures appear as available [20]. For this case, taking into account invariance properties, the theory of objective tensors (different ranks and types) and their mappings has been developed, new continuous sets of objective derivatives are introduced, and generalized theory of stress and finite strain tensor measures is constructed [21,22]; each objective derivative corresponds to a certain pare of conjugate stress and strain measures. On this base, the method for correct generalization of plasticity constitutive relations known at small strains to the case of finite strains was elaborated; each generalization corresponds to a pare of measures [23–25]. The continuous variety of obtained finite plasticity relations is capable to cover experimental data in a wide range of plastic material properties. Special attention should be turned to the role of geometry in modeling bodies themselves, their motions and interactions, especially in models of multi-phase structures and non-classic media. To realize such an approach the method of mechanical (constructive) modeling based on detailed study of a material macroparticle was proposed [26] and applied for deriving the motion and the constitu-

On Geometrical and Analytical Aspects

53

tive equations of Cosserat type structures [9, 27–34] and saturated porous materials [35–42]. For Cosserat structures, it gives a clear transparent look at all kinematic and dynamic characteristics, and of material properties of the model [43–47]. For saturated porous media, the method of mechanical modeling together with the hypothesis of interpenetrative continua and the principles of geometric invariance lead to analytical constitutive expressions (general reduced forms) of interactive forces and moments [48–52]. This article presents authors’ results demonstrating the examples of links between geometrical and analytical aspects in models and problems of certain divisions of continuum mechanics: the generalized theory of strain and stress tensor measures, corresponding new approaches in plasticity at finite strains [20–25], application of the mechanical modeling method to Cosserat type structures [43–47] and saturated porous media [48–52].

2. Generalized theory of strain and stress tensor measures. Plasticity at finite strains 2.1. Objective tensors and time derivatives 2.1.1. Kinematics. The Lagrangean description of a motion has a form [3, 5–7] x = f (x, t),

(2.1)

where x and x are the current (at t-configuration) and reference positions of a body point. The deformation gradient A : = ∇x f has unique polar right and left decompositions A = QX = YQ (det A = 0) (2.2) with orthogonal Q and symmetric positively defined right X and left Y parts. ˙ −1 (where v = f˙ (x, t) is the velocity The stretching tensor D := ∇x v = AA vector) has its skew-symmetric Ω (spin) and symmetric V (strain rate tensor) additive parts: Ω := Ω0 + QΩX QT ,

˙ T, Ω0 := QQ

˙ −1 − X−1 X), ˙ ΩX := 12 (XX

T ˙ −1 + X−1 X)Q ˙ . V := 12 Q(XX

(2.3)

2.1.2. Objective tensors. The change of frame-reference is expressed by the equations which give the dependence of new Eulerian independent variables x∗ , t∗ on old ones x, t: x∗ = x∗0 (t) + Q(t)(x − x0 ), (2.4) t∗ = t + a, where x0 = const and x0 (t) are the reference centers in old and new frames, Q(t) is the orthogonal rotation tensor of the old frame respectively the new one, a = const is a time-shift.

54


Among different mechanical tensor (second rank) processes, the objective tensors [21, 53] of four types can be marked out which are transformed under framechange (2.4) in four different manners: (a) U∗ = U,

(b) Z∗ = QZQT ,

(c) F∗ = QF,

(d) G∗ = GQT .

(2.5)

Tensors of (a) type are called as right or materially oriented, tensors (b) – as left or spatially oriented, tensors (c) and (d) – as tensors of mixed types of objectivity. Objective right U and left Z tensors describing one and the same mechanical process may be connected by the equivalence relation in the form Z = AUBT ,

(2.6)

where A and B are nonsingular objective of (c) type tensors determined by the motion of a particle. They permit unique representations A ≡ QX,

B ≡ QY

(2.7)

with orthogonal tensor Q from polar decomposition (2.2) and right objective tensors X, Y determined by the pre-history of the right pure strain tensor X from (2.2). Right U and left Z tensors satisfying (2.6) are called the analogs of each other. 2.1.3. Objective derivatives (with respect to time). Respectively, the one-to-one ˙ and D[Z] of the right U and the left Z tensors dependence between the rates U may be determined by the equivalence relation [21, 53, 54] ˙ T ≡ A(A−1 ZB−1T )˙ BT . D[Z] = AUB

(2.8)

The differential operator D[Z] acting on the left tensor Z is named as an objective derivative of the spatial type. It has the forms ˙ −1 Z − Z(BB ˙ −1 )T . ˙ − AA D[Z] := A(A−1 ZB−1T )˙ BT ≡ Z

(2.9)

This definition generalizes all known notions of objective derivatives (Jaumann, Oldroyd, Cotter–Rivlin, Truesdell, Hill, Sedov, Dienes etc.) [4, 7, 53–58]. 2.1.4. Co-rotational objective derivatives. Let us put A = B = Q – orthogonal tensor (Q−1 ≡ QT ) permitting the representations (2.7) with orthogonal tensors X ≡ Y ≡ R determined by the pre-history of isochoric part of the pure strain tensor X. Then the left objective derivative takes the forms: ˙ − ΨZ + ZΨ, Dcr [Z] := Q(QT ZQ)˙ QT ≡ Z ˙ T, Ψ = QQ

Q = QR.

(2.10)

It is named as co-rotational objective derivative. Well-known Jaumann and Dienes derivatives are co-rotational [54–58].


55

All co-rotational derivatives have remarkable distinguishing properties: they preserve the symmetry and skew-symmetry of (left) tensors, and they preserve the spherical-deviatoric decomposition of a tensor [54]. 2.2. Generalized theory of stress and strain tensor measures 2.2.1. Axioms of the generalized theory. Axiom 1. Strain measure is a symmetric second rank tensor ε which pre-history determines, completely and in one-to-one manner, a process of a pure deformation – elongation and shear strains of all elementary fibres (independently from a rigid motion accompanying the strains). Axiom 2. Stress measure is a symmetric second rank tensor σ that determines, completely and independently from a rigid motion of a particle (perhaps, together with ε), the inner stress state of a particle – quantities and relative orientation of surface forces on each elementary material area. Axiom 3. In each motion stress and strain measures are energetically conjugate: specific elementary work of inner contact forces is equal to scalar product of the stress tensor σ and full (substantial) increment of the strain tensor ε. Axiom 4. In classic case of “small strains” (when strains and rotations of material fibres are small in an order Δ 1), the strain measure ε, its material derivative ε˙ by time and the stress measure σ asymptotically coincide (with relative accuracy Δ) with classic Cauchy small strain tensor, strain rate tensor and Cauchy stress tensor respectively. The Axioms 1–4 seem to be principal and obligatory for all stress and strain measures to be introduced. Pares of known measures satisfy these axioms, for example, Green strain tensor E and second Piola–Kirchhoff stress tensor P, or Green strain tensor E and Ilyushin stress tensor Σ, or the pare of “rate type” measures E V and ΣV [7, 20, 22, 58]. Important additional postulates are formulated as the following. Axiom 10 . Spherical and deviatoric parts of the strain measure determine independently the processes of volumetric and shear strains respectively. Axiom 20 . Spherical and deviatoric parts of the stress measure determine independently (maybe, together with the same named parts of the strain measure) the hydrostatic and shearing stresses respectively. Remark 2.1. All these axioms are addressed to stress and strain measures of the right (materially oriented) types of objectivity. By the equivalence relations of the (2.6), (2.8) type the axioms can be co-addressed to the left (spatially oriented) type analogs of the right measures [21, 22, 53]. 2.2.2. Lagrangean class of strain and stress measures. On the base of these axioms, the generalized theory was developed and the so-called full Lagrangean class of measures ε and σ was introduced. The subclass of right measures ε = E and σ = Σ with their spatial analogs E and S named as simple Lagrangean class was

56


constructed using the equivalence relations (2.6), (2.8): E˙ = A−1 VB−1T ,

D[E] = V,

Σ = AT SB,

(2.11)

where S is the “true” Cauchy stress tensor and V is the strain rate tensor. These measures respond to the equality for specific elementary work of inner contact forces ˙ W = S : D[E] = Σ : E. (2.12) Remark 2.2. As the choice of the tensors A and B determine uniquely the concrete type of the objective derivative (2.9) as the definition of the measures (2.11) is uniquely tied with the choice of the objective derivative. All Lagrangean measures satisfy Axioms 1–4. 2.2.3. Co-rotational family of stress and strain measures. Parametric subfamily. Choosing the co-rotational family of objective derivatives (2.10) we obtain the co-rotational family of measures [20, 21]: cr E˙ = (QR)T V(QR),

Dcr [Ecr ] = V,

Σcr = (QR)T S(QR).

(2.13)

Co-rotational measures satisfy all the Axioms 1–4 and 10 , 20 . ˙ j RT = jΩX ≡ j (XX ˙ −1 − X−1 X), ˙ one Putting in (2.10) R ≡ Rj , where R j 2 obtains the subfamily of co-rotational derivatives determined by the choice of a parameter j (an objective scalar isotropically depending on isochoric part of the tensor X): T ˙ ˙ ˙ T Dj [Z] := Qj (QT j ZQj ) Qj ≡ Z − ΨZ + ZΨ, Ψ = Qj Qj ,

Qj = QRj . (2.14)

This subfamily includes known Jaumann and Dienes derivatives. It generates the corresponding parametric subfamily of co-rotational stress and strain measures [23–25]: E˙ j = (QRj )T V(QRj ),

Dj [Ej ] = V,

Σj = (QRj )T S(QRj ).

(2.15)

Remark 2.3. All the co-rotational measures are non-holonomic; they demonstrate holonomic properties only in motions with fixed-in-time right principal strain axes. 2.3. Plasticity at finite strains 2.3.1. Method of generalization. The method for generalization of constitutive plasticity relations known at small strains to the case of finite strains consists in reproducing the form of a small-strains relation for material analogs of measures (at finite strains) from the generalized class (family, subfamily). This method provides the correctness of generalization in view of Noll’s objectivity principle [1–3] and Ilyushin’s macroscopic determinability postulate [7]. Let us consider the simplest example of isotropic elastic incompressible body which constitutive equation at small strains has the well-known form σ = 2μ˜ ε + σI

(2.16)


57

(σ is the indefinite scalar, ε˜ is the strain deviator). The corresponding generalized constitutive equation at finite strains for parametric right co-rotational measures Σj and E j (E˜j is the deviatoric part of E j ) has the form Σj = 2μE˜j + σI.

(2.17)

Because of non-holonomy of the measures Σj and E˜j the relation (2.17) describes not the elasticity but hypo-elasticity. In terms of rate tensors the relation (2.17) has the equivalent forms for right and left analogs ˜ + σI. ˙ j = 2μE˜˙ j + σI, Σ ˙ Dj [S] = 2μV ˙ (2.18) 2.3.2. Numerical experiments. For the generalization of the constitutive equation (2.16) to the relation of hypo-elasticity (2.17), (2.18), the parametric subfamily of co-rotational measures (2.15) was chosen with the parameter j of a fading memory [24, 25]: j = j0 exp(−m |ΩX | t) (j0 = const, m = const). (2.19) Numerical experiments were conducted for simultaneous tension and torsion of a cylinder (with initial radius R0 (0) = 1) at finite strains: ⎧ 6

⎪ λ(t), ⎨ r=R ϕ = κ(t) · λ(t) · Z + Φ, ⎪ ⎩ z = λ(t) · Z, where R, Φ, Z and r, ϕ, z are the cylindrical coordinates of a body point at initial and current configurations respectively, κ(t) is the torsion, and λ(t) is the extension (κ(t) 0, λ(t) 0). Putting κ(t) = t, λ(t) = 1 + ut, u = const the plots of non-zero components of Cauchy stress tensor were obtained in different cases with j0 = 1, m =0, 0.2, 0.5, 1, ∞, and μ = 23 105 MPa). Uniaxial tension: κ ≡ 0, λ(t) = 1 + ut, u = 1. Non-zero stress component is one and the same for all j (for all m) (Fig. 1).

Figure 1

58


Pure torsion: κ(t) = t, λ(t) ≡ 1. Graphs of non-zero stress components for fragments of initial radius R0 = 1 are presented in Fig. 2. Stresses σzz are non-zero: Pointing effect.

Figure 2 Graphs of stretching force and torque at pure torsion are shown in Fig. 3.

Figure 3 Simultaneous tension and torsion: κ(t) = t, λ(t) = 1 + ut, u = 1. Non-zero stress components for fragments of initial radius R0 = 1 are illustrated in Fig. 4.

Figure 4 For the case of simultaneous tension and torsion, graphs of stretching force (averaged tension stress) vs time (Fig. 5) and of torque (averaged tangential stress) vs time (Fig. 6) are presented for initial (a) and current (b) radius.


a

59

b Figure 5

a

b Figure 6

2.4. Conclusive remark Thus, the results of the conducted numerical experiments presented by graphs show that the choice of the pare of tensor stress and strain measures stating the geometrical aspect of a problem demonstrate (at one and the same form of constitutive equation) the essential effect on the analytical properties of the model of a material. It gives the principally new possibility for approximation the experimental data on finite plasticity in a wide range.

3. Cosserat models For building-up models of Cosserat type continuum systems the method of mechanical modeling [26] was applied. The main idea of the method consists in detailed analysis of the structure of a media and elaborating the appropriate initial model (often as discrete construction), derivation of constitutive and motion equations of the initial model, and averaging procedure for obtaining equations of a resulting continuum model. Here results of the method application are illustrated on example of onedimensional Cosserat model, namely the model of a supplied beam.

60


3.1. Model of a supplied Cosserat beam 3.1.1. Equations of the model. The initial model consists of a homogeneous beam supplied by identical rigid massive inclusions periodically placed along the longitudinal line of the beam on aces (spindles) parallel to one another and perpendicular to the beam line. We shall consider bending-tension motions of this construction in the plane orthogonal to the aces assuming that the inclusions are capable to rotate around their spindles being connected with the nearby carrying elements of the beam by elastic joints and with their nearest neighbors by elastic belt drives through transmissive weightless pulleys. The undeformed (reference) and deformed configurations of the supplied beam in its motion (in-plane bending-tension of the supporting beam and rotations of rigid massive inclusions) are schematically shown on Fig. 7.

a

(a)

(b)

Figure 7. Supplied beam in pre-deformed (a) and deformed (b) configurations. Massive inclusions are shown as dark circles, transmissive weightless pulleys as transparent circles, belt drive as a dashed line. A cell of the construction is picked out with chain line rectangle. Considering a cell of the construction as an element picked out in Fig. 7 one can derive the motion equations, the expression of the power of inner interactions (forces and moments) showing the pairs of energetically conjugate generalized forces and generalized displacements, and build-up the constitutive relations for the initial discrete construction. Then the averaging procedure at a → 0 preserving the values of linear densities of force and moment parameters and characteristics of the supplied beam leads to equations of the final continuum model. The motion equations of the continuum


61

model have the form ∂F + f − ρ¨r = 0, ∂ξ

∂Mbend + Qλ + m + Mincl→beam = 0, ∂ξ ∂Mint + min − Mincl→beam − J ϕïncl = 0. ∂ξ

(3.1)

Here r is the position vector of a beam point (u and w are its projections on x and y axes), λ is the elongation of the beam element, ϕincl is the absolute angle of the inclusion rotation, ξ is the longitudinal coordinate in undeformed configuration, F is the force vector in a beam section (P and Q are the longitudinal and shearing projections of F), Mbend is the bending moment of the supporting beam, Mint is the interactive moment of the system of inclusions, Mincl→beam is the moment action of inclusions on supporting beam elements, ρ is the linear mass density, J is the linear density of inertia moments of inclusions, and f , m, min are the linear densities of external force, external moment acting on a beam element and external moment acting on inclusions. The expression of the power of internal forces for continuum model takes the form ∂ϕbeam ˙ ∂ϕincl ˙ ˜ ˙ W(i) = P · λ+M +Mint · + Mincl→beam ·(ϕincl − ϕbeam )˙ . bend · ∂ξ ∂ξ (3.2) The simple variant of the constitutive equations expressing elastic properties of the construction holds the form ∂ϕbeam ∂ϕincl , Mint = Cincl , P = Ctens (λ − 1), Mbend = Cbend (3.3) ∂ξ ∂ξ Mincl→beam = Cincl→beam (ϕincl − ϕbeam ) with material constants Ctens , Cbend , Cincl and Cincl→beam . The equations (3.1), (3.3) express the conditions of dynamic equilibrium of elastic one-dimensional Cosserat continuum (in this plane case). In general case, the substitution of the constitutive equations (3.3) into the equations (3.1) leads to the system of one vector and two scalar equations for unknown functions r, ϕincl , Q (quantities λ and ϕbeam are expressed through r). Particular and special models should be pointed out: 1. Disconnected, or uncoupled model. In the case of absence of interactions between the beam and inclusions, i.e., when Mincl→beam = 0, the first two equations (3.1) describe the motion of the beam (weighted by inclusions) and the third one independently describes the dynamic of inclusions’ rotations. 2. Momentless model: Mincl ≡ Mint ≡ 0 (for Cincl = 0). The “belt connections” are vanishing, but Mincl→beam and ϕincl must not be equal to 0. 3. Pseudo-continuum (constrained model). The inner kinematic constrain ϕincl ≡ ϕbeam eliminates ϕincl from the set of decision variables, and the

62

G.L. Brovko, O.A. Ivanova and A.S. Finoshkina reaction (“supporting force”) Mincl→beam becomes undetermined by constitutive equations (it is undergoing to be determined from the whole system of equations).

3.1.2. Linearization at small movements. When the displacements u and w, specific elongation ε = λ − 1 and angles of the beam elements ϕbeam are small then the approximate equations hold (κ is the curvature of the beam line): ∂u ∂ϕbeam ∼ ∂ 2 w ∂w ∂ ∼ ∂ , λ∼ , ϕbeam ∼ , κ∼ ξ∼ . = = 1, ε ∼ = = = = x, = ∂ξ ∂x ∂x ∂x ∂x ∂x2

(3.4)

∂ϕincl , ϕincl − ϕbeam small enough, the consti∂x tutive equations (3.3) (in view of (3.4)) take the form similar to classic [59–61]: For the value ranges of ε, κ,

P = EScross ε,

Mbend = EJcross κ,

Mint = C

Mincl→beam = K (ϕincl − ϕbeam ) ,

∂ϕincl , ∂ξ

(3.5)

where E is the Young module of the beam material, Scross is the value of area, Jcross is the value of moment of inertia of the cross-section of the beam, C is the stiffness coefficient of the belt drive, K is the elasticity constant of hinging (between inclusions and the beam). The substitution of (3.5) to (3.1) leads to the system of equations for four unknown functions u, w, ϕincl , Q of arguments x and t: ∂ 2u ∂2u + f = ρ , x ∂x2 ∂t2 ∂2w ∂Q + fy = ρ 2 , ∂x ∂t

EScross

∂3w ∂w = 0, EJcross 3 + m + Q + K ϕincl − ∂x ∂x ∂ 2 ϕincl ∂w ∂ 2 ϕincl C + m − K ϕ − = J incl in ∂x2 ∂x ∂t2

(3.6)

with the known external forces fx , fy and external moments m, min . The first of the equations (3.6) independently describes the longitudinal motion. Eliminating Q from the last three equations (3.6) one obtains the system of two partial differential equations for two functions – deflection w of the supporting beam and angle ϕincl of inclusions: ∂2w ∂ϕincl ∂m ∂2w ∂4w −EJcross 4 + K 2 − K − + fy = ρ 2 , ∂x ∂x ∂x ∂x ∂t (3.7) ∂ 2 ϕincl ∂w ∂ 2 ϕincl + min = J C − Kϕincl + K . ∂x2 ∂x ∂t2 For disconnected model (K = 0) the system (3.7) becomes simpler and the equations become un-tied: the first one is the equation for beam deflection w,


63

in statics it agrees with the well-known Zhuravskiy equations [60] (taking into account the mass of inclusions), the second equation separately describes the rotations of inclusions (pulleys) under the external linear-specific moment min , and (additionally) in the case of C = 0 (momentless model) the disks rotations are independent from one another. 3.2. Small vibrations 3.2.1. Free vibrations. Concentrating the attention at the system (3.7) (eliminating the consideration of the first equation (3.6)) let us study free linear vibrations (in-plane deflections and inclusions rotations) of the system assuming K = 0 and C = 0 as well as the absence of external forces and moments fy = 0, m = 0 and min = 0, and accepting boundary conditions on the both edges x = 0 and x = l as follows ∂2w ∂ϕincl w = 0, =0 (3.8) = 0, ∂x2 ∂x (pinning of the edges and absence of moment actions on the ends of the beam and end inclusions). Finding the solution of the problem (3.7), (3.8) in the form w = Cw (x) eiωt ,

ϕincl = Cϕ (x) eiωt ,

(3.9)

where Cw (x) and Cϕ (x) are the amplitude functions, and ω is the angular (radian) oscillation frequency, one obtains the system of ordinary differential equations: d2 Cw (x) dCϕ (x) d4 Cw (x) −K +K = ρω 2 Cw (x) , 4 2 dx dx dx d2 Cϕ (x) dCw (x) = Jω 2 Cϕ (x) . −C + KCϕ (x) − K 2 dx dx Search of eigenfunctions for (3.10), (3.8) leads to equalities EJcross

(3.10)

πk , k ∈ N) (3.11) l with arbitrary constants A, B and, for existence of non-trivial solutions, to the characteristic equation biquadratic with respect to the frequency ω ρJω 4 − JEJcross p4 + (ρC + KJ) p2 + Kρ ω 2 + (3.12) +EJcross Cp6 + K (EJcross + C) p4 = 0. Cw (x) = A sin px,

Cϕ (x) = B cos px

(p =

Considering (3.12) as a quadratic equation for ω 2 we have the expression of its discriminant 2

6 D #(p) = (JEJcross ) p8 − 2JEJcross $ (ρC − KJ) p 2

+ (ρC − KJ) − 2ρKJEJcross p4 +2Kρ (ρC + KJ) p2 + ρ2 K 2

(3.13)

and its solutions (both roots of this quadratic equation are real and positive):

JEJcross p4 + (ρC + KJ) p2 + Kρ ± D (p) 2 , (3.14) ω1,2 (p) = 2ρJ

64


hence, the solutions of the equation (3.12) are the real nonzero numbers ±ω1 (p), ±ω2 (p); let us agree that ω1 (p) > 0, ω2 (p) > 0 and ω1 (p) ω2 (p). In general case (K = 0) for each p from (3.11) and each corresponding (by (3.14)) one of two obtained values of ω, the non-trivial solution (3.11) of the system (3.10) holds the proportionality between A and B A = a(p, ω) · Φ,

B = Φ,

(3.15)

where Φ is an arbitrary (nonzero) constant, and a(p, ω) =

EJcross

p4

Kp Cp2 + K − Jω 2 . ≡ 2 2 + Kp − ρω Kp

(3.16)

Thus, each solution of the problem (3.7), (3.8) in the form (3.9) is the pare of functions wj = aj Φj sin pxe±iωj t ,

ϕincl j = Φj cos pxe±iωj t

(j = 1, 2),

(3.17)

where the signs + or − as well as the integer j in the both equalities are chosen as the same, Φj is an arbitrary nonzero constant, p is any of values from (3.11), ωj = ωj (p) and aj = a(p, ωj ) are determined by the formulae (3.14) and (3.16). The linearity of the problem (3.7), (3.8) permits each linear combination of pares of functions (3.17) with constant coefficients be a solution too. Every possible real-valued combination is a pare of sums (with natural k): $ 0 πkx # (k) (k) (k) (k) (k) (k) (k) (k) w = sin a1 Φ1 sin (ω1 t + ϕ1 ) + a2 Φ2 sin (ω2 t + ϕ2 ) , l k $ 0 πkx # (k) (k) (k) (k) (k) (k) Φ1 sin (ω1 t + ϕ1 ) + Φ2 sin (ω2 t + ϕ2 ) , ϕincl = cos l k (3.18) (k) (k) where Φj , ϕj are independent arbitrary constants, while definite constants (k)

(k)

ωj = ωj (p) and aj = a(p, ωj ) (j = 1, 2) are specified by (3.14), (3.16) for each p = πk/l (k ∈ N ). Solutions of form (3.18) are either finite sums or infinite series which uniform convergence is provided by the convergence of majorant series (k) (k) (k) (j = 1, 2). Φj , aj Φj k

k

The pare of functions represents the general form for the solution of the problem (3.7), (3.8) (in frames of made assumptions). The essential feature of the solution consists in the fact that, for each oscillation mode determined by a natural k, there exist exactly two values of frequency and two forms of oscillations. The same feature of oscillations is pointed out for another model of a beam constructed in the frames of other approach based on the micro-polar elasticity theory [62]. In the case of disconnected model (K = 0) the consideration turns simpler, and the solution corresponds to independent oscillations of the beam and system of inclusions.


65

Example. Here are given the calculations of the problem (3.7), (3.8) and its solution (3.18) in application to the considered constructions (Fig. 7) of “antenna type” frame consisting of a metallic supporting rod with periodically placed along its line perpendicular cross rods playing the role of massive inclusions; cross rods are supposed conditionally rigid, elastically joint to the supporting rod, their tips are connected by a rubber thread (as an elastic belt drive). For the family of such constructions of the length l = 1 m similar to one another and characterized by the material constants E = 2 · 1011 mN2 , ρ = 0, 61 kg J = 7.9 · 10−3 kg · m, m, −11 4 Jcross = 5, 2 · 10 m , K = 250 N, C = 25 N · m2 , (k)

(k)

the calculations show that the values (3.14) of the both frequencies ω1 and ω2 (measured in sec−1 ) increase on the interval 1 ≤ k ≤ 100 (see Fig. 8). Values (3.16) (k) (k) of the coefficients a1 and a2 (measured in meters) vs k (expressing the quotients A/B of amplitude coefficients accordingly to (3.15)) are shown on Fig. 9. For lower (k) (k) frequencies ω1 , corresponding values of a1 are all positive and rapidly tend to (k) (k) zero with k increasing. Values of a2 correspond to higher frequencies ω2 , they all are negative and their modules increase infinitely with k.

4500 4000 3500 3000 2500 2000 1500 1000 500 0 0

(a) (k)

2

4

(b)

6

8

10

(k)

Figure 8. Graphs of ω1 and ω2 in ranges (a) 1 k 100 and (b) (k) (k) 1 k 10 (ω1 – circles, ω2 – daggers). For the mode with k = 1 there are exactly two oscillation forms: 1) with an(1) (1) gular (radian) frequency ω1 = 59, 6 sec−1 (oscillation frequency ν1 = 9, 49 Hz) (1) (1) and coefficient a1 = 0, 5972 m, and 2) with circle frequency ω2 = 256, 6 sec−1 (1) (1) (ν2 = 40, 86 Hz) and coefficient a2 = −0, 0214 m.

66

G.L. Brovko, O.A. Ivanova and A.S. Finoshkina 0.6

2000 0

0.4

-2000 -4000

0.2

-6000

0

-8000

-0.2

-10000 -12000

-0.4

-14000

-0.6

-16000 -18000

0

20

40

60

80

100

-0.8

1

1.5

2

2.5

3

3.5

4

4.5

5

(b)

(a) (k)

(k)

Figure 9. Graphs of a1 and a2 in ranges (a) 1 k 100 and (b) (k) (k) 1 k 5 (a1 – circles, a2 – daggers). 0.06

0.2

0.05

0.15 0.1

0.04

0.05

0.03

0 0.02

-0.05

0.01

-0.1

0 -0.01

0

-0.15 0.2

0.4

(a)

0.6

0.8

1

-0.2

0

0.2

0.4

(b)

0.6

0.8

1

Figure 10. Maximal deflections (measured in meters) of the supporting rod (a) and corresponding maximal rotation angles of supporting rod (b) in the first (circles) and in the second (daggers) oscillation forms. The solid line in (b) shows the rotation angles of the rods-inclusions (the same in both oscillation forms) corresponding to maximal deflections (a).

The configurations of the supporting rod respondent to maximal deflections and rotation angles (maximal values) of rods-inclusions and elements of the supporting rod in the first and in the second oscillation forms of the first mode (k = 1) are presented in the Fig. 10. Particularly, the Fig. 10 (b) shows that the first oscillation form (low frequency) corresponds to “concomitant” (in one and the same direction) rotations


67

of inclusions and the elements of the supporting rod (see Fig. 11 (a)) and the second oscillation form (higher frequency) corresponds to “counter” rotations, i.e., in opposite directions (Fig. 11 (b)).

(a)

(b)

Figure 11. “Concomitant” (a) and “counter” (b) forms of oscillations. Calculations show that if maximal rotations of end inclusions are equal to 5◦ then the maximal deflection (in the middle of supporting rod) reaches 5 cm in the first form and only 2 mm in the second form. 3.2.2. Vibrations in a fluid-flow (gas-flow). The same model of the supplied beam was considered in view of equations (3.7) in the case when fy = 0, m = 0, but min = gϕincl ,

(3.19)

with the coefficient g. At g > 0 the moment min tends to increase the angular deviation of inclusions, and at g < 0 it tends to decrease the angular deviation. The condition (3.19) could be interpreted as the influence of homogeneous approach gas- or fluid-flow with the constant velocity v and mass density ρ∗ onto the inclusions equipped by winglets turned towards the flow (g > 0) or along the flow (g < 0). At small angular deviations ϕincl of inclusions the quantity g may be put as constant: g = cρ∗ v 2 , (3.20) where c is an aerodynamic constant of the same sign as g. The same approach as in the case of free vibrations leads us to new expression of discriminant 2 D (p) = (JEJcross ) p8 + 2JEJcross (KJ − ρC) p6 # $ (3.21) + (ρC − KJ)2 − 2ρJEJcross (K − g) p4 2

+ [2ρ (K − g) (ρC + KJ) + 4ρJKg] p2 + ρ2 (K − g) , and expressions for frequencies ω12

JEJcross p4 + (ρC + KJ) p2 + ρ (K − g) − = 2ρJ

D (p)

,

(3.22)

JEJcross p4 + (ρC + KJ) p2 + ρ (K − g) + D (p) = 2ρJ (the case g = 0 coincides with free vibrations). The calculation of the similar “antenna type” construction in the gas-flow shows the dependence of squared frequencies on k for different g (Fig. 12) and on g for different k (Fig. 13). ω22

68

G.L. Brovko, O.A. Ivanova and A.S. Finoshkina 5

x 10

12

g=−100000 10 g=−2000 8 g=−1000 g=0

6

g=400 1

ω2

4 g=1500 2 g=3000

0

g=5000

−2 −4 −6

1

2

3 k

4

5

Figure 12. Dependence of squared low frequency ω12 (sec−2 ) on k (k = 1, 2, . . . , 5) at g = −100000 N, g = −2000 N, g = −1000 N, g = 0 N, g = 400 N, g = 1500 N, g = 3000 N, g = 5000 N. 5

2

6

x 10

1.5

x 10

1.5 1 k=5

k=4 1

2 2

ω1, ω2

2

0.5

0.5

2

ω1, ω2

k=2

0

0 k=1

k=1

−0.5

k=2 k=3

−0.5 −1

−1.5

0

200

400

600

800 g

(a)

1000

1200

1400

1600

−1 −1000

0

1000

2000

3000 g

4000

5000

6000

7000

(b)

Figure 13. Dependence of ω12 and ω22 on g (a) at k = 1, 2 in the range 0 g 1600 N and (b) at k = 1, 2, . . . , 5 in the range −1000 N g 7000 N. In all cases the negative value of a squared frequency discovers the phenomenon of “divergence”: exponential one-way deviation of the system (instead of oscillation regime). 3.3. Closing remarks 3.3.1. Other models. Using the method of mechanical (constructive) modeling, other one-dimensional Cosserat models were constructed demonstrating different properties. For example, the model geometrically similar to considered one, but having elastic – perfect plastic properties is able to have infinite number of equilibrium forms at the same loading factors (in a certain range).


69

The method of mechanical (constructive) modeling was also applied to similar 2-dimensional systems. Motion equations (in Euler description) of corresponding 2-dimensional Cosserat continuum model have appeared in the form: div S + ρf − ρ¨ u = 0,

div M + 2 coax S + ρg − ρj · ω ˙ = 0,

where u and ω are the displacement and the rotational velocity vectors (in-plane movement), S and M are stress and moment-stress tensors, coax S is the coaxial vector to S, f and g are the external force and external moment, ρ is the mass density, j is the moment of inertia (2nd rank tensor). The equivalent equations are derived in Lagrangean description. 3.3.2. Conclusion. Thus, the building of Cosserat continuum models from appropriate initial discrete structures through the indicated method of mechanical (constructive) modeling, demonstrates the productivity of a clear geometric approach to building-up the destination analytical properties of a product model: • on the one hand, it permits to immediately clarify the mechanical sense of all Cosserat characteristics, • on the other hand, it demonstrates the principal availability for real existence (naturally or artificially) of Cosserat type continua and drops a hint at possible ways for technological manufacturing of such materials with predetermined properties.

4. Saturated porous media By using the method of mechanical modeling and the hypothesis of interpenetrative continua, the new model of multi-phase saturated porous media was built on the base of [48] and following results [49–52]. The model is able to circumscribe deformations of a skeleton at arbitrary strains (in Lagrangean description) and arbitrary motions of movable (fluid, gas) components through the skeleton (in Eulerian description). Here we present the variant of a three-phase porous conglomerate consisting of solid skeleton made from an incompressible material saturated by an incompressible fluid and compressible barotropic gas (vacuum cavities are considered as extremely rarefied gas). Lagrangean independent variables for skeleton are x, t and Eulerian variables for fluid and gas are x, t: x ∈ Ω0 , x ∈ Ω where Ω0 and Ω are domains of the reference and current configurations of the skeleton. 4.1. Equations of the model According to the model [48] the state of the conglomerate at each moment is characterized by the following functions (unknown in a problem): vp , vs , vf , vg – specific volumes of porous space, skeleton, fluid and gas phases, ρg m , pp – average in a macro-volume true mass density of the gas and porous pressure, f – position vector of skeleton points in a current configuration (x = f (x, t) is the Lagrangean law of motion), A – deformation gradient of the skeleton, E – Green strain tensor

70


(effective value), Ss – effective Cauchy stress tensor of the skeleton, vs , vf , vg – velocity vectors (average in a macro-volume) of skeleton, fluid and gas particles (39 numeric quantities at all). The motion of the conglomerate is regulated [48] by the equations for the specific volumes vp + vs = 1, vp = vf + vg , (4.1) skeleton kinematic equations d f (x, t) 1 T , A = ∇x f , E = A ·A−I (x = f (x, t)), (4.2) vs (x, t) = dt 2 continuity equations for the skeleton phase (vs0 is the skeleton specific volume in a reference configuration) vs0 vs = (θ = J − 1, J ≡ | det A|), (4.3) 1+θ for fluid and gas phases dvf + vf div vf = 0, (4.4) dt d (vg ρg m ) + vg ρg m div vg = 0, (4.5) dt by motion equations of fluid, gas and skeleton phases accordingly dvf , (4.6) div Sf + ρf gf + ff = ρf dt dvg div Sg + ρg gg + fg = ρg , (4.7) dt dvs div Ss + ρs gs + fs = ρs , (4.8) dt where gf , gg , gs are given external body forces (mass densities), ff , fg , fs are interactive (from other phases) body forces (volume densities) on fluid, gas and skeleton, and, finally, by constitutive equations of barotropic gas (with known function pgas ) (4.9) pp = pgas (ρg m ) and skeleton phase (with known mapping F ) (4.10) Ss = Q(x, t)F [E(x, τ )]t0 τ t ; pp , x QT (x, t) (Q is the orthogonal tensor of polar decomposition of deformation gradient A). Here true densities of fluid media ρf m and skeleton material ρs m are known constants, and effective densities of skeleton, fluid and gas phases are defined as ρs = vs ρs m , ρf = vf ρf m , ρg = vg ρg m . In (4.9) the hypothesis of equality between average true values of fluid and gas pressure is accepted. Effective stresses of fluid, gas and solid phases are defined by Sf , Sg , Ss . Constitutive equations of fluid and gas phases are supposed in the form Sf = −pf I + λf ϑf + 2μf Vf ,

Sg = −pg I + λg ϑg + 2μg Vg ,

(4.11)


71

where λf , μf , λg , μg are the effective viscosity coefficients of fluid and gas phases (frequently assumed constant), pf = vf pp , pg = vg pp are effective values of pressure in fluid and gas phases, Vf = sym ∇x vf , Vg = sym ∇x vg are effective strain rate tensors in these phases, and ϑf = tr Vf ≡ div vf , ϑg = tr Vg ≡ div vg are corresponding rates of relative volume expansion. In order to decline the number of searched functions the quantities Sf , Sg are eliminated from the set of unknown quantities and are used in equations (4.6), (4.7) only as definitions of the form (4.11). Thus, the equations (4.1)–(4.10) taking into account of (4.11) compose the system of 39 numerical equations for 39 previously mentioned unknown functions. Initial and boundary conditions enclose the formulation of a boundary value problem for three-phase model of a saturated porous medium of type [48] in a general case. Simplifications of the system take place in special cases and in the case of small strains of a skeleton and small motions of fluid and gas components. The equations of this three-phase model may be reduced to those for a two-phase model (skeleton-fluid or skeleton-gas). 4.2. Constitutive relations 4.2.1. Effective properties of phases. The formulation of the problem needs the preliminary (experimentally) determination of the concrete effective mechanical properties of a conglomerate: effective viscosity coefficients of fluid and gas phases λf , μf , λg , μg in (4.11), the form of the function pgas in (4.9), the form of the mapping F in the constitutive equation (4.10) of the skeleton phase (taking into account its saturation), and forms of the interactive forces ff , fg , fs in (4.6)–(4.8). The function pgas characterizing the gas barotropy is determined by its thermomechanical nature and in certain cases could be taken as linear. Viscosity coefficients λf , μf , λg , μg depend on natural properties of fluid and gas media, on specific volumes of their phases in a conglomerate, on geometry of porous space, on temperature and penetration velocities; in isothermal processes within certain range of specific volumes and velocities these coefficients could be put as functions of porosity, and at homogeneous porosity as constants. Effective mechanical properties of the skeleton phase possess a more complicated nature. In the case of small strains the constitutive equation for a linearly elastic isotropic saturated skeleton could be taken in the form [63] Ss = −vs pp I + λθI + 2με

(4.12)

already used in [47, 52] for two-phase models. 4.2.2. Interactive forces. The volumetric interactive body forces ff , fg , fs appearing in (4.6), (4.7), (4.8) are supposed consisting of static (acting in statics and in motion) and dynamic (acting only in motion) components fs = fs stat + fs dyn ,

ff = ff stat + ff dyn ,

fg = fg stat + fg dyn .

(4.13)

As phase interactions in statics are regulated only by porous pressure pp and specific volumes, we shall put, after simple geometrical calculations, the static

72


components in the form fs stat = pp grad vs ,

ff stat = pp grad vf ,

fg stat = pp grad vg ,

(4.14)

which shows that static components are generated only by non-homogeneity of porosity and by variability of specific volumes of fluid and gas phases in pores. For dynamic components in (4.13), taking into account pair interactions between phases, the following expressions could be accepted fs = fs←f + fs←g ,

ff = −fs←f + ff←g ,

fg = −fs←g − ff←g ,

(4.15)

where fs←f , fs←g , ff←g denote dynamic forces acting from fluid on skeleton, from gas on skeleton and from gas on fluid. Supposing dependence of these dynamic force components on vectors of velocities vs , vf , vg and accelerations ws , wf , wg in pares of the phases fs←f = fs←f (vs , vf , ws , wf ),

fs←g = fs←g (vs , vg , ws , wg ),

ff←g = ff←g (vf , vg , wf , wg ),

(4.16)

applying the approach of [64] based on the principle of material objectivity [1– 3], developing the results of [47, 50, 52] we derive the general reduced forms of constitutive equations for dynamic interactive forces fs←f = fs←f (vf rel s , wf rel s ) vf0 rel s , fs←g = fs←g (vg rel s , wg rel s ) vg0 rel s ,

(4.17)

ff←g = ff←g (vg rel f , wg rel f ) vg0 rel f , where vf rel s , vg rel s , vg rel f vf0 rel s , vg0 rel s , vg0 rel f are modules and unit directional vectors of relative velocities of the phases, derivatives wf rel s = v˙ f rel s , wg rel s = v˙ g rel s , wg rel f = v˙ g rel f are the projections of the relative acceleration vectors wf rel s , wg rel s , wg rel f on according directional vectors of relative velocities, and fs←f , fs←g , ff←g are material scalar (non-negative) functions (modules of the vectors (4.17)). Then, taking into account the dependence of the modules of interactive forces (4.17) on the typical size d of the porous structure (for example, in reference configuration), on specific volumes vs , vf , vg , modules of relative velocities vf rel s , vg rel s , vg rel f and collinear relative accelerations wf rel s , wg rel s , wg rel f , on effective mass densities ρf , ρg and viscosities λf , μf , λg , μg of fluid and gas phases fs←f = fs←f (d, vs , vf , ρf , λf , μf , vf rel s , wf rel s ), fs←g = fs←g (d, vs , vg , ρg , λg , μg , vg rel s , wg rel s ), ff←g = ff←g (d, vf , vg , ρf , ρg , λf , μf , λg , μg , vg rel f , wg rel f ),

(4.18)


73

applying methods of the measurement theory [65], similarly to [47, 49, 50] one obtains ρf vf2 rel s 1 1 fs←f = , , ϕs←f vs , vf , lf , d Ref rel s Bf rel s ρg vg2 rel s 1 1 ϕs←g vs , vg , lg , fs←g = , , (4.19) d Reg rel s Bg rel s ρg vg2 rel f 1 1 ff←g = ϕf←g vf , vg , lf , lg , rg/f , mg/f , , , d Reg rel f Bg rel f where symbols Re and B denote the Reynolds numbers and new dimensionless parameters ρf vf rel s d ρg vg rel s d ρg vg rel f d Ref rel s = , Reg rel s = , Reg rel f = , μf μg μg (4.20) 2 2 vg rel s vg rel f v2 Bg rel s = , Bg rel f = Bf rel s = f rel s , wf rel s d wg rel s d wg rel f d λ

ρ

μ

and designations lf = λμff , lg = μgg , rg/f = ρgf , mg/f = μgf have been used. Supposing that the functions ϕs←f , ϕs←g , ϕf←g in (4.19) are linear with respect to their last two arguments we obtain the following representations of the forces (4.17) wf rel s F D B fs←f = fs←f vf rel s , + fs←f + fs←f ≡ c0s←f vf rel s + d0s←f + b0s←f vf rel s wg rel s F D B vg rel s , + fs←g + fs←g ≡ c0s←g vg rel s + d0s←g + b0s←g fs←g = fs←g vg rel s wg rel f F D B ff←g = ff←g + ff←g + ff←g ≡ c0f←g vg rel f + d0f←g + b0f←g vg rel f vg rel f (4.21) with the notations ρf ϕF μf ϕD s←f s←f c0s←f = , b0s←f = ρf ϕB , d0s←f = s←f , d d2 ρg ϕF μg ϕD s←g s←g (4.22) , d0s←g = c0s←g = , b0s←g = ρg ϕB s←g , d d2 ρg ϕF μg ϕD f←g f←g , d0f←g = c0f←g = , b0f←g = ρg ϕB f←g , d d2 D B F where material functions ϕF s←f , ϕs←f , ϕs←f depend on vs , vf , lf , functions ϕs←g , D B F D B ϕs←g , ϕs←g depend on vs , vg , lg , and functions ϕf←g , ϕf←g , ϕf←g depend on vf , ρ μ vg , lf , lg , rg/f = ρgf , mg/f = μgf . Thus, the quantities (4.22) are dimensional and depend on arguments of according functions (4.18), besides the last two ones – relative velocities and accelerations. Within a certain range of the arguments the quantities (4.22) may be set to constants.

74


The terms in (4.21) marked by indexes “F”, “D” and “B” have the meaning of interactive forces of frontal resistance (dynamic velocity pressure), viscous resistance (Darcy law) and inertial resistance (of Biot added mass type). Relations (4.21) together with (4.13)–(4.15) lead to the final form of constitutive equations for full (static and dynamic) interactive forces. In fact, the representations of dynamic components of interactive forces may be simplified in concrete cases (see, for instance, [66] and [47, 49, 50]). 4.2.3. Remarks on compound interactions. In assumption of compound dynamic interactions consisting of continuously distributed volumetric forces and moments, acceptable forms of constitutive equations of such interactions for two-phase (solidsolid and solid-fluid) media were studied [47, 50] in view of invariance [67, 68]. For a saturated porous medium (solid-fluid conglomerate), supposing dependence of such a force f and a moment M on vectors of relative velocities vrel = vf − vs and relative accelerations wrel = wf − ws as well as on relative spin (skew-symmetric tensor) Ωrel = Ωf − Ωs , the following forms of constitutive equations for these compound interactions were obtained: f = avrel + bΩrel · vrel + cΩ2rel · vrel + dE : Ωrel , M = AΩrel + Bskw(vrel ⊗ Ωrel · vrel )+ + Cskw(vrel ⊗

Ω2rel

(4.23)

· vrel ) + DE · vrel ,

where a, b, c, d, A, B, C, D are functions of the scalar product wrel · vrel and mutual invariants of vrel and Ωrel , and E is Levy–Civita tensor. The relations (4.23) show, in particular, that, besides the front resistance force, new types of interactions exerted by the fluid on the skeleton may appear, namely, lifting (shifting) forces as well as overturning and rotating (“screw”) moments. Such interactions may be caused forcibly (relatively to fluid flow) by a pore tortuosity [69] organized in a special geometrical order. Accounting of distributed moment interactions leads out the frames of classic continuum mechanics (stress tensor becomes non-symmetric) and needs to involve non-classic theories including Cosserat theory [9].

5. Conclusion The considered models and problems illustrate the deep connection between geometrical an analytical characteristics of media and show the usefulness of its account in building-up constitutive relations – the central determining element in formulations of problems. The method of generalization of plasticity constitutive equations to finite strains (Sect. 2) based on the new theory of tensor strain and stress measures demonstrates the essential effect of geometrical properties of different objective derivatives (even from one subfamily of co-rotational ones) on analytical properties of the constitutive equation (taken in one and the same form), that is, on


75

mechanical properties of the model. The considered example of the hypo-elasticity model in numerical experiments shows that the model behavior has rather different character depending on different choice of an objective derivative and covers a wide range of responses. The method was also applied to different models of plasticity (known at small strains) [23–25]; the obtained constitutive equations of plasticity models at finite strains also demonstrate the strong influence of objective derivatives and a wide range of responses, which seems to be capable to approximate a wide range of experimental data in finite plasticity. Co-rotational measures satisfying all the Axioms 1–4 and 10 , 20 permit the mechanical interpretations of their spherical and deviator parts in the very same manner as in the case of small strains, and their deviators comply the geometrical representation in Ilyushin’s 5-dimensional space. It should be especially marked out that this representation permitted incidentally to make clear the geometrical nature of well-known “anomaly” [70–73] of stress oscillations in simple shear of hypoelastic (and plastic as well) materials modeled with Jaumann objective derivative (included in the considered sub-family): the strain trajectory of this process in Ilyushin’s space has the form of a circle, and stresses vector (proportional to deformations vector) performs periodical oscillatory movement. The method of mechanical modeling applied here to building-up Cosserat type models (Sect. 3) and models of saturated porous media (Sect. 4) consists of three steps: elaboration of the initial model schematically (discretely) representing a studied medium, derivation of equations for initial model, and averaging procedure to obtain equations for continuum model. The first step provides a detailed analysis (first of all, from geometrical point of view) of the structure of a representative volume of the medium, this step is the most creative. The second one is mainly technical, the complexity level of its realization depends on the complexity of a geometric structure of the initial model. The third step is usually based on certain assumptions for averaging procedure, it is the most difficult in substantiation and exploration, and may lead to different results depending on the assumptions. The method of mechanical modeling may be applied to study of media with complicated mechanical properties (in that number non-classic type properties) as well as to elaboration of calculated basis for technological manufacture of new structures and constructive materials with desired predetermined mechanical properties, including micro-systems and nano-materials.

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[61] E.P. Popov, Theory and calculations for flexible elastic rods. Moscow, Nauka, 1986. [62] A.A. Atoyan, S.H. Sargsyan, The study of free vibrations of micro-polar elastic thin plates. Trans. Nat. Acad. Sci. Armenia 104 (2004), No. 4, 287–294. [63] G.L. Brovko, Models and problems for saturated porous media. Bull. Moscow Univ. (Vestnik Moskovskogo Universiteta). Ser. 1. Mathematics, Mechanics (in print). [64] K. Wilmanski, B. Albers, Two notes on continuous modeling of porous media (A note on objectivity of momentum sources in porous materials). Preprint No. 579 ISSN 0946 – 8633. Berlin, Weierstrass Institute for Applied Analysis and Stochastics, 2000. [65] L.I. Sedov, Methods of similarity and measurements in mechanics. Moscow, Nauka, 1977. [66] R.E. Collins, Flow of fluids through porous materials. New York, Reinhold Publishing Co., 1961. [67] A.J.M. Spencer, Theory of invariants. Continuum Physics, vol. I part III, ed. A.C. Eringen. New York–London, Academic, 1971, 239–353. [68] P. Lankaster, Theory of matrices. New York–London, Academic Press, 1969. [69] K. Wilmanski, Tortuosity and objective relative accelerations in the theory of porous materials. Proc. Roy. Soc. A 461 (2005), 1533–1561. [70] J.C. Nagtegaal, J.E. de Jong, Some aspects of nonisotropic work hardening in finite strain plasticity. Plasticity of metals at finite strain: Theory, Experiment and Computation. Stanford Univ. and Dept. Mech. Eng., R.P.I., 1982, 65–102. [71] Y.F. Dafalias, Corotational rates for kinematic hardening at large plastic deformations. Trans. ASME: Journ. Appl. Mech. V. 50 (1983), No. 3, 561–565. [72] E.H. Lee, R.L. Mallett, T.B. Wertheimer, Stress analysis for anisotropic hardening in finite-deformation plasticity. Trans. ASME: Journ. Appl. Mech. V. 50 (1983), No. 3, 554–560. [73] S.N. Atluri, On constitutive relations at finite strain: hypoelasticity and elastoplasticity with isotropic or kinematic hardening. Comp. Meth. Appl. Mech. and Eng. V. 43 (1984), No. 2, 137–171. George L. Brovko, Olga A. Ivanova and Alexandra S. Finoshkina Theory of Elasticity Department Faculty of Mechanics and Mathematics Lomonosov Moscow State University Main Building of MSU Leninskiye Gory 119992 Moscow, Russia e-mail: [email protected] [email protected] [email protected]



Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schr¨ odinger Operators on Bounded Lipschitz Domains Fritz Gesztesy and Marius Mitrea Dedicated to the memory of M.G. Krein (1907–1989).

Abstract. We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schr¨ odinger operators on bounded Lipschitz domains in Rn , n 2, with generalized Robin boundary conditions. Mathematics Subject Classification (2000). Primary: 35J10, 35J25, 35Q40; Secondary: 35P05, 47A10, 47F05. Keywords. Multi-dimensional Schr¨ odinger operators, bounded Lipschitz domains, Robin-to-Dirichlet and Dirichlet-to-Neumann maps.

1. Introduction This paper is a direct continuation of our recent paper [35] in which we studied Schr¨ odinger operators on bounded Lipschitz and C 1,r -domains with generalized Robin boundary conditions and discussed associated Robin-to-Dirichlet maps and Krein-type resolvent formulas. The paper [35], in turn, was a continuation of the earlier papers [32] and [36], where we studied general, not necessarily self-adjoint, Schr¨ odinger operators on C 1,r -domains Ω ⊂ Rn , n ∈ N, n 2, with compact boundaries ∂Ω, (1/2) < r < 1 (including unbounded domains, i.e., exterior domains) with Dirichlet and Neumann boundary conditions on ∂Ω. Our results also applied to convex domains Ω and to domains satisfying a uniform exterior ball condition. In addition, a careful discussion of locally singular potentials V with close to optimal local behavior of V was provided in [32] and [36]. Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306.

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In the current paper and in [35], we are exploring a different direction: Rather than discussing potentials with close to optimal local behavior, we will assume that V ∈ L∞ (Ω; dn x) and hence essentially replace it by zero nearly everywhere in this paper. On the other hand, instead of treating Dirichlet and Neumann boundary conditions at ∂Ω, we now consider generalized Robin and again Dirichlet boundary conditions, but under minimal smoothness conditions on the domain Ω, that is, we now consider Lipschitz domains Ω. Additionally, to reduce some technicalities, we will assume that Ω is bounded throughout this paper. The principal new result in this paper is a derivation of Krein-type resolvent formulas for Schr¨ odinger operators on bounded Lipschitz domains Ω in connection with two different generalized Robin boundary conditions on ∂Ω. In Section 2 we recall our recent detailed discussion of self-adjoint Laplacians with generalized Robin (and Dirichlet) boundary conditions on ∂Ω in [35]. In Section 3 we summarize generalized Robin and Dirichlet boundary value problems and introduce associated Robin-to-Dirichlet and Dirichlet-to-Robin maps following [35]. Section 4 is devoted to Krein-type resolvent formulas connecting Dirichlet and generalized Robin Laplacians with the help of the Robin-to-Dirichlet map. Section 5 contains our principal new results and studies Robin-to-Robin maps and general Krein-type formulas involving Robin-to-Robin maps. Appendix A collects useful material on Sobolev spaces and trace maps for Lipschitz domains. Appendix B summarizes pertinent facts on sesquilinear forms and their associated linear operators. While we formulate and prove all results in this paper for self-adjoint generalized Robin Laplacians and Dirichlet Laplacians, we emphasize that all results in this paper immediately extend to closed Schrödinger operators HΘ,Ω = −ΔΘ,Ω +V , dom HΘ,Ω = dom − ΔΘ,Ω in L2 (Ω; dn x) for (not necessarily real-valued) potentials V satisfying V ∈ L∞ (Ω; dn x), by consistently replacing −Δ by −Δ + V , etc. More generally, all results extend directly to Kato–Rellich bounded potentials V relative to −ΔΘ,Ω with bound less than one. Next, we briefly list most of the notational conventions used throughout this paper. Let H be a separable complex Hilbert space, ( · , · )H the scalar product in H (linear in the second factor), and IH the identity operator in H. Next, let T be a linear operator mapping (a subspace of) a Banach space into another, with dom(T ) and ran(T ) denoting the domain and range of T . The spectrum (resp., essential spectrum) of a closed linear operator in H will be denoted by σ( · ) (resp., σess ( · )). The Banach spaces of bounded and compact linear operators in H are denoted by B(H) and B∞ (H), respectively. Similarly, B(H1 , H2 ) and B∞ (H1 , H2 ) will be used for bounded and compact operators between two Hilbert spaces H1 and H2 . Moreover, X1 → X2 denotes the continuous embedding of the Banach space X1 into the Banach space X2 . Throughout this manuscript, if X denotes a Banach space, X ∗ denotes the adjoint space of continuous conjugate linear functionals on X, that is, the conjugate dual space of X (rather than the usual dual space of continuous linear functionals on X). This avoids the well-known awkward

Robin-to-Robin Maps and Krein-Type Resolvent Formulas

83

distinction between adjoint operators in Banach and Hilbert spaces (cf., e.g., the pertinent discussion in [29, p. 3–4]). Finally, a notational comment: For obvious reasons in connection with quantum mechanical applications, we will, with a slight abuse of notation, dub −Δ (rather than Δ) as the “Laplacian” in this paper.

2. Laplace operators with generalized Robin boundary conditions We recall various properties of general Laplacians −ΔΘ,Ω in L2 (Ω; dn x) including Dirichlet, −ΔD,Ω , and Neumann, −ΔN,Ω , Laplacians, generalized Robin-type Laplacians, and Laplacians corresponding to classical Robin boundary conditions associated with bounded open Lipschitz domains. For details we refer to our recent paper [35]. We start with introducing our precise assumptions on the set Ω and the boundary operator Θ which subsequently will be employed in defining the boundary condition on ∂Ω: Hypothesis 2.1. Let n ∈ N, n ≥ 2, and assume that Ω ⊂ Rn is an open, bounded, nonempty Lipschitz domain. We refer to Appendix A for more details on Lipschitz domains. For simplicity of notation we will denote the identity operators in L2 (Ω; dn x) and L2 (∂Ω; dn−1 ω) by IΩ and I∂Ω , respectively. In addition, we refer to Appendix A for our notation in connection with Sobolev spaces. Hypothesis 2.2. Assume Hypothesis 2.1 and suppose that aΘ is a closed sesquilinear form in the Hilbert space L2 (∂Ω; dn−1 ω) with domain H 1/2 (∂Ω) × H 1/2 (∂Ω), bounded from below by cΘ ∈ R (hence, in particular, aΘ is symmetric). Denote by Θ cΘ I∂Ω the self-adjoint operator in L2 (∂Ω; dn−1 ω) uniquely associated with aΘ 1 ∈ B H 1/2 (∂Ω), H −1/2 (∂Ω) the extension of Θ as discussed (cf. (B.27)) and by Θ in (B.26) and (B.32). Thus one has 7 8 7 8 1g 1f f, Θ = g, Θ , f, g ∈ H 1/2 (∂Ω). 1/2 1/2 7 8 1f f, Θ ≥ cΘ f 2L2(∂Ω;dn−1 ω) , f ∈ H 1/2 (∂Ω). 1/2

(2.1) (2.2)

Here the sesquilinear form · , · s = H s (∂Ω) · , · H −s (∂Ω) : H s (∂Ω) × H −s (∂Ω) → C,

s ∈ [0, 1],

(2.3)

(antilinear in the first, linear in the second factor), denotes the duality pairing between H s (∂Ω) and ∗ H −s (∂Ω) = H s (∂Ω) , s ∈ [0, 1], (2.4)

84


such that

f, gs =

dn−1 ω(ξ) f (ξ)g(ξ),

(2.5)

∂Ω

f ∈ H s (∂Ω), g ∈ L2 (∂Ω; dn−1 ω) → H −s (∂Ω), s ∈ [0, 1], and dn−1 ω denotes the surface measure on ∂Ω. Hypothesis 2.1 on Ω is used throughout this paper. Similarly, Hypothesis 2.2 1 is involved. (Later in this section, is assumed whenever the boundary operator Θ and the next, we will occasionally strengthen our hypotheses.) 0 We introduce the boundary trace operator γD (the Dirichlet trace) by 0 γD : C(Ω) → C(∂Ω),

0 γD u = u|∂Ω .

(2.6)

Then there exists a bounded, linear operator γD (cf., e.g., [58, Theorem 3.38]), γD : H s (Ω) → H s−(1/2) (∂Ω) → L2 (∂Ω; dn−1 ω), γD : H

3/2

(Ω) → H

1−ε

2

n−1

(∂Ω) → L (∂Ω; d

ω),

1/2 < s < 3/2, ε ∈ (0, 1),

(2.7)

0 . That is, the two Dirichlet trace operwhose action is compatible with that of γD ators coincide on the intersection of their domains. Moreover, we recall that

γD : H s (Ω) → H s−(1/2) (∂Ω) is onto for 1/2 < s < 3/2.

(2.8)

While, in the class of bounded Lipschitz subdomains in Rn , the end-point s cases s = 1/2 and s = 3/2 of γD ∈ B H (Ω), H s−(1/2) (∂Ω) fail, we nonetheless have γD ∈ B H (3/2)+ε (Ω), H 1 (∂Ω) , ε > 0. (2.9) See Lemma A.2 for a proof. Below we augment this with the following result: Lemma 2.3. Assume Hypothesis 2.1. Then for each s > −3/2, the restriction to boundary operator (2.6) extends to a linear operator , γD : u ∈ H 1/2 (Ω) Δu ∈ H s (Ω) → L2 (∂Ω; dn−1 ω), (2.10) is compatible with (2.7), and is bounded when {u ∈ H 1/2 (Ω) | Δu ∈ H s (Ω) is equipped with the natural graph norm u → u H 1/2 (Ω) + Δu H s (Ω) . Furthermore, for each s > −3/2, the restriction to boundary operator (2.6) also extends to a linear operator , (2.11) γD : u ∈ H 3/2 (Ω) Δu ∈ H 1+s (Ω) → H 1 (∂Ω). 3/2 The latter - is compatible with (2.7), and is bounded when {u ∈ H (Ω) | Δu ∈ 1+s H (Ω) is equipped with the natural graph norm u → u H 3/2 (Ω) + Δu H 1+s (Ω) .

Next, we introduce the operator γN (the Neumann trace) by γN = ν · γD ∇ : H s+1 (Ω) → L2 (∂Ω; dn−1 ω),

1/2 < s < 3/2,

(2.12)

where ν denotes the outward pointing normal unit vector to ∂Ω. It follows from (2.7) that γN is also a bounded operator. We wish to further extend the action of


85

the Neumann trace operator (2.12) to other (related) settings. To set the stage, assume Hypothesis 2.1 and recall that the inclusion ∗ ι : H s (Ω) → H 1 (Ω) , s > −1/2, (2.13) is well defined and bounded. Then, we introduce the weak Neumann trace operator , γ 1N : u ∈ H 1 (Ω) Δu ∈ H s (Ω) → H −1/2 (∂Ω), s > −1/2, (2.14) as follows: Given u ∈ H 1 (Ω) with Δu ∈ H s (Ω) for some s > −1/2, we set (with ι as in (2.13)) φ, γ 1N u1/2 = dn x ∇Φ(x) · ∇u(x) + H 1 (Ω) Φ, ι(Δu)(H 1 (Ω))∗ , (2.15) Ω

for all φ ∈ H (∂Ω) and Φ ∈ H 1 (Ω) such that γD Φ = φ. We note that this definition is independent of the particular extension Φ of φ, and that γ 1N is a bounded extension of the Neumann trace operator γN defined in (2.12). As was the case of the Dirichlet trace, the (weak) Neumann trace operator (2.14), (2.15) is onto (cf. [35]). For additional details we refer to equations (A.8)–(A.10). Next, we wish to discuss the end-point case s = 1/2 of (2.12). 1/2

Lemma 2.4. Assume Hypothesis 2.1. Then the Neumann trace operator (2.12) also extends to , γN : u ∈ H 3/2 (Ω) Δu ∈ L2 (Ω; dn x) → L2 (∂Ω; dn−1 ω) 1 (2.16) 3/2 2 n in a bounded fashion when the space {u ∈ H (Ω) | Δu ∈ L (Ω; d x) is equipped with the natural graph norm u → u H 3/2 (Ω) + Δu L2(Ω;dn x) . This extension is compatible with (2.14). For future purposes, we shall need yet another extension of the concept of Neumann trace. This requires some preparations (throughout, Hypothesis 2.1 is enforced). First, we recall that, as is well known (see, e.g., [40]), one has the natural identification 1 ∗ , H (Ω) ≡ u ∈ H −1 (Rn ) supp (u) ⊆ Ω . (2.17) Note that the latter is a closed subspace of H −1 (Rn ). In particular, if RΩ u = u|Ω denotes the operator of restriction to Ω (considered in the sense of distributions), then ∗ RΩ : H 1 (Ω) → H −1 (Ω) (2.18) is well defined, linear and bounded. Furthermore, the composition of RΩ in (2.18) with ι in (2.13) is the natural inclusion of H s (Ω) into H −1 (Ω). Next, given z ∈ C, set , ∗ Wz (Ω) = (u, f ) ∈ H 1 (Ω) × H 1 (Ω) (−Δ − z)u = f |Ω in D (Ω) , (2.19) ∗ equipped with the norm inherited from H 1 (Ω) × H 1 (Ω) . We then denote by γ 1N : Wz (Ω) → H −1/2 (∂Ω)

(2.20)

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the ultra weak Neumann trace operator defined by φ, γ1N (u, f )1/2 = dn x ∇Φ(x) · ∇u(x) Ω −z dn x Φ(x)u(x) − H 1 (Ω) Φ, f (H 1 (Ω))∗ ,

(2.21) (u, f ) ∈ Wz (Ω),

Ω

for all φ ∈ H 1/2 (∂Ω) and Φ ∈ H 1 (Ω) such that γD Φ = φ. Once again, this definition is independent of the particular extension Φ of φ. Also, as was the case of the Dirichlet trace, the ultra weak Neumann trace operator (2.20), (2.21) is onto (this is a corollary of Theorem 4.4). For additional details we refer to equations (A.8)–(A.10). The relationship between the ultra weak Neumann trace operator (2.20), (2.21) and the weak Neumann trace operator (2.14), (2.15) can be described as follows. Given s > −1/2 and z ∈ C, denote by jz : {u ∈ H 1 (Ω) Δu ∈ H s (Ω) → Wz (Ω) (2.22) the injection jz (u) = (u, ι(−Δu − zu)),

u ∈ H 1 (Ω), Δu ∈ H s (Ω),

(2.23)

where ι is as in (2.13). Then γ 1N ◦ jz = 1 γN .

(2.24)

Thus, from this perspective, 1 γN can also be regarded as a bounded extension of the Neumann trace operator γN defined in (2.12). Moving on, we shall now describe a family of self-adjoint Laplace operators −ΔΘ,Ω in L2 (Ω; dn x) indexed by the boundary operator Θ. We will refer to −ΔΘ,Ω as the generalized Robin Laplacian. Theorem 2.5. Assume Hypothesis 2.2. The generalized Robin Laplacian, −ΔΘ,Ω , defined by − ΔΘ,Ω = −Δ,

, dom(−ΔΘ,Ω ) = u ∈ H 1 (Ω) Δu ∈ L2 (Ω; dn x); 1 D u = 0 in H −1/2 (∂Ω) , γN + Θγ 1 is self-adjoint and bounded from below in L2 (Ω; dn x). Moreover, dom | − ΔΘ,Ω |1/2 = H 1 (Ω).

(2.25)

(2.26)

In addition, −ΔΘ,Ω , has purely discrete spectrum bounded from below, in particular, σess (−ΔΘ,Ω ) = ∅. (2.27) The important special case where Θ corresponds to the operator of multiplication by a real-valued, essentially bounded function θ leads to Robin boundary conditions we discuss next:


87

Corollary 2.6. In addition to Hypothesis 2.1, assume that Θ is the operator of multiplication in L2 (∂Ω; dn−1 ω) by the real-valued function θ satisfying the condition θ ∈ L∞ (∂Ω; dn−1 ω). Then Θ satisfies the conditions in Hypothesis 2.2 resulting in the self-adjoint and bounded from below Laplacian −Δθ,Ω in L2 (Ω; dn x) with Robin boundary conditions on ∂Ω in (2.25) given by (1 γN + θγD )u = 0 in H −1/2 (∂Ω).

(2.28)

Remark 2.7. (i) In the case of a smooth boundary ∂Ω, the boundary conditions in (2.28) are also called “classical” boundary conditions (cf., e.g., [73]); in the more general case of bounded Lipschitz domains we also refer to [5] and [80, Ch. 4] in this context. Next, we point out that, in [51], the authors have dealt with the case of Laplace operators in bounded Lipschitz domains, equipped with local boundary conditions of Robin-type, with boundary data in Lp (∂Ω; dn−1 ω), and produced nontangential maximal function estimates. For the case p = 2, when our setting agrees with that of [51], some of our results in this section and the following are a refinement of those in [51]. Maximal Lp -regularity and analytic contraction semigroups of Dirichlet and Neumann Laplacians on bounded Lipschitz domains were studied in [83]. Holomorphic C0 -semigroups of the Laplacian with Robin boundary conditions on bounded Lipschitz domains have been discussed in [81]. Moreover, Robin boundary conditions for elliptic boundary value problems on arbitrary open domains were first studied by Maz’ya [56], [57, Sect. 4.11.6], and subsequently in [22] (see also [23] which treats the case of the Laplacian). In addition, Robin-type boundary conditions involving measures on the boundary for very general domains Ω were intensively discussed in terms of quadratic forms and capacity methods in the literature, and we refer, for instance, to [5], [6], [15], [80], and the references therein. 1 = 0), that is, in the case of the Neumann (ii) In the special case θ = 0 (resp., Θ Laplacian, we will also use the notation −ΔN,Ω = −Δ0,Ω .

(2.29)

The case of the Dirichlet Laplacian −ΔD,Ω associated with Ω formally corresponds to Θ = ∞ and so we recall its treatment in the next result. To state it, recall that, given a bounded Lipschitz domain Ω ⊂ Rn , H01 (Ω) = {u ∈ H 1 (Ω) | γD u = 0 on ∂Ω}.

(2.30)

Theorem 2.8. Assume Hypothesis 2.1. Then the Dirichlet Laplacian, −ΔD,Ω , defined by − ΔD,Ω = −Δ,

, dom(−ΔD,Ω ) = u ∈ H 1 (Ω) Δu ∈ L2 (Ω; dn x); γD u = 0 in H 1/2 (∂Ω) , = u ∈ H01 (Ω) Δu ∈ L2 (Ω; dn x) , (2.31) is self-adjoint and strictly positive in L2 (Ω; dn x). Moreover, dom (−ΔD,Ω )1/2 = H01 (Ω).

(2.32)

88


Since Ω is open and bounded, it is well known that −ΔD,Ω has purely discrete spectrum contained in (0, ∞), in particular, σess (−ΔD,Ω ) = ∅.

3. Generalized Robin and Dirichlet boundary value problems and Robin-to-Dirichlet and Dirichlet-to-Robin maps This section is devoted to generalized Robin and Dirichlet boundary value problems associated with the Helmholtz differential expression −Δ − z in connection with the open set Ω. In addition, we provide a detailed discussion of Robin-to(0) Dirichlet maps, MΘ,D,Ω , in L2 (∂Ω; dn−1 ω). Again, the material in this section is taken from [35, Sect. 3]. In this section we strengthen Hypothesis 2.2 by adding assumption (3.1) below: Hypothesis 3.1. In addition to Hypothesis 2.2 suppose that 1 ∈ B∞ H 1 (∂Ω), L2 (∂Ω; dn−1 ω) . Θ

(3.1)

We note that (3.1) is satisfied whenever there exists some ε > 0 such that 1 ∈ B H 1−ε (∂Ω), L2 (∂Ω; dn−1 ω) . Θ

(3.2)

We recall the definition of the weak Neumann trace operator γ 1N in (2.14), (2.15) and start with the Helmholtz Robin boundary value problems: Theorem 3.2. Assume Hypothesis 3.1 and suppose that z ∈ C\σ(−ΔΘ,Ω ). Then for every g ∈ L2 (∂Ω; dn−1 ω), the following generalized Robin boundary value problem, + (−Δ − z)u = 0 in Ω, u ∈ H 3/2 (Ω), (3.3) 1 D u = g on ∂Ω, γN + Θγ 1 has a unique solution u = uΘ . This solution uΘ satisfies γD uΘ ∈ H 1 (∂Ω),

γ 1N uΘ ∈ L2 (∂Ω; dn−1 ω),

γN uΘ L2 (∂Ω;dn−1 ω) , ≤ C g L2 (∂Ω;dn−1 ω) γD uΘ H 1 (∂Ω) + 1

(3.4)

and uΘ H 3/2 (Ω) ≤ C g L2 (∂Ω;dn−1 ω) ,

(3.5)

for some constant constant C = C(Θ, Ω, z) > 0. Finally, ∗ γD (−ΔΘ,Ω − zIΩ )−1 ∈ B L2 (∂Ω; dn−1 ω), H 3/2 (Ω) ,

(3.6)

and the solution uΘ is given by the formula ∗ uΘ = γD (−ΔΘ,Ω − zIΩ )−1 g.

(3.7)

Next, we turn to the Dirichlet case originally treated in [36, Theorem 3.1] but under stronger regularity conditions on Ω.


89

Theorem 3.3. Assume Hypothesis 2.1 and suppose that z ∈ C\σ(−ΔD,Ω ). Then for every f ∈ H 1 (∂Ω), the following Dirichlet boundary value problem, + (−Δ − z)u = 0 in Ω, u ∈ H 3/2 (Ω), (3.8) γD u = f on ∂Ω, has a unique solution u = uD . This solution uD satisfies γ 1N uD ∈ L2 (∂Ω; dn−1 ω)

and

1 γN uD L2 (∂Ω;dn−1 ω) ≤ CD f H 1 (∂Ω) ,

(3.9)

for some constant CD = CD (Ω, z) > 0. Moreover,

Finally,

uD H 3/2 (Ω) ≤ CD f H 1 (∂Ω) .

(3.10)

∗ γ 1N (−ΔD,Ω − zIΩ )−1 ∈ B H 1 (∂Ω), H 3/2 (Ω) ,

(3.11)

and the solution uD is given by the formula ∗ 1N (−ΔD,Ω − zIΩ )−1 f. uD = − γ

(3.12)

In addition to Theorem 3.3, we recall the following result. Lemma 3.4. Assume Hypothesis 2.1 and suppose that z ∈ C\σ(−ΔD,Ω ). Then γ 1N (−ΔD,Ω − zIΩ )−1 ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 ω) , (3.13) and

∗ γ 1N (−ΔD,Ω − zIΩ )−1 ∈ B L2 (∂Ω; dn−1 ω), L2 (Ω; dn x) .

(3.14)

Assuming Hypothesis 3.1, we next introduce the Dirichlet-to-Robin map (0) MD,Θ,Ω (z) associated with (−Δ − z) on Ω, as follows, + H 1 (∂Ω) → L2 (∂Ω; dn−1 ω), (0) z ∈ C\σ(−ΔD,Ω ), (3.15) MD,Θ,Ω (z) : 1 D uD , f → − γ 1N + Θγ where uD is the unique solution of (−Δ − z)u = 0 in Ω,

u ∈ H 3/2 (Ω),

γD u = f on ∂Ω.

(3.16)

Continuing to assume Hypothesis 3.1, we introduce the Robin-to-Dirichlet (0) map MΘ,D,Ω (z) associated with (−Δ − z) on Ω, as follows, + L2 (∂Ω; dn−1 ω) → H 1 (∂Ω), (0) z ∈ C\σ(−ΔΘ,Ω ), (3.17) MΘ,D,Ω (z) : g → γD uΘ , where uΘ is the unique solution of (−Δ − z)u = 0 in Ω,

u ∈ H 3/2 (Ω),

1 D u = g on ∂Ω. γN + Θγ 1

We note that Robin-to-Dirichlet maps have also been studied in [9].

(3.18)

90

F. Gesztesy and M. Mitrea Next we recall one of the main results in [35]:

Theorem 3.5. Assume Hypothesis 3.1. Then (0) MD,Θ,Ω (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 ω) ,

z ∈ C\σ(−ΔD,Ω ),

(3.19)

and

(0) 1 D γ 1 D (−ΔD,Ω − zIΩ )−1 ∗ , MD,Θ,Ω (z) = γ 1N + Θγ 1N + Θγ

z ∈ C\σ(−ΔD,Ω ). (3.20)

Moreover, (0) MΘ,D,Ω (z) ∈ B L2 (∂Ω; dn−1 ω), H 1 (∂Ω) ,

z ∈ C\σ(−ΔΘ,Ω ),

(3.21)

(0) MΘ,D,Ω (z) ∈ B∞ L2 (∂Ω; dn−1 ω) ,

z ∈ C\σ(−ΔΘ,Ω ).

(3.22)

∗ (0) MΘ,D,Ω (z) = γD γD (−ΔΘ,Ω − zIΩ )−1 ,

z ∈ C\σ(−ΔΘ,Ω ).

(3.23)

and, in fact,

In addition,

Finally, let z ∈ C\(σ(−ΔD,Ω ) ∪ σ(−ΔΘ,Ω )). Then MΘ,D,Ω (z) = −MD,Θ,Ω (z)−1 . (0)

(0)

(3.24)

Remark 3.6. In the above considerations, the special case Θ = 0 represents the fre(0) quently studied Neumann-to-Dirichlet and Dirichlet-to-Neumann maps MN,D,Ω (z) (0)

(0)

(0)

(0)

and MD,N,Ω (z), respectively. That is, MN,D,Ω (z) = M0,D,Ω (z) and MD,N,Ω (z) = (0) MD,0,Ω (z).

Thus, as a corollary of Theorem 3.5 we have MN,D,Ω (z) = −MD,N,Ω (z)−1 , (0)

(0)

(3.25)

whenever Hypothesis 2.1 holds and z ∈ C\(σ(−ΔD,Ω ) ∪ σ(−ΔN,Ω )). Remark 3.7. We emphasize again that all results in this section immediately = dom − = −Δ + V , dom H extend to Schr¨ o dinger operators H Θ,Ω Θ,Ω Θ,Ω ΔΘ,Ω in L2 (Ω; dn x) for (not necessarily real-valued) potentials V satisfying V ∈ L∞ (Ω; dn x), or more generally, for potentials V which are Kato–Rellich bounded with respect to −ΔΘ,Ω with bound less than one. Denoting the corresponding M operators by MD,N,Ω (z) and MΘ,D,Ω (z), respectively, we note, in particular, that (3.15)–(3.24) extend replacing −Δ by −Δ+ V and restricting z ∈ C appropriately. Our discussion of Weyl–Titchmarsh operators follows the earlier papers [32] and [36]. For related literature on Weyl–Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc.), we refer, for instance, to [2], [4], [11], [12], [16]–[20], [25]– [28], [31], [34], [37, Ch. 3], [39, Ch. 13], [54], [55], [59], [64], [65], [68]–[71], [79].


91

4. Some variants of Krein’s resolvent formula involving Robin-to-Dirichlet maps In this section we recall some of the principal new results in [35], viz., variants of Krein’s formula for the difference of resolvents of generalized Robin Laplacians and Dirichlet Laplacians on bounded Lipschitz domains. For details on the material in this section we refer to [35]. We start by weakening Hypothesis 3.1 by using assumption (4.1) below: Hypothesis 4.1. In addition to Hypothesis 2.2 suppose that 1 ∈ B∞ H 1/2 (∂Ω), H −1/2 (∂Ω) . Θ

(4.1)

We note that condition (4.1) is satisfied if there exists some ε > 0 such that 1 ∈ B H 1/2−ε (∂Ω), H −1/2 (∂Ω) . (4.2) Θ We wish to point out that Hypothesis 3.1 is indeed stronger than Hypothesis 4.1 since (3.1) implies, via duality and interpolation (cf. the discussion in [35]), that 1 ∈ B∞ H s (∂Ω), H s−1 (∂Ω) , 0 ≤ s ≤ 1. Θ (4.3) In our next two results below (Theorems 4.2–4.4) we discuss the solvability of the Dirichlet and Robin boundary value problems with solution in the energy space H 1 (Ω). Theorem 4.2. Assume Hypothesis 4.1 and suppose that z ∈ C\σ(−ΔΘ,Ω ). Then for every g ∈ H −1/2 (∂Ω), the following generalized Robin boundary value problem, + (−Δ − z)u = 0 in Ω, u ∈ H 1 (Ω), (4.4) 1 D u = g on ∂Ω, γ 1N + Θγ has a unique solution u = uΘ . Moreover, there exists a constant C = C(Θ, Ω, z) > 0 such that uΘ H 1 (Ω) ≤ C g H −1/2 (∂Ω) . (4.5) In particular, ∗ γD (−ΔΘ,Ω − zIΩ )−1 ∈ B H −1/2 (∂Ω), H 1 (Ω) , (4.6) and the solution uΘ of (4.4) is once again given by formula (3.7). Remark 4.3. As a byproduct of Theorem 4.2 (with Θ = 0) we obtain that the weak Neumann trace γ 1N in (2.14), (2.15) is onto. 1 In the following 1 ∗we denote by IΩ the continuous inclusion (embedding) map of H (Ω) into H (Ω) . By a slight abuse of notation, we also denote the continuous ∗ inclusion map of H01 (Ω) into H01 (Ω) by the same symbol I1Ω . We recall the ultra weak Neumann trace operator γ 1N from (2.20), (2.21). Finally, assuming Hypothesis 4.1, we denote by 1 Θ,Ω ∈ B H 1 (Ω), H 1 (Ω) ∗ −Δ (4.7) 1

92


the extension of −ΔΘ,Ω in accordance with (B.26). In particular, 7 8 1 1 Dv dn x ∇u(x) · ∇v(x) + γD u, Θγ , H 1 (Ω) u, −ΔΘ,Ω v(H 1 (Ω))∗ = 1/2 Ω

(4.8)

u, v ∈ H 1 (Ω), 1 Θ,Ω to L2 (Ω; dn x) (cf. (B.27)). and −ΔΘ,Ω is the restriction of −Δ Theorem 4.4. Assume Hypothesis 4.1 and let z ∈ C\σ(−ΔΘ,Ω ). Then for every w ∈ (H 1 (Ω))∗ , the following generalized inhomogeneous Robin problem, + (−Δ − z)u = w|Ω in D (Ω), u ∈ H 1 (Ω), (4.9) 1 D u = 0 on ∂Ω, γN (u, w) + Θγ 1 has a unique solution u = uΘ,w . Moreover, there is a constant C = C(Θ, Ω, z) > 0 such that uΘ,w H 1 (Ω) ≤ C w (H 1 (∂Ω))∗ . (4.10) In particular, the operator (−ΔΘ,Ω − zIΩ )−1 , z ∈ C\σ(−ΔΘ,Ω ), originally defined as a bounded operator on L2 (Ω; dn x), (4.11) (−ΔΘ,Ω − zIΩ )−1 ∈ B L2 (Ω; dn x) , ∗ 1 1 can be extended to a mapping in B H (Ω) , H (Ω) , which in fact coincides with 1 Θ,Ω − z I1Ω −1 ∈ B H 1 (Ω) ∗ , H 1 (Ω) . (4.12) −Δ Remark 4.5. In the context of Theorem 4.4, it is useful to observe that for any ∗ 1 Θ,Ω − z I1Ω −1 w ∈ H 1 (Ω) satisfies w ∈ H 1 (Ω) , the function u = − Δ (−Δ − z)u = w|Ω in D (Ω),

(4.13)

where the restriction of w to Ω is interpreted by regarding w as a distribution in H −1 (cf. associates ∗ (2.17)). Indeed, the identification (2.17) 1to na functional ∗ (Ω) 1 −1 n 2 = w ◦ RΩ ∈ H (R ) = H (R ) (which w ∈ H (Ω) the distribution w happens to be supported in Ω). Consequently, if for an arbitrary test function ϕ ∈ C0∞ (Ω) we denote by ϕ 1 ∈ C0∞ (Rn ) the extension of ϕ by zero outside Ω, we then have 2 Ω D (Ω) D(Ω) ϕ, w|

= D(Rn ) ϕ, 1 w 2 D (Rn ) = H 1 (Rn ) ϕ, 1 w 2 (H 1 (Rn ))∗ = H 1 (Ω) RΩ (ϕ), 1 w(H 1 (Ω))∗ 7 8 1 Θ,Ω − z I1Ω u 1 = H 1 (Ω) ϕ, w(H 1 (Ω))∗ = H 1 (Ω) ϕ, − Δ (H (Ω))∗ 7 8 1 Θ,Ω )u 1 = H 1 (Ω) ϕ, − Δ − z (ϕ, u)L2 (Ω;dn x) (H (Ω))∗ = (∇ϕ, ∇u)(L2 (Ω;dn x))n − z (ϕ, u)L2 (Ω;dn x) = ((−Δ − z)ϕ, u)L2 (Ω;dn x) ,

on account of (4.8). This justifies (4.13).

(4.14)


93

Remark 4.6. Similar (yet simpler) considerations also show that the operator (−ΔD,Ω − zIΩ )−1 , z ∈ C\σ(−ΔD,Ω ), originally defined as bounded operator on L2 (Ω; dn x), (4.15) (−ΔD,Ω − zIΩ )−1 ∈ B L2 (Ω; dn x) , extends to a mapping 1 D,Ω − z I1Ω −1 ∈ B H −1 (Ω); H 1 (Ω) . −Δ (4.16) 0 1 1 D,Ω ∈ B H (Ω), H −1 (Ω) is the extension of −ΔD,Ω in accordance with Here −Δ 0 (B.26). Indeed, the Lax–Milgram lemma applies and yields that 1 D,Ω − z I1Ω : H 1 (Ω) → H 1 (Ω) ∗ = H −1 (Ω) −Δ (4.17) 0 0 is, in fact, an isomorphism whenever z ∈ C\σ(−ΔD,Ω ). Corollary 4.7. Assume Hypothesis 4.1 and let z ∈ C\σ(−ΔΘ,Ω ). Then the oper (0) ator MΘ,D,Ω (z) ∈ B L2 (∂Ω; dn−1 ω) in (3.17), (3.18) extends (in a compatible manner ) to 9(0) (z) ∈ B H −1/2 (∂Ω), H 1/2 (∂Ω) , z ∈ C\σ(−ΔΘ,Ω ). (4.18) M Θ,D,Ω 9(0) (z) permits the representation In addition, M Θ,D,Ω (0) ∗ 1 1 −1 γD 9 , z ∈ C\σ(−ΔΘ,Ω ). (4.19) M Θ,D,Ω (z) = γD − ΔΘ,Ω − z IΩ (0) (0) The same applies to the adjoint MΘ,D,Ω (z)∗ ∈ B L2 (∂Ω; dn−1 ω) of MΘ,D,Ω (z), ∗ −1/2 (0) 9 ∈ B H (∂Ω), H 1/2 (∂Ω) , resulting in the bounded extension M Θ,D,Ω (z) z ∈ C\σ(−ΔΘ,Ω ). Lemma 4.8. Assume Hypothesis 4.1 and let z ∈ C\(σ(−ΔΘ,Ω ) ∪ σ(−ΔD,Ω )). Then ∗ the following resolvent relation holds on H 1 (Ω) , 1 Θ,Ω − z I1Ω −1 = − Δ 1 D,Ω − z I1Ω −1 ◦ RΩ −Δ (4.20) 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn . 1 Θ,Ω − z I1Ω −1 γ ∗ γ + −Δ 1N − Δ D

We also recall the following regularity result for the Robin resolvent. Lemma 4.9. Assume Hypothesis 3.1 and suppose that z ∈ C\σ(−ΔΘ,Ω ). Then , (4.21) (−ΔΘ,Ω − zIΩ )−1 : L2 (Ω; dn x) → u ∈ H 3/2 (Ω) Δu ∈ L2 (Ω; dn x) , is a well-defined bounded operator. Here we equip the space u ∈ H 3/2 (Ω) Δu ∈ 2 n L (Ω; d x) with the natural graph norm u → u H 3/2 (Ω) + Δu L2 (Ω;dn x) . Under Hypothesis 4.1, (4.12) and (2.7) yield 1 Θ,Ω − z I1Ω −1 ∈ B (H 1 (Ω))∗ , H 1/2 (∂Ω) . γD − Δ Hence, by duality, 1 Θ,Ω − z I1Ω −1 ∗ ∈ B H −1/2 (∂Ω), H 1 (Ω) . γD − Δ

(4.22)

(4.23)

94

F. Gesztesy and M. Mitrea Next we complement this with the following result.

Corollary 4.10. Assume Hypothesis 3.1 and suppose that z ∈ C\σ(−ΔΘ,Ω ). Then γD (−ΔΘ,Ω − zIΩ )−1 ∈ B L2 (Ω; dn x), H 1 (∂Ω) . (4.24) In particular, ∗ γD (−ΔΘ,Ω − zIΩ )−1 ∈ B H −1 (∂Ω), L2 (Ω; dn x) → B L2 (∂Ω; dn−1 Ω), L2 (Ω; dn x) . In addition, the operator (4.25) is compatible with (4.23) in the sense that ∗ 1 Θ,Ω − z I1Ω −1 ∗ f in L2 (Ω; dn x), γD (−ΔΘ,Ω − zIΩ )−1 f = γD − Δ f ∈ H −1/2 (∂Ω). As a consequence, ∗ 1 Θ,Ω − z I1Ω −1 ∗ f in L2 (Ω; dn x), γD (−ΔΘ,Ω − zIΩ )−1 f = γD − Δ f ∈ L2 (∂Ω; dn−1 ω).

(4.25)

(4.26)

(4.27)

We will need a similar compatibility result for the composition between the Neumann trace and resolvents of the Dirichlet Laplacian. To state it, we recall the restriction operator RΩ in (2.18). Also, we denote by IRn the identity operator (for spaces of functions defined in Rn ). Finally, we recall the space (2.19) and the ultra weak Neumann trace operator γ 1N in (2.20), (2.21). Lemma 4.11. Assume Hypothesis 2.1. Then 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn : (H 1 (Ω))∗ → Wz (Ω), −Δ

z ∈ C\σ(−ΔD,Ω ), (4.28)

is a well-defined, linear and bounded operator. Consequently, 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn ∈ B (H 1 (Ω))∗ , H −1/2 (∂Ω) , γ 1N − Δ z ∈ C\σ(−ΔD,Ω ), and, hence, 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn ∗ ∈ B H 1/2 (∂Ω), H 1 (Ω) , γN − Δ 1 z ∈ C\σ(−ΔD,Ω ).

(4.29)

(4.30)

Furthermore, the operators (4.29), (4.30) are compatible with (3.13) and (3.14), respectively, in the sense that for each z ∈ C\σ(−ΔD,Ω ), 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn f in H −1/2 (∂Ω), γ 1N (−ΔD,Ω − zIΩ )−1 f = 1 γN − Δ f ∈ L2 (Ω; dn x), and

∗ 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn ∗ f γN (−ΔD,Ω − zIΩ )−1 f = γ 1 1N − Δ in L2 (Ω; dn x), for every element f ∈ H 1/2 (∂Ω).

(4.31)

(4.32)


95

This yields the following L2 -version of Lemma 4.8. Lemma 4.12. Assume Hypothesis 3.1 and let z ∈ C\(σ(−ΔΘ,Ω ) ∪ σ(−ΔD,Ω )). Then the following resolvent relation holds on L2 (Ω; dn x), (−ΔΘ,Ω − zIΩ )−1

(4.33) ∗ = (−ΔD,Ω − zIΩ )−1 + γD (−ΔΘ,Ω − zIΩ )−1 γ 1N (−ΔD,Ω − zIΩ )−1 ∗ 1N (−ΔD,Ω − zIΩ )−1 γD (−ΔΘ,Ω − zIΩ )−1 . = (−ΔD,Ω − zIΩ )−1 + γ

We note that the special case Θ = 0 in Lemma 4.12 was discussed by Nakamura [61] (in connection with cubic boxes Ω) and subsequently in [32, Lemma A.3] (in the case of a Lipschitz domain with a compact boundary). We also recall the following useful result. Lemma 4.13. Assume Hypothesis 4.1 and suppose that z ∈ C\σ(−ΔΘ,Ω ). Then (0) ∗ 9 9(0) (z) M =M (4.34) Θ,D,Ω (z) Θ,D,Ω −1/2 (∂Ω); H 1/2 (∂Ω) . In particular, assuming Hypothesis 3.1, as operators in B H then (0) ∗ (0) MΘ,D,Ω (z) = MΘ,D,Ω (z). (4.35) Next we briefly recall the Herglotz property of the Robin-to-Dirichlet map. We recall that an operator-valued function M (z) ∈ B(H), z ∈ C+ (where C+ = {z ∈ C | Im(z) > 0), for some separable complex Hilbert space H, is called an operator-valued Herglotz function if M ( · ) is analytic on C+ and Im(M (z)) 0,

z ∈ C+ .

(4.36)

Here, as usual, Im(M ) = (M − M ∗ )/(2i). Lemma 4.14. Assume Hypothesis 4.1 and suppose that z ∈ C+ . Then for every g ∈ H −1/2 (∂Ω), g = 0, one has 8 1 7 9 9Θ,D (z)∗ g g, MΘ,D (z) − M = Im(z) uΘ 2L2 (Ω;dn x) > 0, (4.37) 1/2 2i where uΘ satisfies + (−Δ − z)u = 0 in Ω, u ∈ H 1 (Ω), (4.38) 1 D u = g on ∂Ω. γ 1N + Θγ In particular, assuming Hypothesis 3.1, then (0) Im MΘ,D,Ω (z) 0,

z ∈ C+ ,

(4.39)

(0)

and hence MΘ,D,Ω ( · ) is an operator-valued Herglotz function on L2 (∂Ω; dn−1 ω). The following result represents a first variant of Krein’s resolvent formula 1 Θ,Ω and Δ 1 D,Ω recently proved in [35]: relating Δ

96


Theorem 4.15. Assume Hypothesis 4.1 and suppose that z ∈ C\(σ(−ΔΘ,Ω ) ∪ ∗ σ(−ΔD,Ω )). Then the following Krein formula holds on H 1 (Ω) , 1 Θ,Ω − z I1Ω −1 = − Δ 1 D,Ω − z I1Ω −1 ◦ RΩ −Δ 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn ∗ M 9(0) (z) + γ 1N − Δ (4.40) Θ,D,Ω −1 1 D,Ω − z I1Ω × γ 1N − Δ ◦ RΩ , IRn . The following result details the L2 (Ω; dn x)-variant of Krein’s formula: Theorem 4.16. Assume Hypothesis 3.1 and suppose that z ∈ C\(σ(−ΔΘ,Ω ) ∪ σ(−ΔD,Ω )). Then the following Krein formula holds on L2 (Ω; dn x): (−ΔΘ,Ω − zIΩ )−1 = (−ΔD,Ω − zIΩ )−1 ∗ (0) + γ 1N (−ΔD,Ω − zIΩ )−1 MΘ,D,Ω (z) γ 1N (−ΔD,Ω − zIΩ )−1 .

(4.41)

It should be noted that, by Lemma 3.4, the composition of operators in the right-hand side of (4.41) acts in a well-defined manner on L2 (Ω; dn x). An attractive feature of the Krein-type formula (4.41) lies in the fact that (0) MΘ,D,Ω (z) encodes spectral information about ΔΘ,Ω . This will be pursued in future work. Assuming Hypothesis 2.1, the special case Θ = 0 then connects the Neumann and Dirichlet resolvents, 1 N,Ω − z I1Ω −1 = − Δ 1 D,Ω − z I1Ω −1 ◦ RΩ −Δ 1 D,Ω − z I1Ω −1 ◦ RΩ , IRn ∗ M 9(0) (z) + 1 γN − Δ (4.42) N,D,Ω −1 1 D,Ω − z I1Ω × 1 γN − Δ ◦ RΩ , IRn , z ∈ C\(σ(−ΔN,Ω ) ∪ σ(−ΔD,Ω )), 1 ∗ on H (Ω) , and similarly, (−ΔN,Ω − zIΩ )−1 = (−ΔD,Ω − zIΩ )−1 ∗ (0) + γ 1N (−ΔD,Ω − zIΩ )−1 MN,D,Ω (z) γ 1N (−ΔD,Ω − zIΩ )−1 ,

(4.43)

z ∈ C\(σ(−ΔN,Ω ) ∪ σ(−ΔD,Ω )), 9(0) (z) M N,D,Ω

(0)

and MN,D,Ω (z) denote the associated Neumannon L2 (Ω; dn x). Here to-Dirichlet operators. Due to the fundamental importance of Krein-type resolvent formulas (and more generally, Robin-to-Dirichlet maps) in connection with the spectral and inverse spectral theory of ordinary and partial differential operators, abstract versions, connected to boundary value spaces (boundary triples) and self-adjoint extensions of closed symmetric operators with equal (possibly infinite) deficiency spaces, have received enormous attention in the literature. In particular, we note that Robin-to-Dirichlet maps in the context of ordinary differential operators reduce to the celebrated (possibly, matrix-valued) Weyl–Titchmarsh function, the basic object of spectral analysis in this context. Since it is impossible to cover


97

the literature in this paper, we refer, for instance, to [1, Sect. 84], [3], [7], [8], [11], [13], [14], [18], [20], [21], [33], [34], [39, Ch. 13], [41], [43]–[50], [53], [54], [59], [62]–[69], [72], [75]–[77], and the references cited therein. We add, however, that the case of infinite deficiency indices in the context of partial differential operators (in our concrete case, related to the deficiency indices of the operator closure of −Δ C0∞ (Ω) in L2 (Ω; dn x)), is much less studied and the results obtained in this section, especially, under the assumption of Lipschitz (i.e., minimally smooth) domains, to the best of our knowledge, are new. Finally, we emphasize once more that Remark 3.7 also applies to the content of this section (assuming that V is real-valued in connection with Lemmas 4.13 and 4.14).

5. Some variants of Krein’s resolvent formula involving Robin-to-Robin maps In this section we present our principal results, variants of Krein’s formula for the difference of resolvents of generalized Robin Laplacians corresponding to two different Robin boundary conditions on bounded Lipschitz domains. To the best of our knowledge, the results in this section are new. Hypothesis 5.1. Assume that the conditions in Hypothesis 2.2 are satisfied by two sesquilinear forms aΘ1 , aΘ2 and, in addition, 1 2 ∈ B∞ H 1/2 (∂Ω), H −1/2 (∂Ω) . 1 1, Θ (5.1) Θ Lemma 5.2. Assume Hypothesis 5.1 and let z ∈ C\(σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω )). ∗ Then the following resolvent relation holds on H 1 (Ω) , 1 Θ2 ,Ω − z I1Ω −1 1 Θ1 ,Ω − z I1Ω −1 = − Δ −Δ (5.2) ∗ 1 1 Θ2 ,Ω − z I1Ω −1 . 1 Θ1 ,Ω − z I1Ω −1 γD 1 2 γD − Δ Θ1 − Θ + −Δ Proof. To set the stage, we recall (2.14) and (2.15). Together with (4.12) and (4.16), these ensure that the composition of operators appearing on the righthand side of (4.20) is well defined. Next, let φ1 , φ2 ∈ L2 (Ω; dn x) be arbitrary and define ψ1 = (−ΔΘ1 ,Ω − zIΩ )−1 φ1 ∈ dom(ΔΘ1 ,Ω ) ⊂ H 1 (Ω), (5.3) ψ2 = (−ΔΘ2 ,Ω − zIΩ )−1 φ2 ∈ dom(ΔΘ2 ,Ω ) ⊂ H 1 (Ω). As a consequence of our earlier results, both sides of (4.20) are bounded operators ∗ from (H 1 (Ω))∗ into H 1 (Ω). Since L2 (Ω; dn x) → H 1 (Ω) densely, it therefore suffices to show that the following identity holds: (φ1 , (−ΔΘ1 ,Ω − zIΩ )−1 φ2 )L2 (Ω;dn x) − (φ1 , (−ΔΘ2 ,Ω − zIΩ )−1 φ2 )L2 (Ω;dn x) (5.4) ∗ 1 1 2 γD (−ΔΘ2 ,Ω − zIΩ )−1 φ2 2 Θ1 − Θ . = φ1 , (−ΔΘ1 ,Ω − zIΩ )−1 γD L (Ω;dn x)

98


We note that according to (5.3) one has, (φ1 , (−ΔΘ1 ,Ω − zIΩ )−1 φ2 )L2 (Ω;dn x) = ((−ΔΘ1 ,Ω − zIΩ )ψ1 , ψ2 )L2 (Ω;dn x) , (5.5) ∗ (φ1 , (−ΔΘ2 ,Ω − zIΩ )−1 φ2 )L2 (Ω;dn x) = (−ΔΘ2 ,Ω − zIΩ )−1 φ1 , φ2 L2 (Ω;dn x) = ((−ΔΘ2 ,Ω − zIΩ )−1 φ1 , φ2 )L2 (Ω;dn x) = (ψ1 , (−ΔΘ2 ,Ω − zIΩ )ψ2 )L2 (Ω;dn x) ,

(5.6)

and, further, ∗ 1 1 2 γD (−ΔΘ2 ,Ω − zIΩ )−1 φ2 2 Θ1 − Θ φ1 , (−ΔΘ1 ,Ω − zIΩ )−1 γD L (Ω;dn x) 7 8 ∗ 1 1 2 γD (−ΔΘ2 ,Ω − zIΩ )−1 φ2 Θ1 − Θ = H 1 (Ω) (−ΔΘ1 ,Ω − zIΩ )−1 φ1 , γD (H 1 (Ω))∗ 8 7 11 − Θ 1 2 γD (−ΔΘ2 ,Ω − zIΩ )−1 φ2 = γD (−ΔΘ1 ,Ω − zIΩ )−1 φ1 , Θ 1/2 8 7 11 − Θ 1 2 γD ψ2 = γD ψ1 , Θ . (5.7) 1/2 Thus, matters have been reduced to proving that ((−ΔΘ1 ,Ω − zIΩ )ψ1 , ψ2 )L2 (Ω;dn x) − (ψ1 , (−ΔΘ2 ,Ω − zIΩ )ψ2 )L2 (Ω;dn x) 8 7 11 − Θ 1 2 γD ψ2 = γD ψ1 , Θ . 1/2

(5.8)

Using (A.11) for the left-hand side of (5.8) one obtains ((−ΔΘ1 ,Ω − zIΩ )ψ1 , ψ2 )L2 (Ω;dn x) − (ψ1 , (−ΔΘ2 ,Ω − zIΩ )ψ2 )L2 (Ω;dn x) = −(Δψ1 , ψ2 )L2 (Ω;dn x) + (ψ1 , Δψ2 )L2 (Ω;dn x) = (∇ψ1 , ∇ψ2 )L2 (Ω;dn x)n − 1 γN ψ1 , γD ψ2 1/2 − (∇ψ1 , ∇ψ2 )L2 (Ω;dn x)n + γD ψ1 , γ 1N ψ2 1/2 = −1 γN ψ1 , γD ψ2 1/2 + γD ψ1 , γ 1N ψ2 1/2 .

(5.9)

1 j γD ψj since ψj ∈ dom(ΔΘj ,Ω ), j = 1, 2, one concludes Observing that γ 1N ψj = −Θ (5.8). 9 Assuming Hypothesis 5.1 we introduce the Robin-to-Robin map M Θ1 ,Θ2 ,Ω (z) as follows, + H −1/2 (∂Ω) → H −1/2 (∂Ω), (0) 9 z ∈ C\σ(−ΔΘ1 ,Ω ), MΘ1 ,Θ2 ,Ω (z) : 1 2 γD uΘ1 , f → − γ 1N + Θ (5.10) (0)

where uΘ1 is the unique solution of (−Δ − z)u = 0 in Ω,

u ∈ H 1 (Ω),

1 1 γD u = f on ∂Ω. γ 1N + Θ

Theorem 5.3. Assume Hypothesis 5.1. Then −1/2 9(0) M (∂Ω) , Θ1 ,Θ2 ,Ω (z) ∈ B H

z ∈ C\σ(−ΔΘ1 ,Ω ),

(5.11)

(5.12)

Robin-to-Robin Maps and Krein-Type Resolvent Formulas and

9(0) 1 1 9(0) M Θ1 ,Θ2 ,Ω (z) = −I∂Ω + Θ1 − Θ2 MΘ1 ,D,Ω (z),

z ∈ C\σ(−ΔΘ1 ,Ω ).

99

(5.13)

In particular,

1 1 1 1 1 1 9(0) 1 1 9(0) M Θ1 ,Θ2 ,Ω (z) Θ1 − Θ2 = − Θ1 − Θ2 + Θ1 − Θ2 MΘ1 ,D,Ω (z) Θ1 − Θ2 , z ∈ C\σ(−ΔΘ1 ,Ω ),

and (0) 9 1 1 ∗=M 9(0) 1 1 M Θ1 ,Θ2 ,Ω (z) Θ1 − Θ2 Θ1 ,Θ2 ,Ω (z) Θ1 − Θ2 ,

(5.14)

z ∈ C\σ(−ΔΘ1 ,Ω ). (5.15)

Also, if z ∈ C\(σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω )), then −1 9(0) 9(0) . M Θ1 ,Θ2 ,Ω (z) = MΘ2 ,Θ1 ,Ω (z)

(5.16)

Proof. The membership in (5.12) is a consequence of (5.10) and Theorem 4.2. To see (5.13), assume that f ∈ H −1/2 (∂Ω) and denote by uΘ1 ∈ H 1 (Ω) the unique 1 1 γD )uΘ1 = f on ∂Ω. Then γN + Θ function satisfying (−Δ − z)uΘ1 = 0 in Ω and (1 9(0) 1 2 γD uΘ1 = − γ 1 1 γD uΘ1 + Θ 11 − Θ 1 2 γD uΘ1 M γN + Θ 1N + Θ Θ1 ,Θ2 ,Ω (z)f = − 1 (0) 9 11 − Θ 12 M = −f + Θ (5.17) Θ1 ,D,Ω (z)f, proving (5.13). Going further, (5.14) is a direct consequence of (5.13), and (5.15) is clear from (5.14) and Lemma 4.13. Finally, as far as (5.16) is concerned, if f ∈ H −1/2 (∂Ω) and uΘ2 ∈ H 1 (Ω) is the unique function satisfying (−Δ − z)uΘ2 = 0 in 1 2 γD uΘ2 = f on ∂Ω, then M 9(0) 1 1 γD uΘ2 . As a Ω and γ 1N + Θ 1N + Θ Θ2 ,Θ1 ,Ω (z)f = − γ consequence, if uΘ1 ∈ H 1 (Ω) is the unique function satisfying (−Δ − z)uΘ2 = 0 in 1 1 γD uΘ1 = − γ 1 1 γD uΘ2 on ∂Ω, it follows that uΘ2 = −uΘ1 1N + Θ Ω and γ 1N + Θ 9(0) 9(0) 1 2 γD uΘ1 = γ 1 2 γD uΘ2 = f . so that M 1N + Θ 1N + Θ Θ1 ,Θ2 ,Ω (z)MΘ2 ,Θ1 ,Ω (z)f = − γ 9(0) 9(0) (z)M (z)f = f , so (5.16) is proved. In a similar fashion, M Θ2 ,Θ1 ,Ω

Θ1 ,Θ2 ,Ω

Theorem 5.4. Assume Hypothesis 5.1 and suppose that z ∈ C\(σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω )). Then the following Krein formula holds: 1 Θ1 ,Ω − z I1Ω )−1 = (−Δ 1 Θ2 ,Ω − z I1Ω )−1 (−Δ (0) 9 1 1 Θ2 ,Ω − z I1Ω )−1 ∗ M 1 + γD (−Δ Θ1 ,Θ2 ,Ω (z) + I∂Ω Θ1 − Θ2 1 Θ2 ,Ω − z I1Ω −1 , × γD − Δ ∗ as operators on H 1 (Ω) .

(5.18)

Proof. We first claim that 11 − Θ 1 Θ1 ,Ω − z I1Ω −1 1 2 γD − Δ Θ (5.19) −1 ∗ −1 1 Θ1 ,Ω − z I1Ω 11 − Θ 1 Θ2 ,Ω − z I1Ω 1 2 γD − Δ 1 2 γD − Δ 11 − Θ γD Θ , = Θ

100


as operators in B (H 1 (Ω))∗ , H 1 (Ω) . To prove this, consider an arbitrary w ∈ 1 ∗ H (Ω) , then introduce ∗ 1 1 Θ2 ,Ω − z I1Ω −1 w ∈ H 1 (Ω) ∗ , 1 2 γD − Δ Θ1 − Θ (5.20) v = γD and observe that, under the identification (2.17), (A.13) yields supp (v) ⊆ ∂Ω.

(5.21)

As far as (5.19) is concerned, the goal is to show that 1 Θ1 ,Ω − z I1Ω −1 w = Θ 1 Θ1 ,Ω − z I1Ω −1 v. (5.22) 1 2 γD − Δ 11 − Θ 1 2 γD − Δ 11 − Θ Θ To this end, we observe from (5.2) that 1 Θ2 ,Ω − z I1Ω −1 w + − Δ 1 Θ1 ,Ω − z I1Ω −1 v. (5.23) 1 Θ1 ,Ω − z I1Ω −1 w = − Δ −Δ Hence, by linearity, 1 Θ1 ,Ω − z I1Ω −1 w, w = γ 1 Θ2 ,Ω − z I1Ω −1 w, w γ 1N − Δ 1N − Δ (5.24) 1 Θ1 ,Ω − z I1Ω −1 v, 0 . +γ 1N − Δ 1 Θj ,Ω − A word of explanation is in order here: First, by Remark 4.5, −Δ −1 w, w ∈ Wz (Ω) for j = 1, 2, so the terms in the first line of (5.24) are z I1Ω

well defined in H −1/2 (∂Ω) (cf. (2.20)). Second, thanks to (5.21), we have that 1 Θ1 ,Ω − z I1Ω −1 v, 0 ∈ Wz (Ω), so the last term in (5.24) is also well defined in −Δ 1 Θj ,Ω − z I1Ω −1 w, j = 1, 2, H −1/2 (∂Ω). Next, from the fact that the functions − Δ satisfy homogeneous Robin boundary conditions, one infers 1 Θj ,Ω − z I1Ω −1 w, w = −Θ 1 Θj ,Ω − z I1Ω −1 w, j = 1, 2. (5.25) 1 j γD − Δ γ 1N − Δ In a similar fashion, 1 Θ1 ,Ω − z I1Ω −1 v, 0 = γ 1 Θ1 ,Ω − z I1Ω −1 v, v − γ γ 1N − Δ 1N − Δ 1N 0, v (5.26) 1 Θ1 ,Ω − z I1Ω −1 v − γ 1 1 γD − Δ = −Θ 1N 0, v . To compute γ 1N 0, v , pick an arbitrary φ ∈ H 1/2 (∂Ω) and assume that Φ ∈ H 1 (Ω) is such that γD Φ = φ. Then, based on (2.21) and (5.20), one has φ, γ 1N 0, v 1/2 = −H 1 (Ω) Φ, v(H 1 (Ω))∗ 7 8 ∗ 1 1 Θ2 ,Ω − z I1Ω −1 w 1 2 γD − Δ Θ1 − Θ = −H 1 (Ω) Φ, γD (H 1 (Ω))∗ −1 8 7 1 Θ2 ,Ω − z I1Ω 11 − Θ 1 2 γD − Δ = − γD Φ, Θ w 1/2 −1 8 7 1 Θ2 ,Ω − z I1Ω 11 − Θ 1 2 γD − Δ = − φ, Θ w 1/2 . (5.27) This shows that 11 − Θ 1 Θ2 ,Ω − z I1Ω −1 w. 1 2 γD − Δ γ 1N 0, v = − Θ

(5.28)


101

By plugging (5.25), (5.26), and (5.28) back into (5.24), one then arrives at 1 1 γD − Δ 1 Θ1 ,Ω − z I1Ω −1 w = −Θ 1 Θ2 ,Ω − z I1Ω −1 w 1 2 γD − Δ −Θ (5.29) 1 Θ1 ,Ω − z I1Ω −1 v + Θ 1 Θ2 ,Ω − z I1Ω −1 w. 11 − Θ 1 2 γD − Δ 1 1 γD − Δ −Θ Upon recalling from (5.23) that 1 Θ2 ,Ω − z I1Ω −1 w = − Δ 1 Θ1 ,Ω − z I1Ω −1 w − − Δ 1 Θ1 ,Ω − z I1Ω −1 v, −Δ

(5.30)

now (5.22) readily follows from (5.29), (5.30) and some simple algebra. This finishes the proof of (5.30). Next, since (see (4.19)) −1 ∗ 9(0) M γD , Θ1 ,D,Ω (z) = γD (−ΔΘ1 ,Ω − zIΩ )

z ∈ C\σ(−ΔΘ1 ,Ω ),

(5.31)

we may then transform (5.19) into 11 − Θ 1 Θ1 ,Ω − zIΩ −1 1 2 γD − Δ Θ (0) −1 9 1 11 − Θ 12 M 1 1 = Θ Θ1 ,D,Ω (z) Θ1 − Θ2 γD − ΔΘ2 ,Ω − zIΩ −1 (0) 1 1 9 1 = M , Θ1 ,Θ2 ,Ω (z) + I∂Ω Θ1 − Θ2 γD − ΔΘ2 ,Ω − zIΩ

(5.32)

where the last line is based on (5.13). Taking adjoints in (5.32) (written with z in place of z) then leads to 1 Θ1 ,Ω − zIΩ −1 γ ∗ Θ 11 − Θ 12 −Δ D (0) 1 Θ2 ,Ω − zIΩ −1 ∗ M 9 1 1 ∗ = γD − Δ Θ1 ,Θ2 ,Ω (z) + I∂Ω Θ1 − Θ2 (0) 1 Θ2 ,Ω − zIΩ −1 ∗ M 9 1 1 (5.33) = γD − Δ Θ1 ,Θ2 ,Ω (z) + I∂Ω Θ1 − Θ2 , by (5.15). Replacing this back in (5.2) then readily yields (5.18).

We are interested in proving an L2 -version of Krein’s formula in Theorem 5.4. This requires the following strengthening of Hypothesis 5.1. Hypothesis 5.5. Assume that the conditions in Hypothesis 2.2 are satisfied by two sesquilinear forms aΘ1 , aΘ2 and suppose in addition that, 1 1, Θ 1 2 ∈ B∞ H 1 (∂Ω), L2 (∂Ω; dn−1 ω) . Θ (5.34) We recall (cf. (4.3)) that Hypothesis 5.5 is indeed stronger than Hypothesis 5.1. As a preliminary matter, we first discuss the L2 -version of Theorem 5.3. Theorem 5.6. Assume Hypothesis 5.5. Then the Robin-to-Robin map, originally −1/2 9(0) consider as an operator M (∂Ω) , z ∈ C\σ(−ΔΘ1 ,Ω ), extends Θ1 ,Θ2 ,Ω (z) ∈ B H (in a compatible fashion ) to an operator (0) MΘ1 ,Θ2 ,Ω (z) ∈ B L2 (∂Ω; dn−1 ω) , z ∈ C\σ(−ΔΘ1 ,Ω ), (5.35)

102


which, for every z ∈ C\σ(−ΔΘ1 ,Ω ), satisfies (0) 11 − Θ 1 2 M (0) MΘ1 ,Θ2 ,Ω (z) = −I∂Ω + Θ (5.36) Θ1 ,D,Ω (z), (0) (0) 11 − Θ 11 − Θ 11 − Θ 12 = − Θ 12 + Θ 12 M 1 1 MΘ1 ,Θ2 ,Ω (z) Θ Θ1 ,D,Ω (z) Θ1 − Θ2 , (5.37) ∗ (0) (0) 11 − Θ 12 11 − Θ 12 . (5.38) = MΘ1 ,Θ2 ,Ω (z) Θ MΘ1 ,Θ2 ,Ω (z) Θ Furthermore, if z ∈ C\(σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω )), then also MΘ1 ,Θ2 ,Ω (z) = MΘ2 ,Θ1 ,Ω (z)−1 . (0)

(0)

(5.39)

9(0) Proof. We note that for each z ∈ C\σ(−ΔΘ1 ,Ω ) the mapping M Θ1 ,Θ2 ,Ω (z) ∈ −1/2 (0) 2 (∂Ω) extends to an operator MΘ1 ,Θ2 ,Ω (z) ∈ B L (∂Ω; dn−1 ω) is a B H consequence of Theorem 3.2 and (5.34). This justifies the claim about (5.35). Properties (5.36)–(5.39) then follow from (5.35), Theorem 5.3, and a density argument. With these preparatory results in place we are ready to state and prove the following L2 -version of Krein’s formula. Theorem 5.7. Assume Hypothesis 5.5 and suppose that z ∈ C\(σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω )). Then the following Krein formula holds on L2 (Ω; dn x): (−ΔΘ1 ,Ω − zIΩ )−1 = (−ΔΘ2 ,Ω − zIΩ )−1 ∗ (0) 11 − Θ 12 MΘ1 ,Θ2 ,Ω (z) + I∂Ω Θ + γD (−ΔΘ2 ,Ω − zIΩ )−1 × γD (−ΔΘ2 ,Ω − zIΩ )−1 .

(5.40)

Proof. We start by observing that the following operators are well defined, linear and bounded: (−ΔΘj ,Ω − zIΩ )−1 ∈ B L2 (Ω; dn x) , j = 1, 2, (5.41) 2 −1 n 1 γD (−ΔΘ2 ,Ω − zIΩ ) ∈ B L (Ω; d x), H (∂Ω) , (5.42) 1 1 2 ∈ B H (∂Ω), L2 (∂Ω; dn−1 ω) , 11 − Θ (5.43) Θ 2 (0) n−1 ω) , (5.44) MΘ1 ,Θ2 ,Ω (z) + I∂Ω ∈ B L (∂Ω; d ∗ (5.45) γD (−ΔΘ2 ,Ω − zIΩ )−1 ∈ B L2 (∂Ω; dn−1 ω), L2 (Ω; dn x)) . Indeed, (5.41) follows from the fact that z ∈ C\ σ(−ΔΘ1 ,Ω ) ∪ σ(−ΔΘ2 ,Ω ) , (5.42) is covered by (4.24), (5.43) is taken care of by (5.34), (5.44) follows from (5.35), and (5.45) is a consequence of (4.25). Altogether, this shows that both sides of (5.40) are bounded operators on L2 (Ω; dn x). With this in hand, the desired conclusion follows from Theorem 5.4, (4.27) and the fact that the operators (4.11) and (4.12) are compatible.


103

We conclude by establishing the following Herglotz property for the Robin11 − Θ 1 2 . Specifically, we have the to-Robin map composed (to the right) by Θ following result: Theorem 5.8. Assume Hypothesis 5.1 and suppose that z ∈ C\σ(−ΔΘ1 ,Ω ). Then (0) 9(0) (z) Θ 9 11 − Θ 11 − Θ 1 2 , and hence M 1 2 , have the operator M + IΩ Θ Θ1 ,Θ2 Θ1 ,Θ2 (z) the Herglotz property when considered as operators in B H −1/2 (∂Ω) . Consequently, if Hypothesis 5.5 is assumed and z ∈ C\σ(−ΔΘ1 ,Ω ), then (0) 11 − Θ 11 −Θ 1 2 and M (0) (z) + IΩ Θ 1 2 also have the Herglotz propMΘ1 ,Θ2 (z) Θ Θ1 ,Θ2 erty when considered as operators in B L2 (∂Ω; dn−1 ω) . Proof. By Theorem 5.6 it suffices to prove only the first part in the statement. To this end, we recall (5.13) in Theorem 5.3. Composing the latter on the right by 11 − Θ 1 2 ) then yields (Θ (0) 11 − Θ 9 11 − Θ 1 2 = −(Θ 1 2) + Θ 11 − Θ 12 M 1 1 9(0) (z) Θ M Θ1 ,Θ2 Θ1 ,D (z) Θ1 − Θ2 , (5.46) z ∈ C\σ(−ΔΘ1 ,Ω ). Consequently, (0) (0) 11 − Θ 9 1 1 12 M 1 1 9 = Im Θ Im M Θ1 ,Θ2 (z) Θ1 − Θ2 Θ1 ,D (z) Θ1 − Θ2 (0) 11 − Θ 9 1 2 Im M 1 1 = Θ Θ1 ,D (z) Θ1 − Θ2 ,

(5.47)

z ∈ C\σ(−ΔΘ1 ,Ω ). Now one can use Lemma 4.14 in order to conclude that (0) 9 11 − Θ 1 2 ≥ 0, Im M (z) Θ Θ1 ,Θ2

(5.48)

as desired.

We note again that Remark 3.7 also applies to the content of this section (assuming that V is real-valued in connection with (5.38) and Theorem 5.8).

Appendix A. Properties of Sobolev spaces and boundary traces for Lipschitz domains The purpose of this appendix is to recall some basic facts in connection with Sobolev spaces corresponding to Lipschitz domains Ω ⊂ Rn , n ∈ N, n ≥ 2, and their boundaries. For more details we refer again to [35]. In this manuscript we use the following notation for the standard Sobolev Hilbert spaces (s ∈ R), : 2 2 (ξ)2 1 + |ξ|2s < ∞ , (A.1) H s (Rn ) = U ∈ S (Rn ) U H s (Rn ) = dn ξ U n R

H (Ω) = {u ∈ D (Ω) | u = U |Ω for some U ∈ H s (Rn )} , s

H0s (Ω)

= {u ∈ H (R ) | supp (u) ⊆ Ω}. s

n

(A.2) (A.3)

104


Here D (Ω) denotes the usual set of distributions on Ω ⊆ Rn , Ω open and nonempty (with D(Ω) standing for the space of test functions in Ω), S (Rn ) is the space of 2 denotes the Fourier transform of U ∈ S (Rn ). tempered distributions on Rn , and U It is then immediate that H s1 (Ω) → H s0 (Ω) for − ∞ < s0 ≤ s1 < +∞,

(A.4)

continuously and densely. Next, we recall the definition of a Lipschitz-domain Ω ⊆ Rn , Ω open and nonempty, for convenience of the reader: Let N be a space of real-valued functions in Rn−1 . One calls a bounded domain Ω ⊂ Rn of class N if there exists a finite open covering {Oj }1≤j≤N of the boundary ∂Ω of Ω with the property that, for every j ∈ {1, . . . , N }, Oj ∩ Ω coincides with the portion of Oj lying in the over-graph of a function ϕj ∈ N (considered in a new system of coordinates obtained from the original one via a rigid motion). If N is Lip (Rn−1 ), the space of real-valued functions satisfying a (global) Lipschitz condition in Rn−1 , is called a Lipschitz domain; cf. [74, p. 189], where such domains are called “minimally smooth”. The classical theorem of Rademacher of almost everywhere differentiability of Lipschitz functions ensures that, for any Lipschitz domain Ω, the surface measure dn−1 ω is well defined on ∂Ω and that there exists an outward pointing normal vector ν at almost every point of ∂Ω. For a Lipschitz domain Ω ⊂ Rn it is known that ∗ s (A.5) H (Ω) = H −s (Ω), −1/2 < s < 1/2. See [78] for this and other related properties. We also refer to our convention of using the adjoint (rather than the dual) space X ∗ of a Banach space X as described near the end of the introduction. Next, assume that Ω ⊂ Rn is the domain lying above the graph of a Lipschitz function ϕ : Rn−1 → R. Then for 0 ≤ s 1, the Sobolev space H s (∂Ω) consists of functions f ∈ L2 (∂Ω; dn−1 ω) such that f (x , ϕ(x )), as a function of x ∈ Rn−1 , belongs to H s (Rn−1 ). In this scenario we set ∗ (A.6) H s (∂Ω) = H −s (∂Ω) , −1 s 0. To define H s (∂Ω), 0 ≤ s 1, when Ω is a Lipschitz domain with compact boundary, we use a smooth partition of unity to reduce matters to the graph case. More precisely, if 0 ≤ s ≤ 1 then f ∈ H s (∂Ω) if and only if the assignment Rn−1 x → (ψf )(x , ϕ(x )) is in H s (Rn−1 ) whenever ψ ∈ C0∞ (Rn ) and ϕ : Rn−1 → R is a Lipschitz function with the property that if Σ is an appropriate rotation and translation of {(x , ϕ(x )) ∈ Rn | x ∈ Rn−1 }, then (supp (ψ)∩∂Ω) ⊂ Σ (this appears to be folklore, but a proof will appear in [60, Proposition 2.4]). Then Sobolev spaces with a negative amount of smoothness are defined as in (A.6) above. From the above characterization of H s (∂Ω) it follows that any property of Sobolev spaces (of order s ∈ [−1, 1]) defined in Euclidean domains, which are invariant under multiplication by smooth, compactly supported functions as well as composition by bi-Lipschitz diffeomorphisms, readily extends to the setting of


105

H s (∂Ω) (via localization and pull-back). As a concrete example, for each Lipschitz domain Ω with compact boundary, one has H s (∂Ω) → H s−ε (∂Ω) compactly if 0 < ε ≤ s ≤ 1.

(A.7)

For additional background information in this context we refer, for instance, to [9], [10], [29, Chs. V, VI], [38, Ch. 1], [58, Ch. 3], [82, Sect. I.4.2]. Moving on, we next consider the following bounded linear map ⎧, 1 ∗ 2 n n −1/2 ⎪ ⎨ (w, f ) ∈ L (Ω; d x) × H (Ω) div(w) = f |Ω → H (∂Ω) ∗ = H 1/2 (∂Ω) ⎪ ⎩ w → ν · (w, f ) (A.8) by setting dn x ∇Φ(x)·w(x)+ H 1 (Ω) Φ, f (H 1 (Ω))∗ (A.9) H 1/2 (∂Ω) φ, ν ·(w, f )(H 1/2 (∂Ω)∗ = Ω

whenever φ ∈ H (∂Ω) and Φ ∈ H 1 (Ω) is such that γD Φ = φ. Here the pairing ∗ 1 1 ∗ Φ, f in (A.9) is the natural pairing between functionals in H 1 (Ω) H (Ω) (H (Ω)) and elements in H 1 (Ω) (which, in turn, is compatible with the (bilinear) distributional pairing). It should be remarked that the above definition is independent of the particular extension Φ ∈ H 1 (Ω) of φ. Going further, one can introduce the ultra weak Neumann trace operator γ 1N as follows: +, ∗ (u, f ) ∈ H 1 (Ω) × H 1 (Ω) Δu = f |Ω → H −1/2 (∂Ω) γN : 1 u → γ 1N (u, f ) = ν · (∇u, f ), (A.10) with the dot product understood in the sense of (A.8). We emphasize that the ultra weak Neumann trace operator 1 γN in (A.10) is a re-normalization of the operator γN introduced in (2.12) relative to the extension of Δu ∈ H −1 (Ω) to an element ∗ 1 f of the space H (Ω) = {g ∈ H −1 (Rn ) | supp (g) ⊆ Ω}. For the relationship between the weak and ultra weak Neumann trace operators, see (2.22)–(2.24). In addition, one can show that the ultra weak Neumann trace operator (A.10) is onto (indeed, this is a corollary of Theorem 4.4). We note that (A.9) and (A.10) yield the following Green’s formula 1/2

γD Φ, 1 γN (u, f )1/2 = (∇Φ, ∇u)L2 (Ω;dn x)n + H 1 (Ω) Φ, f (H 1 (Ω))∗ , (A.11) ∗ valid for any u ∈ H 1 (Ω), f ∈ H 1 (Ω) with Δu = f |Ω , and any Φ ∈ H 1 (Ω). The ∗ pairing on the left-hand side of (A.11) is between functionals in H 1/2 (∂Ω) and elements in H 1/2 (∂Ω), whereas last pairing on the right-hand side in (A.11) is the ∗ between functionals in H 1 (Ω) and elements in H 1 (Ω). For further use, we also note that the adjoint of (2.7) maps boundedly as follows ∗ ∗ ∗ γD : H s−1/2 (∂Ω) → (H s (Ω) , 1/2 < s < 3/2. (A.12)

106


∗ Identifying H s (Ω) with H0−s (Ω) → H −s (Rn ) (cf. Proposition 2.9 in [40]), it follows that ∗ ran(γD ) ⊆ {u ∈ H −s (Rn ) | supp (u) ⊆ ∂Ω},

1/2 < s < 3/2.

(A.13)

Remark A.1. While it is tempting to view γD as an unbounded but densely defined operator on L2 (Ω; dn x) whose domain contains the space C0∞ (Ω), one should note ∗ that in this case its adjoint γD is not densely defined: Indeed (cf. [32, Remark A.4]), ∗ dom(γD ) = {0} and hence γD is not a closable linear operator in L2 (Ω; dn x). We conclude this appendix by recalling the following result from [36]. Lemma A.2 (cf. [36], Lemma A.6). Suppose Ω ⊂ Rn , n ≥ 2, is an open Lipschitz domain with a compact, nonempty boundary ∂Ω. Then the Dirichlet trace operator γD (originally considered as in (2.7)) satisfies (2.9).

Appendix B. Sesquilinear forms and associated operators In this appendix we describe a few basic facts on sesquilinear forms and linear operators associated with them. A slightly more expanded version of this material appeared in [35, Appendix B]. Let H be a complex separable Hilbert space with scalar product ( · , · )H (antilinear in the first and linear in the second argument), V a reflexive Banach space continuously and densely embedded into H. Then also H embeds continuously and densely into V ∗ . (B.1) V → H → V ∗ . ∗ Here the continuous embedding H → V is accomplished via the identification H u → ( · , u)H ∈ V ∗ ,

(B.2)

and we recall the convention in this manuscript (cf. the discussion at the end of the introduction) that if X denotes a Banach space, X ∗ denotes the adjoint space of continuous conjugate linear functionals on X, also known as the conjugate dual of X. In particular, if the sesquilinear form V · ,

· V ∗ : V × V ∗ → C

(B.3)

∗

denotes the duality pairing between V and V , then V u, vV ∗

= (u, v)H ,

u ∈ V, v ∈ H → V ∗ ,

(B.4)

∗

that is, the V, V pairing V · , · V ∗ is compatible with the scalar product ( · , · )H in H. Let T ∈ B(V, V ∗ ). Since V is reflexive, (V ∗ )∗ = V, one has T : V → V ∗,

T ∗ : V → V∗

(B.5)

and V u, T vV ∗

= V ∗ T ∗ u, v(V ∗ )∗ = V ∗ T ∗ u, vV = V v, T ∗ uV ∗ .

(B.6)


107

Self-adjointness of T is then defined by T = T ∗ , that is, V u, T vV ∗

= V ∗ T u, vV = V v, T uV ∗ ,

u, v ∈ V,

(B.7)

nonnegativity of T is defined by V u, T uV ∗

≥ 0,

u ∈ V,

(B.8)

and boundedness from below of T by cT ∈ R is defined by V u, T uV ∗

≥ cT u 2H,

u ∈ V.

(B.9)

(By (B.4), this is equivalent to V u, T uV ∗ ≥ cT V u, uV ∗ , u ∈ V.) Next, let the sesquilinear form a( · , · ) : V × V → C (antilinear in the first and linear in the second argument) be V-bounded, that is, there exists a ca > 0 such that |a(u, v)| ca u V v V , u, v ∈ V. (B.10) 1 Then A defined by + ∗ 1: V → V , (B.11) A 1 v → Av = a( · , v), satisfies

1 ∈ B(V, V ∗ ) and A

V

7 8 1 u, Av = a(u, v), V∗

u, v ∈ V.

(B.12)

Assuming further that a( · , · ) is symmetric, that is, a(u, v) = a(v, u),

u, v ∈ V,

(B.13)

and that a is V-coercive, that is, there exists a constant C0 > 0 such that a(u, u) ≥ C0 u 2V ,

u ∈ V,

(B.14)

respectively, then, 1 : V → V ∗ is bounded, self-adjoint, and boundedly invertible. A

(B.15)

1 in H defined by Moreover, denoting by A the part of A , 1 ∈ H ⊆ H, A = A 1 dom(A) = u ∈ V | Au : dom(A) → H, dom(A)

(B.16)

then A is a (possibly unbounded) self-adjoint operator in H satisfying A ≥ C0 IH , dom A1/2 = V.

(B.17) (B.18)

In particular,

A−1 ∈ B(H). (B.19) The facts (B.1)–(B.19) are a consequence of the Lax–Milgram theorem and the second representation theorem for symmetric sesquilinear forms. Details can be found, for instance, in [24, §VI.3, §VII.1], [29, Ch. IV], and [52]. Next, consider a symmetric form b( · , · ) : V × V → C and assume that b is bounded from below by cb ∈ R, that is, b(u, u) ≥ cb u 2H,

u ∈ V.

(B.20)

108


Introducing the scalar product ( · , · )V(b) : V × V → C (and norm · V(b) ) by (u, v)V(b) = b(u, v) + (1 − cb )(u, v)H ,

u, v ∈ V,

(B.21)

turns V into a pre-Hilbert space (V; ( · , · )V(b) ), which we denote by V(b). The form b is called closed if V(b) is actually complete, and hence a Hilbert space. The form b is called closable if it has a closed extension. If b is closed, then |b(u, v) + (1 − cb )(u, v)H | u V(b) v V(b) ,

u, v ∈ V,

(B.22)

and |b(u, u) + (1 − cb ) u 2H | = u 2V(b),

u ∈ V,

(B.23)

show that the form b( · , · ) + (1 − cb )( · , · )H is a symmetric, V-bounded, and Vcoercive sesquilinear form. Hence, by (B.11) and (B.12), there exists a linear map + ∗ 1c : V(b) → V(b) , (B.24) B b 1c v = b( · , v) + (1 − cb )( · , v)H , v → B b with 1c ∈ B(V(b), V(b)∗ ) and B b

V(b)

7 8 1c v u, B = b(u, v) + (1 − cb )(u, v)H , b V(b)∗

Introducing the linear map 1=B 1c + (cb − 1)I1: V(b) → V(b)∗ , B b

u, v ∈ V. (B.25) (B.26)

where I1: V(b) → V(b)∗ denotes the continuous inclusion (embedding) map of V(b) 1 to H, into V(b)∗ , one obtains a self-adjoint operator B in H by restricting B , 1 ∈ H ⊆ H, B = B 1 : dom(B) → H, (B.27) dom(B) = u ∈ V Bu dom(B) satisfying the following properties: B ≥ cb IH , dom |B|1/2 = dom (B − cb IH )1/2 = V, b(u, v) = |B|1/2 u, UB |B|1/2 v H = (B − cb IH )1/2 u, (B − cb IH )1/2 v H + cb (u, v)H 7 8 1 , u, v ∈ V, = V(b) u, Bv V(b)∗ b(u, v) = (u, Bv)H ,

u ∈ V, v ∈ dom(B),

(B.28) (B.29) (B.30) (B.31) (B.32) (B.33)

dom(B) = {v ∈ V | there exists an fv ∈ H such that b(w, v) = (w, fv )H for all w ∈ V}, Bu = fu ,

(B.34)

u ∈ dom(B),

dom(B) is dense in H and in V(b).

(B.35)

Properties (B.34) and (B.35) uniquely determine B. Here UB in (B.31) is the partial isometry in the polar decomposition of B, that is, B = UB |B|,

|B| = (B ∗ B)1/2 .

(B.36)


109

The operator B is called the operator associated with the form b. The facts in (B.20)–(B.35) comprise the second representation theorem of sesquilinear forms (cf. [29, Sect. IV.2], [30, Sects. 1.2–1.5], and [42, Sect. VI.2.6]). Acknowledgments. We wish to thank Gerd Grubb for questioning an inaccurate claim in an earlier version of the paper and Maxim Zinchenko for helpful discussions on this topic. Fritz Gesztesy would like to thank all organizers of the international conference on Modern Analysis and Applications (MAA 2007), and especially, Vadym Adamyan, for their kind invitation, the stimulating atmosphere during the meeting, and the hospitality extended to him during his stay in Odessa in April of 2007. He is also indebted to Vyacheslav Pivovarchik for numerous assistance before and during this conference.

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[71] V. Ryzhov, Weyl–Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control. Complex Anal. Operator Theory, to appear. [72] Sh.N. Saakjan, On the theory of the resolvents of a symmetric operator with infinite deficiency indices. Dokl. Akad. Nauk Arm. SSR 44 (1965), 193–198. (Russian) [73] B. Simon, Classical boundary conditions as a tool in quantum physics. Adv. Math. 30 (1978), 268–281. [74] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [75] A.V. Straus, Generalized resolvents of symmetric operators. Dokl. Akad. Nauk SSSR 71 (1950), 241–244. (Russian) [76] A.V. Straus, On the generalized resolvents of a symmetric operator. Izv. Akad. Nauk SSSR Ser. Math. 18 (1954), 51–86. (Russian) [77] A.V. Straus, Extensions and generalized resolvents of a non-densely defined symmetric operator. Math. USSR Izv. 4 (1970), 179–208. [78] H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Rev. Mat. Complut. 15 (2002), 475–524. [79] E.R. Tsekanovskii and Yu.L. Shmul’yan, The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Russ. Math. Surv. 32:5 (1977), 73–131. [80] M. Warma, The Laplacian with general Robin boundary conditions. Ph.D. Thesis, University of Ulm, 2002. [81] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains. Semigroup Forum 73 (2006), 10–30. [82] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge, 1987. [83] I. Wood, Maximal Lp -regularity for the Laplacian on Lipschitz domains. Math. Z. 255 (2007), 855–875. Fritz Gesztesy and Marius Mitrea Department of Mathematics University of Missouri Columbia, MO 65211, USA e-mail: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html e-mail: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/mitream.html



On Behavior of Weak Solutions of Operator Differential Equations on (0, ∞) M.L. Gorbachuk and V.I. Gorbachuk To the memory of M.G. Krein

Abstract. The aim of this work is to describe the weak solutions of a firstorder differential equation on the interval (0, ∞) in a Banach space and their behavior when approaching to the ends of this interval. Mathematics Subject Classification (2000). Primary 34G10; Secondary 47D06. Keywords. Closed operator, spectrum and resolvent, C0 -semigroup, differential equation in a Banach space, classical solution, weak solution, analytic vector, entire vector, asymptotical stability, exponential stability.

1. Introduction In this paper we present a survey of the results concerning the structure of weak solutions of a differential equation of the form y (t) = Ay(t),

t ∈ (0, ∞),

(1.1)

where A is the generating operator of a semigroup of linear operators in a complex Banach space B, and their behavior near zero and infinity. Previously the similar questions were considered for the solutions continuous at zero (see [1, 2]). We don’t impose any conditions on their behavior in a neighborhood of 0. This makes possible for elliptic and parabolic type partial differential equations to solve from the uniform (operator) point of view the problems of smoothness of their solutions inside a domain and existence of their boundary values in various function spaces. Note that there are a lot of works devoted to the consideration of such problems. In the particular case of harmonic (analytic) functions, the detailed investigations of such kind are contained, for instance, in the monograph [3] and the paper [4]. As is known, the problems of behavior at infinity of the solutions of equation (1.1) are in the reality those of the stability theory for this equation. In the case, This work was completed with the support of NASU Research Fund (Program 0107U002333).

116

M.L. Gorbachuk and V.I. Gorbachuk

when A is an arbitrary bounded operator in an infinite-dimensional Banach space, they were developed by M.G. Krein who devoted to this subject a number of works. The well-known monographs [5–7] are among them. And although he did not considered himself as “a genuine expert in the stability theory”, this field was always somewhat like hobby for him. These works and the personal contacts with M.G. Krein had a strong hold over our choice of the subject, mentioned above.

2. Subspaces of infinitely differentiable vectors for a closed operator Let A be a closed operator with dense domain in a Banach space B with norm · . We put C n (A) = D(An ). C n (A) is a Banach space with respect to the graphnorm of An . Denote by C ∞ (A) the space of infinitely differentiable vectors of the operator A, that is, ; < C ∞ (A) = C n (A) = proj lim C n (A), N0 = N {0}. n→∞

n∈N0

In general, C ∞ (A) = B. But if, for example, the resolvent set ρ(A) of the operator A is not empty, then C ∞ (A) = B. In what follows we assume C ∞ (A) to be dense in B. Let {mn }n∈N0 be a nondecreasing sequence of positive numbers. We put < α α C{mn } (A) = ind lim Cm (A) = Cm (A), n n α→∞

α>0

α (A) = C(mn ) (A) = proj lim Cm n α→0

where

;

α Cm (A), n

α>0

α (A) = {x ∈ C ∞ (A)∃c = c(x) > 0, ∀n ∈ N0 : An x ≤ cαn mn } Cm n

is a Banach space with the norm α (A) = sup x Cm n

k∈N0

Ak x . αk mk

Recall that the convergence in C{mn } (A) (C(mn ) (A)) is that in some (any) space α Cm (A). n In particular, if mn = n!, we obtain the well-known spaces A(A) = C{n!} (A) and Ac (A) = C(n!) (A) of analytic and entire vectors of the operator A, respectively (see [8,9]). The spaces G{β} (A) = C{nnβ } (A) and G(β) (A) = C(nnβ ) (A) are known as the Gevrey classes for the operator A (see, e.g., [10]).

On Weak Solutions to Operator Differential Equations on (0, ∞)

117

The space ExpA = G{0} (A) is called [11] the space of vectors of exponential type for the operator A. In the more specific situation when B = C([a, b]) (a < b ∈ R1 ), Au =

du , D(A) = C 1 ([a, b]), dx

the spaces C ∞ (A), A(A), Ac (A), ExpA coincide with the usual spaces of infinitely differentiable on [a, b], analytic on [a, b], entire, entire of exponential type functions, respectively; G{β} (A) and G(β) (A) (β > 1) are the Roumieu and Beurling type Gevrey classes. If B = L2 (R1 ), A = A0 , A0 u = −

d2 u + x2 u, dx2

D(A0 ) = C0∞ (R1 ),

then [12] C ∞ (A) = S, G{β} (A) = Sβ/2 (β > 1), β/2

where Sαβ

= f ∃h > 0, ∃c > 0 :

and

sup x∈R1 ; m,n∈N0

: |xm f (n) (x)| 1 − 2θ π . It follows from here that if θ = π2 , then G(β) (A) = B for any β > 0. As for the space ExpA = G{0} (A), there exist semigroups for which G{0} (A) = {0}.But under the additional condition on the resolvent RA (z) of the operator A 1 ln ln M (s) ds < ∞, 0

we have G{0} (A) = B (see [17]).

M (s) =

sup RA (z) , z:| z|≥s

118


If A is the generating operator of an one-parameter C0 -group {U (t)}t∈R1 in B, then, as was proved in [17], G(β) (A) = B for any β > 0. Moreover, in order that G{0} (A) = B, it is sufficient that ∞ −∞

ln U (t) dt < ∞. 1 + t2

3. The description of weak solutions Now consider equation (1.1), where A is a closed linear operator in B, D(A) = B, D(A∗ ) = B∗ . A vector-valued function y(t) : [0, ∞) → D(A) is called a strong (classical) solution of equation (1.1) on [0, ∞) if it is continuously differentiable on [0, ∞) and satisfies this equation there. By a weak solution of equation (1.1) on (0, ∞) we mean a a continuous vector-valued function y(t) : (0, ∞) → B such that for an arbitrary g ∈ D(A∗ ), the scalar function y(t), g is differentiable on (0, ∞) and satisfies there the equation d y(t), g = y(t), A∗ g, (3.1) dt which is equivalent to the following: y ∈ C((0, ∞), B), and for any t, t0 > 0 t

t y(s) ds ∈ D(A) and A

t0

y(s) ds = y(t) − y(t0 ). t0

Here ·, · denotes the pairing between the space B and its dual B∗, and C((0, ∞),B) is the space of strongly continuous B-valued functions on (0, ∞). If A is the generating operator of a C0 -semigroup of bounded linear operators in B, then all strong solutions of equation (1.1) are described [1] by the formula x ∈ D(A), (3.2) , tA denotes here and everywhere further the semigroup with generawhere e t≥0 tor A. It is not hard to see that a vector function of the form y(t) = etA x,

y(t) = etA x,

x ∈ B,

(3.3)

is a weak solution of equation (1.1), continuous at the point 0. As was shown in [18], the formula (3.3) gives all such solutions when x runs over the whole B. Our purpose is to describe all weak solutions of the equation (1.1) on (0, ∞) and investigate their behavior near 0 and ∞. Note once more, that no condition on a weak solution at 0 is preassigned. In what follows, we suppose the operator A to be the generator of a C0 semigroup of contractions, and ker etA = {0} as t > 0.


119

Let B−t be the completion of B in the norm x −t = etA x . 1 (t) from B onto As is easily seen, the operator etA admits a continuous extension U B−t . Taking into account that B−t ⊆ B−t when t < t , we introduce the space ; B−t , B− = proj lim B−t = t→0

t>0

and define in B− the operator family {U (t)}t≥0 as 1 (t)x if t > 0, U (0)x = x. U (t)x = U 1 (t ) to B−t coincides with U 1 (t) when t > t, the above Since the restriction of U definition is correct. The next assertion [19] characterizes the properties of U (t). Theorem 3.1. The family {U (t)}t≥0 forms an equicontinuous C0 -semigroup in the space B− such that: 1. ∀t > 0 : U (t)B− ⊆ B. 2. ∀x ∈ B : U (t)x = etA x. 3. ∀x ∈ B− , ∀t, s > 0 : U (t + s)x = etA U (s)x = esA U (t)x. 1 In 1 be the generator of the semigroup {U (t)}t≥0 . Obviously, A ⊆ A. Let A tA general, it is possible that B− = B and so U (t) = e (for example, when A is the generator a C0 -group). But if the semigroup {etA }t≥0 is differentiable, then 1 is continuous in B− , A 1 = A in B− , and the semigroup B− ⊃ B, the operator A {U (t)}t≥0 is differentiable on [0, ∞). The constructions of the space B− and the semigroup {U (t)}t≥0 enable us to describe all weak solutions of equation (1.1) on (0, ∞) (see [19]). Theorem 3.2. Every weak solution y(t) of equation (1.1) on (0, ∞) has a boundary value y0 at zero in the space B− (y(t) → y0 in the B− -topology), and y(t) = U (t)y0 .

(3.4)

Conversely, for any element y0 ∈ B− , the vector-valued function (3.4) is a weak solution of equation (1.1) on (0, ∞). The next assertion follows immediately from Theorems 3.1 and 3.2. Corollary 3.3. Let A be the generator of a differentiable (analytic) C0 -semigroup in B. Then every weak solution of equation (1.1) on (0, ∞) is an infinitely differentiable (analytic) C ∞ (A)-valued (A(A)-valued) function. Corollary 3.3 implies a number of classical theorems on smoothness inside a domain of weak solutions of elliptic or parabolic partial differential equations. By the weak Cauchy problem for equation (1.1) we mean the problem of finding a weak solution of equation (1.1) on (0, ∞), which satisfies the condition lim y(t) = y0 ∈ B− ,

t→0

where the limit is taken in the B− -topology.

(3.5)

120

M.L. Gorbachuk and V.I. Gorbachuk Theorems 3.1 and 3.2 imply

Corollary 3.4. Whatever vector y0 ∈ B− , the weak Cauchy problem (1.1), (3.5) is uniquely solvable. The solution has the form (3.4).

4. Boundary values of weak solutions As it follows from Theorem 3.2, each weak solution y(t) of equation (1) on (0, ∞) has a boundary value at 0 in the space B− , that is, lim y(t) exists in the B− t→0

topology. The question arises of finding the weak solutions whose boundary values exist in the input space B. Theorem 4.1. Suppose the operator A to be the generator of a C0 -semigroup of contractions in a reflexive Banach space B and ker etA = {0} as t > 0. Let y(t) be a weak solution of equation (1.1) on (0, ∞). Then y(t) → y0 in B as t → 0 ⇐⇒ y(t) ≤ c < ∞. Thus, the boundedness of a weak solution y(t) of equation (1.1) in a neighborhood of zero in the norm of B is equivalent to the continuity of y(t) at 0 in B (analog of Fatou’s and Riesz’s theorems (see, e.g., [3]) for harmonic functions in a disk or a half-plane). Note that the reflexivity of B plays here the essential role. There are examples of nonreflexive B for which Theorem 4.1 is not correct. For instance, the L1 -boundedness of a harmonic in a disk or a half-plane function on concentric circles or lines parallel to the real axis, respectively, does not yet imply the existence of the L1 -limit of this function when approaching the boundary of a domain (see [20]). Note also that Theorem 3.1 contains a number of well-known results from the theory of boundary values of solutions of partial differential equations in various classical function spaces (see the survey [21]). Let A be a self-adjoint operator in a Hilbert space H and EΔ be its spectral measure. Then ExpA = {f = EΔ h, ∀h ∈ H, Δ is an arbitrary compact set in R1}. If {mn }n∈N0 is such that mn ∀α > 0 : lim n = ∞, n→∞ α then the following dense and continuous embeddings hold: ExpA ⊆ C(mn ) (A) ⊆ C{mn } (A) ⊆ C ∞ (A) ⊆ C n (A) ⊆ H. Denote by C −n (A), C −∞ (A), C(m (A), C{m (A) the duals of C n (A), C ∞ (A), n) n} C(mn ) (A), C{mn } (A), respectively. Then the embeddings H ⊆ C −n (A) ⊆ C −∞ (A) ⊆ C{m (A) ⊆ C(m (A) ⊆ C{1} (A) n} n)

are dense and continuous, too. As was shown in [22], the following assertion takes place.


121

Theorem 4.2. Let B = H be a Hilbert space, and A be a nonpositive self-adjoint operator in H. Then

B− = A (A) := C{n!} (A). Under the conditions of Theorem 4.2 on the operator A, Theorems 3.2 and 4.2 imply the existence of a weak solution y(t) of equation (1.1) on (0, ∞) such that lim y(t) = ∞. t→0

The question arises: is it possible to characterize the growth of y(t) when approaching zero in terms of the singularity order of the vector y0 appearing in the representation (3.4)? The answer is given by the next two theorems. Theorem 4.3. Let A be a nonpositive self-adjoint operator in H, and y(t) be a weak solution of equation (1.1) on (0, ∞). Then y(0) = y0 ∈ C

−n

1 (A) ⇐⇒

t2n−1 y(t) 2 dt < ∞ 0

and y(0) = y0 ∈ C −∞ (A) ⇐⇒ ∃α > 0, ∃c > 0 : y(t) ≤ ct−α ,

t ∈ (0, 1].

In the first case y(t) → y0 (t → 0) in the space C −n (A) and in the second one in C −∞ (A)-topology. Let γ(t) be a continuous function on [0, b], b < ∞, γ(t) > 0 for t > 0, and ∃c0 > 0, ∃β0 < 1 : tγ(t) > c0 γ(β0 t). We put

⎛ G(λ) = ⎝

b

⎞−1/2 γ(t)e−λt dt⎠

.

0

Theorem 4.4. Let A be a nonpositive self-adjoint operator in H. For a weak solution y(t) of equation (1.1) on (0, ∞), the following equivalences hold: y0 ∈ C{m (A) n} y0 ∈ C(m (A) n)

⇐⇒ ⇐⇒

∀α > 0, ∃c = c(α) > 0 : y(t) ≤ cγ −1 (α−1 t); ∃α > 0, ∃c > 0 : y(t) ≤ cγ −1 (α−1 t).

Here λn . λ>0 G(λ)

mn = sup

In the first case y(t) → y0 (t → 0) in the space C{m (A) and in the second one in n} the C(m (A)-topology. n)

122

M.L. Gorbachuk and V.I. Gorbachuk If γ(t) = exp(−t−q ) (q > 0),

then mn = nnβ , β =

q+1 , q

and we arrive at the following assertion. Corollary 4.5. Let A be a nonpositive self-adjoint operator in H. If y(t) is a weak solution of equation (1.1) on (0, ∞), then y0 ∈ G{β} (A) y0 ∈ G(β) (A)

⇐⇒ ∀α > 0, ∃c = c(α) > 0 : y(t) ≤ c exp(αt−q ); ⇐⇒ ∃α > 0, ∃c > 0 : y(t) ≤ c exp(αt−q ).

Here y(t) → y0 (t → 0) in the G{β} (A)- and G(β) (A)-topology, respectively. Theorems 4.3 and 4.4 and Corollary 4.5 enable us to consider from the uniform point of view a lot of known results concerning boundary values of solutions of partial differential equations in various classes of distributions.

5. Behavior of weak solutions at infinity Now we shall discuss the behavior at ∞ of weak solutions of equation (1.1) on (0, ∞) in the case, when A is the generator of a C0 -semigroup of contractions in B. Recall that equation (1.1) is called exponentially stable if there exists a number ω > 0 such that for any weak solution y(t) of this equation on (0, ∞), lim eωt y(t) = 0.

(5.1)

t→∞

The equation (1.1) is called asymptotically stable if lim y(t) = 0.

(5.2)

t→∞

If dim B < ∞, then both the concepts coincide. This is, generally, not true if dim B = ∞. The next criterion for the exponential and asymptotical stability of equation (1.1) is valid. Proposition 5.1. If A is the generating operator of a bounded analytic C0 -semigroup, then the exponential stability of equation (1.1) ⇐⇒ the asymptotical stability of equation (1.1) ⇐⇒

0 ∈ ρ(A); 0 ∈ σc (A).

where σc (A) is the continuous spectrum of the operator A. In the case where the operator A is bounded, M.G. Krein has established the following theorem (see [23]).


123

Theorem 5.2. Assume the operator A to be bounded. The equation (1.1) is exponentially stable if and only if there exists a number p ≥ 1 such that ∞ y(t) p dt < ∞ (5.3) 0

for any weak solution of this equation on (0, ∞). It should be noted that in this case all the weak solutions admit the extensions to exponential type entire B-valued functions. We propose a generalization of M.G. Krein’s Theorem 5.2. As follows from Theorem 3.2, in the case of unbounded A generating a C0 semigroup {etA }t≥0 in B, it is possible for some kind of weak solutions of equation (1.1) not to be continuous. They can have arbitrary degree singularities at the point 0. Therefore the integrals (5.3) with such solutions can be diverged. But it turns out to be that in order to investigate the stability of equation (1.1), it is sufficient to confine ourselves to y ∈ C([0, ∞), B), that is, the solutions y(t) which admit a representation of the form y(t) = etA y0 ,

y0 ∈ B.

We obtain from here the next assertion. Theorem 5.3. The equation (1.1) is asymptotically stable if and only if ∀y0 ∈ B :

lim etA y0 = 0.

t→∞

(5.4)

In order equation (1.1) be exponentially stable, it is necessary and sufficient that ∀y0 ∈ B :

lim eωt etA y0 = 0.

t→∞

(5.5)

In the case when the semigroup {etA }t≥0 is differentiable (analytic), it is sufficient for equality (5.4) (equality (5.5)) to be fulfilled only for y0 ∈ C ∞ (A) (y0 ∈ A(A)). Using this theorem, we obtain the next statement. Theorem 5.4. Let A be the generator of a C0 -semigroup in B. The equation (1.1) is exponentially stable if and only if for every x ∈ B there exists a number p(x) > 0 such that ∞ etA x p(x) dt < ∞. (5.6) 0

If the semigroup {etA }t≥0 if differentiable (analytic), then it is sufficient that the relation (5.6) be valid only for x ∈ C ∞ (A) (x ∈ G{1} (A)). This theorem generalizes the well-known Pazy theorem as well (see [24]), where it is required for p(x) to be independent of x : p(x) ≡ p ≥ 1. If equation (1.1) is asymptotically (not exponentially) stable, then the degree of convergence to 0 at infinity of a weak solution can be arbitrary. Therefore it is

124


reasonable to ask: is it possible in this case to characterize the degree of this convergence by means of the “smoothness” property for the initial data of solutions? We solve this problem when the operator A generates a bounded analytic C0 semigroup. As Proposition 5.1 shows, in this case 0 ∈ σc (A) and (see [25, 26]) the operator A−1 is also the generating operator of a bounded analytic C0 -semigroup of the same angle. Theorem 5.5. Let A be the generator of a bounded analytic semigroup in B and 0 ∈ σc (A). Then 1 1 y(0) ∈ D(A−n ) =⇒ y(t) = o n and y(t) = O n+ε =⇒ y(0) ∈ D(A−n ). t t Moreover, ∀n ∈ N : lim (tn y(t) ) = 0 ⇐⇒ y(0) ∈ C ∞ (A−1 ). t→∞

Theorem 5.5 is contained in [27]. For the characterization of solutions whose decrease at infinity is not power, we use the class (R) which was studied in detail by M.G. Krein [28], that is, the class of functions ϕ(t), t ∈ [0, ∞), of the form ∞ t ϕ(t) = dσ(s), t+s 0

such that

∞

dσ(s) < ∞, 1+s

lim

t→∞

ϕ(t) = 0, t

lim

t→∞

ϕ(t) = ∞. ln t

0

The function sin πα t = π

∞

α

t α−1 s ds, 0 < α < 1, t+s

0

belongs to (R). As was proved in [27], the following theorem holds. Theorem 5.6. Let A be the generating operator of a bounded analytic semigroup in B, 0 ∈ σc (A), and ϕ ∈ (R). If y(t) is a weak solution of equation (1.1), then ∃α > 0, ∃c > 0 : y(t) ≤ ce−αϕ(t) ∀α > 0, ∃c = c(α) > 0 : y(t) ≤ ce−αϕ(t) where mn =

⇐⇒ y(0) ∈ C{mn } (A−1 ), ⇐⇒ y(0) ∈ C(mn ) (A−1 ),

1 max tn e−ϕ(t) . n! t

In particular (ϕ(t) = tβ , 0 < β < 1), we have ∃α > 0, ∃c > 0 : y(t) ≤ ce−αt β ∀α > 0, ∃c = c(α) > 0 : y(t) ≤ ce−αt β

⇐⇒ y(0) ∈ C{mn } (A−1 ), ⇐⇒ y(0) ∈ C(mn ) (A−1 )

On Weak Solutions to Operator Differential Equations on (0, ∞) with mn = nn(β

−1

−1)

125

.

If β = 1/2, then we obtain the description of all analytic and entire vectors of the operator A−1 in terms of behavior of the corresponding weak solutions of (1.1). For β = 1, we have ∃α > 0, ∃c > 0 : y(t) ≤ ce−αt ⇐⇒ y(0) ∈ ExpA−1 .

References [1] S.G. Krein, Linear Differential Equations in Banach Space. Nauka, Moscow, 1963. [2] W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Birkh¨ auser, Basel, 1999. [3] P. Koosis, Lectures on Hp Spaces. Cambridge University Press, London, 1980. [4] H. Komatsu, Ultradistributions. J. of the Facul. of Sci. Univ. Tokyo 20 (1973), 25– 105. [5] M.G. Krein, Lectures on the Theory of Stability of solutions of Differential Equations in a Banach Space. Inst. of Math. Akad. Nauk Ukrain. SSR, Kiev, 1964. [6] M.G. Krein, Introduction to Geometry of Indefinite J-Spaces and the Operator Theory in this Spaces. The Second Summer Math School (Katsiveli, June–July 1964). Part 1. Inst. of Math. Akad. Nauk Ukrain. SSR, Kiev, 1965, 15–92. [7] Yu.L. Daletsky and M.G. Krein, Stability of Solutions of Differential Equations in a Banach Space. Nauka, Moscow, 1970. [8] E. Nelson, Analytic vectors. Ann. Math. 70 (1959), 572–615. [9] R. Goodman, Analytic and entire vectors for representations of Lie groups. Trans. Amer. Math. Soc. 143 (1969), 55–76. [10] V.I. Gorbachuk and M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht, 1991. [11] Ya.V. Radyno, The space of vectors of exponential type. Dokl. Akad. Nauk BSSR 27 (1983), 791–793. [12] A.I. Kashpirovski ˘i, Analytic representation of generalized functions of S-type. Dokl. Akad. Nauk Ukrain. SSR. Ser. A. 4 (1980), 12–14. [13] I.M. Gel’fand, On one-parameter groups of operators in a normed space. Dokl. Akad. Nauk SSSR 25 (1939), 713–718. [14] A. Nussbaum, Quasi-analytic vectors. Ark. Mat. 6 (1965), 179–192. [15] R.Beals, Semigroups and abstract Gevrey spaces. J. Funct. Anal. 10 (1972), 300–308. [16] M.L. Gorbachuk, Yu.G. Mokrousov, On density of some sets of infinitely differentiable vectors of a closed operator on a Banach space. Methods Funct. Anal. Topology 8 (2002), 23–29. [17] M.L. Gorbachuk, V.I. Gorbachuk, On approximation of smooth vectors of a closed operator by entire vectors of exponential type. Ukrain. Mat. Zh. 47 (1995), 616–628. [18] J.M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 63 (1977), 370–373.

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[19] M.L. Gorbachuk, V.I. Gorbachuk, On a generalization of the Berezanski˘i evolution criterion of self-adjointness of an operator. Ukrain. Mat. Zh. 52 (2000), 608–615. [20] K. Hoffman, Banach Spaces of Analytic Functions. Prentice-Hall, INC. Englewood Cliffs, N. J., 1962. [21] V.I. Gorbachuk, A.V. Knyazyuk, Boundary values of solutions of operator differential equations. Uspekhi Mat. Nauk 44 (1989), 55–91. [22] V.I. Gorbachuk, M.L. Gorbachuk, Boundary values of solutions of some classes of differential equations. Mat. Sbornik 102 (1977), 124–150. [23] M.G. Krein, A remark on a theorem in the paper of V.A. Yakubovich titled “A frequency theorem for the case where . . . ”. Sibirsk. Mat. Zh. 18 (1977), 1411–1413. [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983. [25] M.L. Gorbachuk, I.T. Matsishin, The behavior at infinity of solutions of a first-order parabolic differential equations in Banach space. Dokl. Akad Nauk SSSR 312 (1990), 521–524. [26] R. deLaubenfels, Inverses of generators. Proc. Amer. Math. Soc. 104 (1988), 443– 448. [27] V.I. Gorbachuk, M.L. Gorbachuk, On behavior at infinity of orbits of uniformly stable semigroups. Ukrain. Mat. Zh. 58 (2006), 148–159. [28] M.G. Krein, On a generalization of Stieltjes’ investigations. Dokl. Akad. Nauk SSSR 87 (1952), 881–884. M.L. Gorbachuk and V.I. Gorbachuk Institute of Mathematics National Academy of Sciences of Ukraine 3 Tereshchenkivska St. 01601 Kyiv-4, Ukraine e-mail: [email protected]


Some Modern Methods in Mechanics of Cracks Mikhail A. Grekov and Nikita F. Morozov Abstract. An application of some modern methods to studying cracks in aligned composites and an interaction of curvilinear cracks with an interface is presented. The first method based on Novozhilov’s hybrid model is focused on estimation of variation intervals of sizes of a partially bridged equilibrium crack and bridging zone. The problem of a slightly curved crack near an interface is solved by a combination of some methods which are Muskhelishvili’s method of complex potentials, an original superposition method reduced to Fredholm integral equations of the second type, and the boundary perturbation method. Mathematics Subject Classification (2000). 74R10; 30E25. Keywords. Hybrid model, equilibrium crack, bridging zone, boundary perturbation method, curvilinear crack, interface, integral equations.

1. Introduction The aim of this work is to present some methods in mechanics of cracks that we have developed for the last time. The first one relates to an equilibrium crack theory based on Novozhilov’s hybrid model [1], [2] and applied to a crack in aligned composite. In general, the hybrid model means two-level analysis of fracture. At the lower level, it is taking into account discrete structure of a body by means of considering a bridging zone of a crack near a tip. At the upper one, a macrocrack is situated in continuum. According to Novozholov, the equilibrium state of a crack in a brittle body depends on the opening displacements in the bridging zone that is a zone of interaction of atoms laying at the adjacent atomic planes, and an average stress in a small region (fracture zone) around the tip. In the most hybrid models, the critical state of a crack is estimated by the stress intensity factors (SIF) instead This work was supported by Russian Foundation for Basic Research under grants 05-01-00274 and 06-01-00452.

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M.A. Grekov and N.F. Morozov

of analyzing the fracture zone. The first such models were given by Leonov and Panasyuk [3], Dugdale [4] and Barenblatt [5]. Then, a lot of investigators have used similar models applied to aligned composite as well (e.g., Marshall and Cox [6], Budiansky at al. [7]). At the same time Novozhilov’s approach was generalized and developed by Morozov and his disciples and followers [8]–[11]. In many cases the nature of the bridging zone is not so clear as for the crack between two rows of atoms that Novozhilov considered in his works. For a crack in metals, the bridging zone is often taken to be part of a plastic response of continuum material [3], [4]. But concept of the bridging zone is natural for a crack in material reinforced by fibers such as ceramics. Unlike a lot of studies of a bridged crack in aligned composites where a size of a bridging zone has been accepted as a given parameter, we estimate variation intervals of the size of an equilibrium crack and bridging zone before the crack begin to grow. There are a lot of problems of great importance in continuum mechanics, which can be solved by means of the perturbation method. For the last years, we have developed this method applying it to an analysis of curvilinear defects, including cracks, in different structures [12]–[16]. Unlike many works constructing only the first-order perturbation solution, we create an algorithm for finding any order solution of each problem considered. The application of the perturbation technique to 2-D problem of a slightly curved crack located near an interface is presented in Section 3.

2. Model of equilibrium penny-shaped crack in material reinforced by fibers In this section we study equilibrium states of a circular crack in an anisotropic inhomogeneous material reinforced by unidirectional fibers. This inhomogeneity, or structural imperfection of the composite, is taken into account by means of Novozhilov hybrid method [1], [2]. Namely, a crack growth is determined by two events: the breakage of outermost fibers in a bridging zone, and failure of a brittle matrix in a fracture zone adjoining the edge of the crack. We assume also that before unstable crack growth, the size of the bridging zone becomes much smaller than the radius of the crack. The same approach was used by Morozov et al. [9] for plane strain. Note, that this assumption is not sine qua non and one can repeat reasoning presented in the current study and obtain analogue results without it. 2.1. The statement of an axisymmetric problem Consider an elastic composite (e.g., ceramics) consisting of a brittle matrix and brittle unidirectional fibers. This composite contains a penny-shaped crack ρ2 = x21 + x22 ≤ a2 , x3 = 0 normal to the fibers (Fig. 1). The crack is opened by remote ∞ loading σ33 = p and two concentrated forces P normal to the crack and applied at the center of the crack faces. The most fibers that had bonded the crack surfaces have been broken and only a narrow annulus (bridging zone) b ≤ ρ ≤ a of width

Some Modern Methods in Mechanics of Cracks

129

Δ = a − b is bridged by the fibers (Fig. 1). We represent their resistance to the crack opening by an action of the normal traction σ0 uniformly and continuously distributed over the bridging zone. The stress in the outermost bridging fibers (for ρ = b) ) is equal to the fiber tensile strength Rf and corresponding crack opening displacement 2w is 2w(b) = 2w0 ,

(2.1)

where w0 is a parameter of the composite. The inequality w(ρ) > w0 means that the fibers are broken along the circle of radius ρ. So, w(ρ) > w0 if ρ < b, and w(ρ) < w0 if b < ρ ≤ a. If w(b) < w0 , the crack is assumed to be closed. Stress σ0 is determined from the bridging law σ = σ(w), related to pulling a fiber out of the matrix, by means of the formula w0 c σ0 = σ(t)dt. (2.2) w0 − wB wB

Figure 1. The model of a partially bridged penny-shaped crack under applied load in an aligned composite. Here σ(w) is a tensile stress in a fiber, the displacement wB of the upper and lower crack faces corresponds to the beginning of the nonlinear part of the curve σ(w), c is the fiber volume fraction. The values of wB and w0 depend on properties of the fiber-to-matrix adhesion and the elastic constants of each component. Equation (2.2) means that the work of traction σ0 at the interval ( wB , w0 ) is equal to the work of the stress σ(w) arising in the fibers at the boundary of the bridging zone. Note that substitution of the actual function σ0 (ρ) by constant value σ0 from this equation is similar to the Novozhilov’s substitution of the descending branch of the atom interaction law by step-function [1], [2]. Equation (2.2) is the simplest approximation of the genuine function σ0 (ρ) that, in reality, changes from

130


the value σ0 (b) = Rf to the value σ0 (a) = p, but as it is shown in [6], this changes can be neglected over a larger part of the bridging zone. Following Novozhilov’s fracture criterion, we assume that fracture of the matrix near a crack tip does not occur if a+D m σ33 (ρ)dρ ≤ DRm ,

(2.3)

a m where σ33 is a stress in the matrix, Rm is the matrix tensile strength, D is a size of the matrix fracture zone. Violation of condition (2.3) indicates that the crack is growing. The equality in (2.3) means that the crack is in a critical state. When there is no sliding between fibers and the matrix in the unbroken part f m of the composite, the stress σ33 is related to the stress in a fiber, σ33 , and the average stress σ33 by the equations [6] f m σ33 /Em = σ33 /Ef = σ33 /E3 ,

(2.4)

where Ef , Em are Young’s modules of the fiber and the matrix respectively, E3 = cEf + (1 − c)Em . The quantity D is defined as m 2 2 K1c D = kDm , Dm = . (2.5) π Rm m is the fracture toughness of the matrix, k is a characteristic of an Here K1c influence of the composite structural inhomogeneity upon the fracture process in the matrix. The equality D = Dm follows from a formal correlation of Novozhilov brittle fracture criterion and Irvin criterion for homogeneous isotropic material. In the case of a fiber-reinforced composite this equality means that, in spite of influence on anisotropic properties and stress field that drives the crack, reinforcement does not affect the fracture process in the matrix.

2.2. Variation interval of a size of an equilibrium crack Instead of the unidirectionally fiber-reinforced composite, consider an equivalent transversely isotropic homogeneous material. According to Panasyuk [17], the solution in the plane of the equilibrium penny-shaped crack x3 = 0 can be written for such a material as

a 2 − ρ2 P 2 2 ρ ≤ a, w(ρ) = 2H 2p a − ρ − 2σ0 F (ρ) + arctg , (2.6) πρ ρ = √ 2 ap − σ0 a2 − b2 πp a2 − b 2 a

+ σ0 arcsin σ33 (ρ) = + − p arcsin π 2 ρ ρ2 − b 2 ρ2 − a2 +

π 2 ρ2

Pa

, ρ2 − a2

ρ > a,

(2.7)

Some Modern Methods in Mechanics of Cracks where 1 − νf2 H= , π E3

a = F (ρ) = r

t2 − b 2 dt, t2 − ρ 2

r=

131

b, 0 ≤ ρ ≤ b, ρ, b ≤ ρ ≤ a.

Substituting expression (2.6) into (2.1) and (2.7) into (2.3) under conditions Δ/a 1, D/a 1 leads f (y) = g1 (α),

f (y) ≤ g2 (α),

(2.8)

where f (y) = Ay 1/2 + Cy −3/2 , y=

2p d 4P , A= , C= D σ0 πσ0 D2

and √ β g1 (α) = √ +2 α, 2 α

(2.9)

g2 (α) = π

√ √ R − 1 + 2 α + 2(1 + α) arcctg α. (2.10) σ0

In expressions (2.9) and (2.10), d = 2a is the crack diameter, α = Δ/D is the normalized size of the bridging zone and β = w0 /(HDσ0 ) is a parameter of the composite. Denote the lower and upper boundaries of diameters of the equilibrium crack by dl and du . The last value corresponds to the critical state of the crack when both equalities in (2.8) take place. So, in this case we have equation g1 (α) = g2 (α)

(2.11)

the solution of which gives the normalized critical size αc of the bridging zone. In order to find √ the lower boundary dl , note that function g1 (α) reaches the minimum value 2 β at the point α = α0 = β/4. Thus, as it follows from (2.8), the diameter of the equilibrium crack satisfies two inequalities f1 (z) ≥ 0,

f2 (z) ≤ 0,

(2.12)

where (2.13) fj (z) = Az 4 − Bj z 3 + C (j = 1, 2) √ √ √ and z = y, B1 = 2 β, B2 = 4αc + β/(2 αc ). The analysis shows that each equation fj (z) = 0 (j = 1, 2) has no more than two roots in the region y > 0. The values of dl and du depend on composite characteristics and applied load. An example of dependence of functions fj on the diameter d is plotted in Fig. 2 where one can see the corresponding variation interval of the diameter of the equilibrium crack. Quantities dl and du are easy determined in two special cases. According to ∞ (2.12) and (2.13), for the crack under remote loading σ33 =p 2 βσ 2 σ0 g1 (αc ) (2.14) dl = D 20 , du = D p 2p

132


Figure 2. The feasible variation interval (dl , du ) of the diameter of the equilibrium crack. and for the crack opened only by concentrated forces P 2/3 2/3 4P 2P √ dl = D , d = D . u πσ0 D2 g1 (αc ) πσ0 D2 β

(2.15)

2.3. Discussion and numerical results One can now imagine a possible behavior of the crack in the composite considered. If, under fixed remote load p, the crack diameter d reaches the value du , while the size of the bridging zone Δ reaches the value Δc = αc D, the fracture process transfers to the final stage of catastrophic fracture. Before this instant, a stable growth of the crack may occur with partially broken fibers if condition (2.3) is violate but equality (2.1) is not satisfied. The outermost fibers in the bridging zone begin to burst when equality (2.1) is satisfied. If the size of the bridging zone falls in the range Δc ≤ Δ ≤ Δ0 (Δ0 = α0 D) and equality (2.1) is satisfied, the crack will be in an equilibrium state for a fixed value of p. For the case of concentrated forces P applied at the center of the crack and p = 0, the crack growth with the bridging zones is the same as without them, namely, the crack growth is always stable. It follows from the second equality of (2.15) that in order to sustain the fracture process, it is necessary to increase continuously a value of P . It is important to note that besides the presence of the bridging zone, the effect of the fibers on the behavior of the crack in the composite also manifests itself in inequalities E3 = Em and D = Dm . According to relations (2.14) and (2.15), the presence of fibers affects the values of dl and du even when Δc = 0. To illustrate an application of the above theory to a real composite, consider two ceramics: silicon carbide (SiC) or calcium-aluminosilicate (CAS) matrix reinforced with silicon carbide fibers (SiC). The bridging low σ = σ(w) and all


133

experimental data except parameter D are presented in [7]. Some results of calculations based on these data are shown in Table 1 for three values of the fracture zone size D where df is a fiber diameter (df = 7 μm). Table 1. Basic parameters of equilibrium and critical states of a bridged crack vs. the size of the fracture zone D. Material D/Dm Δ0 /df Δc /df dl /du

0,1 58,9 56,3 0,999

SiC/SiC 1 58,9 51,2 0,995

10 58,9 37,8 0,952

0,1 63,0 60,0 0,999

SiC/CAS 1 63,0 53,9 0,994

10 63,0 37,6 0,936

The problem of determining the size of the fracture zone D should be highlighted. Clearly, the equality D = Dm is justified for the crack in an isotropic body the material of which is identical with the material of the matrix of the composite. For a fiber-reinforced material, a value of D may be different since the damage accumulation in the matrix happens in the neighborhood of the fibers the presence of which influences the fracture process. Thus, D is a characteristic of matrix damage and a characteristic of heterogeneity of the composite simultaneously. The simplest way to take into account an influence of material heterogeneity on D is to use linear dependence (2.5). According to Table 1, the critical size of the bridging zone Δc is equal to approximately 51df for SiC/SiC and 54df for SiC/CAS if D = Dm . Since Δ/a 1, the critical size of the crack has to satisfy conditions du ! 104df (du ! 728μm) for SiC/SiC and du ! 108df (du ! 754μm) for SiC/CAS. Hence, our model allows evaluating the critical size of the crack du for these composites if du is more than 1 mm. In conclusion of this section, note that Δc is the same for the plane strain and axisymmetric problem and does not depend on a type of loading [11]. Relations (2.12)–(2.15) remain valid in the case of the plane strain for a crack of length L if we replace β by 4β/π 2 and P by P L.

3. Model of a slightly curved crack near an interface To solve the correspondent 2-D problem, we use the boundary perturbation method suggested by Grekov [12] for studying a curved crack in a homogeneous plane and the superposition method elaborated by Grekov and Germanovich [18] and Grekov [19] for analyzing a rectilinear crack located near a boundary of a half-plane. Based on Goursat-Kolosov’s complex potentials and Muskhelishvili’s representations [20], the solution of the problem considered is reduced to the successive solution of Fredholm integral equations of the second type. This integral equations for any order approximations differ only in the right-hand members depending on all solutions of previous levels of approximation.

134


3.1. The statement of the problem Consider a two-component elastic body with a flat interface under plane strain or plane stress. Thus, we may formulate the corresponding 2-D problem for a plane of the complex variable z = x1 + ix2 , consisting of two half-planes Ωk = {z : (−1)k x2 > 0} (k = 1, 2). The region Ω1 contains the curvilinear crack Γc = {z : z = zc } which is a small deviation of a reference rectilinear crack of the length 2l inclined to the interface Γ by the angle α1 (Fig. 3).

Figure 3. A model of a curvilinear crack near an interface. In local Cartesian coordinates ξ1 , ξ2 the crack Γc is defined by ζ ≡ ζc = ξ1 + iεf (ξ1 ), |ξ1 | < l.

(3.1)

Assume that ε > 0, ε 1, the function f (ξ1 ) is continuous and max |f (ξ1 )| = l and |f (ξ1 )| < M (M = const). Cartesian coordinates x1 , x2 and ξ1 , ξ2 are related by the equality ζ = (z + ih) exp−iα1 , (3.2) where h > l sin α1 , 0 ≤ α1 ≤ π. Displacements and tractions are continuous across the interface u− (x1 ) = u+ (x1 ),

σ − (x1 ) = σ + (x1 ), x1 ∈ Γ,

(3.3)

and the boundary conditions at the crack Γc are σ + (zc ) = σ − (zc ) = p0 (ζc ), ±

Here u (x1 ) = lim

ζ→ζc ±i0

lim

z→x1 ±i0

±

u(z), σ (x1 ) =

lim

z→x1 ±i0

z c ∈ Γc .

(3.4) ±

σ(z) in conditions (3.3), σ (zc ) =

σ(z) in conditions (3.4), u = u1 + iu2 , σ = σnn + iσnt ; u1 , u2 are displace-

ments along the corresponding axes x1 , x2 ; σnn , σnt are a normal and tangential


135

tractions at an element with unit normal n (vector n in conditions (3.3) and (3.4) is perpendicular to the correspondent curve and a direction of unit vector t coincides with a positive direction of a tangent). Traction p0 and all its derivatives is assumed to be Holder class functions almost everywhere at Γc . Conditions at infinity are k∞ , lim σij (z) = σij

|z|→∞

lim ω(z) = ω k∞ ,

|z|→∞

z ∈ Ωk ,

(3.5)

where σij is a stress tensor component in coordinates x1 , x2 and ω is a turning angle. 3.2. Superposition method Following the superposition principle [19], the solution of the problem is represented as + 0, z ∈ Ω2 , b σ(z) = σ (z) + σ c (z), z ∈ Ω1 , + 0, z ∈ Ω2 , b (3.6) u(z) = u (z) + c u (z), z ∈ Ω1 , where σ c (z), uc (z) are a traction and a displacement arisen in a homogeneous plane with elastic properties of the half-plane Ω1 under action of some unknown traction at the crack Γc in this plane; σ b (z), ub (z) are a traction and displacement arisen in a two-component continuous plane under conditions at infinity (3.5) and jumps of tractions Δσ b = σ b+ − σ b− and displacements Δub = ub+ − ub− at the interface Γ. Substituting (3.6) into (3.3) yields Δσ b = σ c , Δub = uc , z ∈ Γ.

(3.7)

Introduce the following notations: + + ηk = 1, σ b (z), σ(z), ηk = 1, G(z) = G (z) = b b du du −2μk dz , ηk = −κk , −2μk , ηk = −κk , dz + σ c (z), η1 = 1, c (3.8) Gc (z) = −2μ1 du , η1 = −κ1 , dz where κk = (3 − νk )/(1 + νk ) for plane stress and κk = (3 − 4νk ) for plane strain; νk , μk are Poisson ratio and the shear modulus of Ωk respectively. Equalities (3.6) and (3.8) give us the basic relation of superposition principle G(z) = Gb (z) + Gc (z)δk1 , z ∈ Ωk . Here δk1 = 1 if k = 1 and δk1 = 0 if k = 2.

(3.9)

136


3.3. Curvilinear crack in homogenous plane Consider the crack Γc in a homogeneous plane with properties of Ω1 . The boundary condition at the crack is σ c± (zc ) = q(ζc ),

z c ∈ Γc

(3.10)

Assume that q(ζc ) is the Holder-continuous function at Γc , and stresses and turning angles are equal zero at infinity. According to Grekov [12], traction σ c at an element with unit normal n and displacement uc at any point ζ outside Γc are related to Goursat-Kolosov’s complex potentials Φ and Υ by the following equality Gc (z) = η1 Φ(ζ) + Φ(ζ) + Υ(ζ) − Φ(ζ) + (ζ − ζ)Φ (ζ) e−2iβ , (3.11) where β is an angle between the direction of the element defined by vector t and ξ1 -axis. Functions Φ(ζ), outside the finite region bounded Υ(ζ) are holomorphic 1 1 by the curve Γc ∪ Γc Γc = {ζ : ζ = ζ c } . Deriving relation (3.11) is the first step in our boundary perturbation method. Relation (3.11) is approximate. It becomes an exact one in the case of the recti1 c = Γc . linear crack for which Γ Let ζ → ζc ∈ Γc and β → βc in (3.11), where βc is the inclination of the tangent to Γc in coordinates ξ1 , ξ2 . Then, taking into account the evident equation e−2iβc = 1 −

2iεf (ξ1 ) 1 + iεf (ξ1 )

and conditions (3.11), we obtain 1 − iεf (ξ ) 1 = q(ζc ), Φ± (ζc ) + Φ± (ζc ) + Υ∓ (ζ c ) − Φ± (ζc ) + 2iεf (ξ1 )Φ± (ζc ) 1 + iεf (ξ1 ) (3.12) where Φ± (ζc ) = lim Φ(ζ), Υ± (ζ c ) = lim Υ(ζ). ζ→ζc ±i0

ζ→ζc ±i0

In accordance with the perturbation technique [12], expand functions Φ(ζ), Υ(ζ) and q(ζc ) in power series’s of the small parameter ε Φ(ζ) =

∞ ∞ ∞ εn εn εn Φn (ζ), Υ(ζ) = Υn (ζ), q(ζc ) = qn (ζc ) n! n! n! n=0 n=0 n=0

(3.13)

and boundary values of functions Φn (ζ), Υn (ζ) at Γc and functions qn (ζc ) into Maclaurin series’s in the vicinity of ξ2 = 0, considering ξ1 as a parameter Φ± n (ζc )

∞ ∞ (iξ2 )m (m)± (−iξ2 )m (m)∓ ∓ Φn Υn = (ξ1 ), Υn (ζc ) = (ξ1 ), m! m! m=0 m=0 ∞ (iξ2 )m (m) q (ξ1 ). qn (ζc ) = m! n m=0

(3.14)


137

As |εf | < 1, one can write 1−

∞ 2iεf (ξ1 ) m+1 = 1 + 2 (−iεf ) ≡ S. 1 + iεf (ξ1 ) m=0

Then, substituting (3.13) and (3.14) into (3.12) yields ∞ ∞ k εn (iεf (ξ1 )) ±(k) Φ±(k) (ξ1 ) + (−1)k Φn (ξ1 )+ n n! k! n=0 k=0 ±(k) ±(k+1) (ξ ) − Φ (ξ ) + 2iεf (ξ ) Φ (ξ ) + (−1)k S Υ∓(k) n n 1 1 1 1 n

=

∞ ∞ k εn (iεf (ξ1 )) (k) qn . n! k! n=0 k=0

Equating polynomial coefficient of power εm (m = 0, 1, . . . ) to zero, one derives the following sequence of boundary conditions ∓ ± Φ± n (ξ1 ) + Υn (ξ1 ) = qn (ξ1 ) + Hn (ξ1 ), |ξ1 | ≤ l.

(3.15)

Here H0± (ξ1 ) = 0, Hn± (ξ1 )

k n! (if (ξ1 )) ±(k) Φm (ξ1 ) + (−1)k Υ∓(k) =− m (ξ1 ) m! k! m=0 (if (ξ1 ))k−j ±(k) (if (ξ1 ))j + 2(−1)k−1 k Φm (ξ1 ) + 2(−1)k−1 (k − 1)! 1≤j≤k ±(k−j) × (2k − 2j + 1) Φm (ξ1 ) − Υ∓(k−j) (ξ ) 1 m n−1

+

n−1

k

n! (if (ξ1 )) (k) qm (ξ1 ), n > 0, k = n − m. m! k! m=0

(3.16)

Conditions (3.15) lead to two boundary Riemann-Gilbert problems for finding functions Φn (ζ) and Υn (ζ) [12]. The solution of these problems can be written as Φn (ζ) = Φnu (ζ) + Φnk (ζ), Υn (ζ) = Υnu (ζ) + Υnk (ζ) 1 Φnu (ζ) = Υnu (ζ) = 2πiX(ζ) Φnk (ζ) =

l −l

X(t)qn (t) dt, t−ζ

1 (In1 (ζ) + In2 (ζ)) , Υnk (ζ) = Φnk (ζ) − In1 (ζ), 2

(3.17) (3.18)

(3.19)

138

M.A. Grekov and N.F. Morozov 1 In1 (ζ) = 2πi

l −l

Hn+ (t) − Hn− (t) dt, t−ζ

1 In2 (ζ) = 2πiX(ζ) where X(ζ) =

l −l

X(t)[Hn+ (t) + Hn− (t)] dt, t−ζ

ζ 2 − l2 , X(t) = ±X ± (t) = i

(3.20)

l2 − t2 , |t| < l.

Functions Hn± in (3.16) depend on all previous approximations and, so, in n-order approximation, functions Φnk , Υnk are known and Φnu , Υnu must be found. 3.4. Two-component plane with rectilinear interface Express now the function Gb in term of unknown functions Φ and Υ. According to Grekov [19], the traction σ b at the element with the normal n and the displacement ub in a problem on a joint deformation of two homogeneous planes having different elastic properties are defined by Gb (z) = ηk Ξk (z) + Ξk (z) − Ξk (z) + Ξk (z) − (z − z)Ξk (z) e−2iα , z ∈ Ωk (3.21) where Ξk (k = 1, 2) are functions holomorphic outside the interface Γ, α is the angle between vector t and x1 -axis, α = α1 + β. Under conditions (3.5) and (3.22), functions Ξk are equal ⎧ μ κ Σ(z) + μ U (z) ⎪ + a12 , z ∈ Ω2 , ⎨ 1 2μ2 + μ1 κ22 Ξ2 (z) = Σ(z) − Ξ1 (z), Ξ1 (z) = ⎪ ⎩ μ1 Σ(z) − μ2 U (z) + a , z ∈ Ω1 . 11 μ +μ κ 1

Here Σ(z) =

1 2πi

∞ −∞

σc (t) dt, t−z

2 1

U (z) = −

μ1 πi

∞ −∞

uc (t) dt t−z

(3.22)

and akj = lim Ξk (z), z ∈ Ωj , k, j = 1, 2, |z|→∞

k∞ ∞ + σ22 2μk σ11 +i ω ∞ , k = j. 4 κk + 1 k Introduce (3.11) into equations (3.22). Then, using properties of Cauchy type integrals [19], we obtain + z ∈ Ω2 , K1 Φ(ζ) # + a12 , $ Ξ1 (z) = 2iα1 + a11 , z ∈ Ω1 , −K2 Φ(w) + Υ(w) − Φ(w) − (w − ζ)Φ (w) e ∞ ∞ − iσ12 , akk = akk − akj = σ22

Some Modern Methods in Mechanics of Cracks + Ξ2 (z) =

+ a22, K3 Φ(ζ) #

K4 Φ(w) + Υ(w) − Φ(w) − (w −

ζ)Φ (w)

139

z ∈ Ω2 , $ 2iα1 + a21 , z ∈ Ω1 , e

where

μ1 κ2 − μ2 κ1 μ1 − μ2 , K2 = , μ2 + μ1 κ2 μ1 + μ2 κ1 K3 = 1 − K1 , K4 = K2 − 1. Taking into account the last relations, function Gb in (3.21) is defined as w = (z − ih) eiα1 , K1 =

Gb (z) = M (Υ, Φ, z, α, η1 ) + G1∞ b ,

z ∈ Ω1 ,

(3.23)

and if z ∈ Ω2 ,

Gb (z) = η2 K3 Φ(ζ) + K3 1 − e−2iα − K4 e−2iα 1 − e2iα1 Φ(ζ) −K4 e−2iβ Υ ζ + K4 ζ − w + K3 eiα1 (z − z) e−2iα Φ (ζ) + G2∞ (3.24) b .

and G2∞ are values of corresponding functions at infinity, Constants G1∞ b b that are found from conditions (3.5). Besides, the following notification was introduced in (3.23) M (Υ, Φ, z, α, η1 ) = −η1 K2 e2iα1 Υ(w) − K2 e−2iα1 1 − e−2iα Υ(w) −η1 K2 1 − e2iα1 Φ(w) + K2 e−2iα1 1 − e−2iα w − ζ Φ (w) − K2 1 − e−2iα1 1 − e−2iα + K1 e−2iα Φ(w) + η1 K2 (w − ζ) e2iα1 Φ (w) # $ −K2 (z − z) Υ (w) + 2 e2iα1 − 1 Φ (w) − w − ζ Φ (w) e−i(2α+3α1 ) (3.25) The value μ2 = 0 corresponds to a crack in the half-plane Ω1 when Ω2 is absent. In this case, relation (3.23) coincides to an accuracy of sign and the constant G1∞ with equality (10.31) in [19], and the right-hand side of (3.24) is b equal zero. 3.5. Integral equation of n-order approximation Functions Φ and Υ are expressed in term of unknown function q by means of equations (3.13) and (3.17)–(3.20). In order to find q, introduce (3.11) and (3.23) into (3.9) taking η1 = 1, and pass to limit in the obtained equation under z → zc , α → αc . Then, taking into account boundary condition (3.4) and equality (3.12), we derive the following equation q(ζc ) + σ b (zc , αc ) = p0 (ζc ), zc ∈ Γc

(3.26)

b

Replace σ (zc , αc ) in (3.26) by its expression in (3.23). Then, using expansions (3.13) in (3.26) yields ∞ εn qn (ζc ) + M (Υn , Φn , zc , αc , 1) = p0 (ζc ) − σ 1∞ (αc ), n! n=0

where

(3.27)

−2iαc 1∞ ∞ 1∞ ∞ 1∞ 1∞ ∞ = σ22 e + iσnt + σ11 + σ22 − σ11 − 2iσ12 . 2σ 1∞ (αc ) = 2 σnn

140


Taking into account that αc = α1 + βc zc = z0 + iεf (ξ1 )eiα1 , wc = w0 + iεf (ξ1 )e2iα1 , z0 = ξ1 eiα1 − ih and w0 = (z0 − ih) eiα1 , expand functions qn , Φn , Υn and p0 into Maclaurin series’s like (3.14). Then equation (3.27) is transformed to the following ∞ ∞ k εn (iεf ) # (k) qn (ξ1 ) − K2 e2i(k+1)α1 Υ(k) n (w0 ) n! k! n=0 k=0

(k) (k) −K2 (−1)k e−2i(k+2)α1 e2iα1 − S Υn (w0 ) − K2 e2ikα1 1 − e2iα1 Φn (w 0 ) −(−1)k e−2i(k+1)α1 K2 1 − e−2iα1 e2iα1 − S + K1 S Φ(k) n (w 0 ) (k+1) (w 0 ) +K2 e2i(k+1)α1 w0 − ξ1 − iεf 1 − e2iα1 Φn +K2 (−1)k e−2i(k+2)α1 e2iα1 − S w0 − ξ1 + iεf 1 − e−2iα1 Φ(k+1) (w0 ) n iα1 k −i(2k+5)α1 −iα1 −K2 (−1) e z0 − z 0 + iεf e + e S (k+1) (w0 ) + 2 e2iα1 − 1 Φ(k+1) (w 0 ) × Υn n $ w0 − ξ1 + iεf 1 − e−2iα1 Φ(k+2) (w0 ) n ∞ −2iα1 ∞ m+1 1∞ ∞ = −σ 1∞ (α1 ) + σ22 e − σ11 − 2iσ12 (−iεf (ξ1 )) m=0

+

∞ (iεf (ξ1 ))k k=0

k!

(k)

p0 (ξ1 )

(3.28)

Collecting polynomial coefficients of power εn (n = 0, 1, . . . ) yields a sequence of Fredholm integral equations of the second type qn (ξ1 ) + M (Υnu , Φnu , z0 , α1 , 1) = Fn (ξ1 ), |ξ1 | ≤ l, n = 0, 1, . . .

(3.29)

It follows from (3.18) and (3.25) that M in (3.29) is Fredholm operator with continuous kernels, acting on functions qn and qn . If μ2 /μ1 = 0 or μ2 /μ1 → ∞, this operator coincides to an accuracy of sign with operator (10.35) in [19], and equation (3.29) in zero-order approximation coincides in these cases (when α1 = 0) with corresponding integral equations applied in [19] to the rectilinear crack parallel to the free and rigid boundary of the half-plane. In zero-order approximation, equation (3.29) corresponds to the rectilinear crack near the interface. This equation was derived in [19] for the case of zeroth stresses and turning angles at infinity. It can be shown that the homogeneous equation corresponding to equation (3.29) has only the trivial solution and, so, equation (3.29) has the unique solution for any-order approximation. It is not difficult to derive functions Fn from (3.29) for any value of n. For the first two approximations, this functions are defined as −2iα1 1 ∞ 1 ∞ 1∞ 1∞ ∞ σ22 − σ11 F0 (ξ1 ) = p0 (ξ1 ) − (σ22 e + σ11 )− − 2iσ12 , 2 2


141

∞ 1∞ ∞ F1 (ξ1 ) = if (ξ1 )p0 (ξ1 ) − i σ22 f (ξ1 ) e−2iα1 − σ11 − 2iσ12 % −M (Υ1k , Φ1k , z0 , α1 , 1) + 2if (ξ1 ) K2 1 − e−2iα1 + K1 e−2iα1 Φ0 (w 0 ) # $& % +K2 e−4iα1 Υ0 (w0 ) − (w 0 − ξ1 )Φ0 (w 0 ) − if (ξ1 ) q0 (ξ1 ) − K2 e4iα1 Υ0 (w0 ) +K2 1 − e−2iα1 e−4iα1 Υ0 (w0 ) + 2e4iα1 Φ0 (w 0 ) +e−4iα1 K1 − 2K2 1 − e−4iα1 Φ0 (w 0 ) + K2 e4iα1 (w0 − ξ1 )Φ0 (w 0 ) +K2 e−4iα1 1 − e−2iα1 (z0 − z 0 )e−iα1 − (w 0 − ξ1 ) Φ0 (w 0 ) & # $ +K2 (z0 − z 0 ) Υ0 (w0 ) + 2 e2iα1 − 1 Φ0 (w 0 ) − (w 0 − ξ1 )Φ0 (w 0 ) e−7iα1 −iK2 2(z0 − z 0 )f (ξ1 ) − f (ξ1 ) eiα1 + e−iα1 # $ Υ0 (w0 ) + 2 e2iα1 − 1 Φ0 (w 0 ) − (w 0 − ξ1 )Φ0 (w 0 ) e−5iα1 . (3.30) Actually, formulae (3.9), ((3.11), (3.13), (3.17)–(3.20), (3.23)–(3.25), (3.28) and (3.29) give the algorithm for approximate computing stresses and displacements in two-component body containing a slightly curved crack near the interface at any-order approximation. First, as the right-hand side F0 is known, we find q0 by solving integral equation (3.29) in the zeroth-order approximation. Then, using equations (3.17), (3.19) and (3.20), we determine the function F1 by means of equation (3.30). After that, solution of equation (3.29) gives the function q1 . The next-order approximations are found in the same way. In order to obtain a numerical solution of the integral equation (3.29), one can use the combined method of solving such equations, described in [19]. As it was shown in [19], this method was very effective in the case of a rectilinear crack in a half-plane and a strip under different type of loading including concentrated forces applied to the crack faces and the boundary of the half-plane.

References [1] V.V. Novozhilov, On the necessary and sufficient criterion for brittle strength. J. Appl. Math. Mech. 33 (1969), 201–210. [2] V.V. Novozhilov, On the foundation of a theory of equilibrium cracks in elastic solids. J. Appl. Math. Mech. 33 (1969), 777–790. [3] M.Y. Leonov, V.V. Panasyuk, A development of the smallest cracks in a solid. Applied Mechanics 5 (1959). [4] D.S. Dugdale, Yielding of sheets containing slits. J. Mech. Phys. Solids. 8 (1960), 100–104. [5] G.I. Barenblatt, On equilibrium crack arising under brittle fracture. Prikl. Mat. Mekh. 23 (1959), 434–444, 707–721, 893–900. [6] D.B. Marshall, B.N. Cox, Tensile fracture of brittle matrix composites: influence of fiber strength. Acta Metall. 35 (1987), 2607–2619.

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[7] B. Budiansky, A.G. Evans, J.W. Hutchinson, Fiber-matrix debonding effects on cracking in aligned fiber ceramic composites. Intern. J. Solids and Structures. 32 (1995), 315–328. [8] N.F. Morozov, M.V. Paukshto, Discrete and hybrid models at fracture mechanics, St. Petersburg State Univ., 1994. [9] N. Morozov, M. Paukshto, N. Ponikarov, On the problem of equilibrium length of bridged crack. Trans. ASME. J. Appl. Mech. 64 (1997), 427–430. [10] M.A. Grekov, N.F. Morozov, Disk-shaped equilibrium cracks. J. Appl. Math. Mech. 64 (1999), 169–172. [11] M.A. Grekov, N.F. Morozov, Equilibrium cracks in composites reinforced with unidirectional fibers. J. Appl. Math. Mech. 70 (2006), 945–955. [12] M.A. Grekov, A slightly curved crack in an isotropic body.Vestn. St. Petersburg Univ., Ser. 1, Vip. 3 (2002), 74–80. [13] M.A. Grekov, The perturbation approach for two-component composite with slightly curved interface. Vestn. St. Petersburg Univ., Ser. 1, Vip. 1 (2004), 81–88. [14] M.A. Grekov, S.N. Makarov, Stress concentration near a slightly curved part of an elastic body surface. Mech. of Solids. 39 (2004), 40–46. [15] M.A. Grekov, Y.V. Malkova, The force and energy parameters of an elastic field near a tip of a curvilinear interface crack. Vestn. St. Petersburg Univ., Ser. 10, Vip. 3 (2006), 17–27. [16] I.D. Volkov, M.A. Grekov, Greens functions for dissimilar materials with slightly curved interface. Vestn. St. Petersburg Univ., Ser. 1, Vip. 3 (2007), 126–136. [17] V.V. Panasyuk, The Limit Equilibrium of Brittle Bodies with Cracks, Naukova Dumka, 1968. [18] M.A. Grekov, L.N. Germanovich, A boundary integral method for closely spaced fracture. Modeling and Simulation Based Engineering, S. Atlury, ed. 2 (1998), 166–171. [19] M.A. Grekov, Singular problems in elasticity. St. Petersb. State Univ., 2001. [20] N.I. Muskhelishvili. Some basic problems of the mathematical theory of elasticity. Noordhoof, 1975. Mikhail A. Grekov Faculty of Applied Mathematics St. Petersburg State University Universitetski pr., 35 St. Petersburg, 198504, Russia e-mail: [email protected] Nikita F. Morozov Faculty of Mathematics and Mechanics St. Petersburg State University Universitetski pr., 28 St. Petersburg, 198504, Russia e-mail: [email protected]


A Method of the Green’s Functions for Quasistatic Thermoelasticity Problems in Layered Thermosensitive Bodies under Complex Heat Exchange Roman Kushnir and Borys Protsiuk Abstract. An approach for solution construction of the one-dimensional nonstatic heat conduction problems and corresponding quasi-static thermoelasticity problems for layered bodies of canonical form under convective-radial heating is proposed accounting temperature dependence of physical-mechanical material properties. This approach is based on employment of the Kirchhoff’s substitution, distribution technique, and constructed the Green’s functions for corresponding linear non-static heat conduction problems and static elasticity problems. The solution of the heat conduction problems is reduced to the solution of integro-differential equations. In analysis of thermoelasticity problems for a layered cylinder and sphere, the elastic moduli and Poisson’s ratios are assumed to be continuous within each phase and are approximated by transient piecewise-constant functions. Mathematics Subject Classification (2000). Primary 74B05; Secondary 80A20. Keywords. Layered bodies, temperature- dependent properties, quasi-static thermoelasticity problems, convective-radial heat exchange, Green’s functions, integro-differential equations.

1. Formulation of non-static heat conduction problems and solution method Let the multilayered bodies (plate, cylinder or sphere) are heated by convectiveradial heat exchange with the surroundings, the temperature of which is function of time τ . The bodies are assumed to be free of external force loadings. The layers of the body are in ideal contact. The physical-mechanical properties of each layer

144

R. Kushnir and B. Protsiuk

strongly depend on the temperature; the initial temperature is different and varies only through the thickness. Introducing the dimensionless coordinate x and time F o, the temperature field of the i-layer ( xi−1 < x < xi ) is determined from a set of heat conduction equations 1 ∂ ∂ti ∂ti k (i) x λt (ti ) = a1 c(i) , i = 1, n, (1.1) v (ti ) k x ∂x ∂x ∂F o under boundary conditions ∂tj+1 ∂tj (j) − λt (tj ) = 0, x = xj , j = 1, n − 1; (1.2) ∂x ∂x ,4 ∂t1 (1) − − − 4 − lα− = 0, x = x0 , λt (t1 ) 0 (t1 )[t1 − tc (F o)] − lγ (t1 ) t1 − [tc (F o)] ∂x ,4 ∂tn (n) + + + 4 +lα+ = 0, x = xn ; (1.3) λt (tn ) 0 (tn )[tn −tc (F o)]+lγ (tn ) tn − [tc (F o)] ∂x (1.4) ti = t0i (x), F o = 0. (j+1)

tj+1 − tj = 0, λt

(i)

(tj+1 )

(i)

(i)

Here λt (ti ) = λ0 Λi (ti ) and cv (ti ) = c0 Ci (ti ) are the coefficients of conduction ± and volumetric heat capacity; α± 0 (tj ) are the heat transfer coefficients; γ (tj ) = ± ± γ0 ε (tj ; γ0 is the Stefan-Boltzmann constant of radiation; ε (tj ) denote the6coef (1) (1) ficients of blackness; F o = a1 τ l2 ; l is a characteristic linear size; a1 = λ0 c0 ; t± c (F o), t0i (z) are given functions; n is the layer number; k = 0, 1, 2 for a layered plate, cylinder, and sphere, respectively. Generalizing the results of [1, 2, 3, 4], the non-linear problems (1.1)–(1.4) can be reduced to the solution of integro-differential equations by the following scheme below: • Apply the Kirchhoff substitution θi =

1

ti

(i)

λt (ζ)dζ

(i)

λ0

(1.5)

0

to equations (1.1). It must be assumed, therewith, that the functions θi = θi (ti ) have the inversions, ti = ti (θi ). As a result, we obtain the set of equations: 1 ∂ (i) Ci [ti (θi )] ∂θi k (i) ∂θi x = a 1 c0 . (1.6) λ 0 k x ∂x ∂x Λi [ti (θi )] ∂F o • Formulate the contact conditions and boundary conditions for the Kirchhoff variables. Let us assume the heat conduction coefficient to be depending on the temperature linearly, Λi (ti ) = 1 + βi ti , βi = const. (1.7) Then, by means of expression ti = ti (θi ) =

1

1 + 2βi θi − 1 βi

(1.8)

A Method of the Green’s Functions. . .

145

found from (1.5), as well as the condition of temperature equality at the interfaces, we obtain the first contact condition for the Kirchhoff variables θj+1 − θj = Fj+1 (θj+1 ), x = xj , where

(1.9)

1 + 2βj+1 θj+1 − 1 βj Fj+1 (θj+1 ) = 1 − . θj+1 − βj+1 βj+1

With regard for (1.5), the second contact condition takes the form (j+1) ∂θj+1

(j) ∂θj − λ0 = 0, x = xj . (1.10) ∂x ∂x Applying the Kirchhoff substitution to the boundary conditions (1.3), (1.4), we obtain ∂θ1 − Bi0 [θ1 − θc− (F o)] = 0, x = x0 , ∂x ∂θn + Bin [θn − θc+ (F o)] = 0, x = xn ; (1.11) ∂x (1.12) θi = θ0i (x), F o = 0,

λ0

where

± α± 0 [θ (F o)] ± [θ (F o) − t± c (F o)] α± γ ± [θ± (F o)] , ± 4 − [θ (F o)]4 − [t± ; c (F o)] ± α lα− lα+ Bi0 = (1) , Bin = (n) ; λ0 λ0

θc± (F o) = θ∗± (F o) −

θ∗− (F o) = θ1 (x0 , F o), θ− (F o) = t1 [θ∗− (F o)], θ∗+ (F o) = θn (xn , F o), θ+ (F o) = tn [θ∗+ (F o)]; ± θ0i (x) = t0i (x) + βi t20i (x) 2; α± is chosen [5] from the change-interval α± 0 [θ (F o)]. • Using the Heaviside functions S(α) and presenting functions θ(x, F o) and coefficients λ0 (x), c0 (x) in the form θ = θ1 (x, F o) +

n−1

[θj+1 (x, F o) − θj (x, F o)] S(x − xj ),

(1.13)

j=1

the system of equations (1.6) can be replaced by the single equation 1 ∂ ∂θ ∂θ k x λ0 (x) = a1 c0 (x) − wt k x ∂x ∂x ∂F o with generalized derivative with respect of x. Here wt = c0 (x)Wt (x, F o) −

n−1 1 (j+1) Fj+1 (θj+1 )|x=xj xkj λ0 δ (x − xj ), xk j=1

(1.14)

146

R. Kushnir and B. Protsiuk (1)

Wt (x, F o) = Wt [θ1 (x, F o)] +

n−1 #

(i+1)

Wt

$ (i) [θi+1 (x, F o)] − Wt [θi (x, F o)] S(x − xi ),

i=1

: ∂θj Cj [tj (θj )] (j) . Wt [θj (x, F o)] = a1 1 − Λj [tj (θj )] ∂F o Note that the equation (1.14) is equivalent to the system of equations (1.6) with contact conditions (1.9), (1.10). It can be proved by employing the Leibnitz rule for differentiation of the product of two piecewise-continuous functions and operation of non-commutative and associative multiplication [6]. • Write the integral expressions for the solutions of the problems (1.14), (1.11), (1.12) in terms of the Green’s functions [7, 8, 9] in the form xn c0 (ρ) k (k) θ (x, F o) = ρ θ0 (ρ)G(k) (x, ρ, F o)dρ + T (k) (x, F o) (1) c0 x0

+

Fo (j+1) ∂G(k) (x, ρ, F o k λ0 xj (1) ∂ρ λ0 j=1 0

n−1

+

1

− ξ)

Fj+1 [θj+1 (xj , ξ)]dξ

ρ=xj+0

F oxn

(1) λ0 0 x0

ρk G(k) (x, ρ, F o − ξ)Wt (ρ, ξ) dρdξ,

where θ0 (x) = θ01 (x) +

n−1

(1.15)

[θ0,j+1 (x) − θ0j (x)] S(x − xj ),

j=1

T

(k)

(x, F o) =

xk0 Bi0

F o G(k) (x, x0 , F o − ξ)θc− (ξ)dξ 0

F o ∗(n) k +xn Kλ Bin G(k) (x, xn , F o − ξ)θc+ (ξ)dξ, 0

G(k) (x, ρ, F o) = 2y (k) (x, ρ)

∞ (k) (k) Φ(k) (μm , x)Φ(k) (μm , ρ) (k) N (k) (μm )

m=1

y (0) (x, ρ) = y (1) (x, ρ) = 1, y (2) (x, ρ) = (k)

Φ(k) (μ, x) = Φ1 (μ, x) +

n−1

(k)

Fo

1 , xρ

(k)

[Φi+1 (μ, x) − Φi (μ, x)]S(x − xi );

i=1 (2) Φ1 (μ, x)

(k) 2

e−μm

= cos μ(x − x0 ) + Bi0 μ−1 sin μ(x − x0 ), (2)

,

A Method of the Green’s Functions. . . (2) (2) Φj (μ, x) = Φj−1 (μ, xj−1 ) cos εj (x − xj−1 ) +

βj∗ xj−1 εj

147

sin εj (x − xj−1 )

(j)

+

Kλ (2) Φ (μ, xj−1 ) sin εj (x − xj−1 ), εj j−1 μ2 N (2) (μ)

2

=μ

n (j) c

(2) 0 L (μ)hj (1) j c j=2 0

+

(2) μ2 Lj (μ)

+

n−1

∗(j+1) Kλ

−

∗(j) Kλ

j=1

=

(2) 2 Φj−1 (μ, xj−1 )

Φ(2) 2 (μ, xj ) j

xj

βj∗ 2 μ2 + Aj x2j−1

+ X (2) (μ),

# $ 1 (j) (2) (2) (j) Kλ Φj−1 (μ, xj−1 ) 2βj∗ Φj−1 (μ, xj−1 ) + xj−1 Kλ Φj−1 (2) (μ, xj−1 ) , Aj xj−1 j = 2, n, (2)

(2) 2 (2) ∗(n) Bin εn Ln (μ) , (2) 2 ε2n + Bin

(2)

X (2) (μ) = μ2 L1 (μ)h1 + Bi0 + Kλ

(0) (0) (2) Φ1 (μ, x) = cos μx + Bi0 μ−1 sin μx, Φj (μ, x) = Φj (μ, x) μ2 N (0) (μ) = μ2

n (i) c0 (1)

j=2

(0) (2) Lj (μ) = Lj (μ)

βj∗ =0

c0

(2) 2

(2)

L1 (μ) = 1 + Bi0

βj∗ =0

6 μ2 ;

,

(0)

Lj (μ)hj + X (0) (μ),

(0) , j = 2, n, L1 (μ) = 1 + Bi20 μ2 , 2 (0) ∗(n) Bin εn Ln (μ) ; ε2n + Bi2n

(0)

X (0) (μ) = μ2 L1 (μ)h1 + Bi0 + Kλ (1)

Φ1 (μ, x) = Bi0 ψ00 (μ, x0 , x) + ψ10 (μ, x0 , x), (1)

Φj (μ, x) = (1)

(j)

(1)

xj−1 [Φj−1 (μ, xj−1 )ψ10 (εj , xj−1 , x) + Kλ Φj−1 (μ, xj−1 )ψ00 (εj , xj−1 , x)], 2ψνp (β, x, y) = πβ |ν−p| [Jν (βx)Yp (βy) − Yν (βx)Jp (βy)], ν, p = 0, 1; μ2 N (1) (μ) =

n−1

% & (j) (1) (j) (1) 2 x2j k1 [Φj (μ, xj )]2 + μ2 k2 Φj (μ, xj ) + X (1) (μ),

j=1 ∗(n)

X (1) (μ) = Kλ

(Bi2n + ε2n )x2n Φ2n (μ, xn ) − (Bi20 + μ2 );

148


6 6 6

(1) (j) (j) (1) (j) (j−1) (j) ∗(j) (j) (1) λ0 , Kλ εj = μ Aj , Aj = λ0 c0 (λ0 c0 ), Kλ = λ0 = λ0 λ0 , βj∗ = 1 − Kλ , k1 (j)

xj −

(k) xj−1 ; μm

(j)

∗(j)

= Kλ

(j+1)

(1 − Kλ

∗(j)

(j)

), k2

= Aj Kλ

∗(j+1)

− Aj+1 Kλ

, hj =

are the roots of equations

(k) (k) Φ(k) n (μ, xn ) + Bin Φn (μ, xn ) = 0; 1 1 (2) Bi0 = Bi0 + , Bi(2) , n = Bin − x0 xn (0)

(1.16)

(1)

(1) Bi0 = Bi0 = Bi0 , Bi(0) n = Bin = Bin ; the prime denotes the derivatives with respect to x. In (1.15), the expressions (k) (k) (k) for ∂θ(k) (x, F o) ∂F o, θ1 (x0 , F o), θj+1 (xj , F o), θn (xn , F o) are unknown. In the case when the thermal conductivity coefficient of each layer is independent of the (k) (k) (k) temperature, the functions θ1 (x0 , F o), θj+1 (xj , F o), θn (xn , F o) are unknown in consequence with Wt (x, F o) = 0. If the linear boundary conditions of the first, sec(k) ond, or third kind are given at the interfaces, there is no need to find θ1 (x0 , F o), (k) θn (xn , F o). One way to find the unknowns in relations (1.15) consists in the following procedure. For each k, the Kirchhoff variable θ(x, F o) is given in the form

F o θ(k) (x, F o) =

∂θ(k) (x, ξ) dξ + θ0 (x). ∂ξ

(1.17)

0

Each of the intervals [ xi−1 , xi ] is divided into n∗i parts. We denote the left and right n 0 boundaries of new intervals as x ¯j−1 , x¯j , respectively (j = 1, Nn∗ , Nn∗ = n∗p ), and p=1

x ¯0 = x0 , x ¯Ni∗ = xi . Thus, the resulting integral over ρ from x0 to xn , can be given ¯j . Function Wt (ρ, F o) in these intervals as the sum of integrals from x ¯j−1 to x (j) (k) is replaced by functions Wt [θj (x∗j , F o)], where x∗j = (¯ xj−1 + x ¯j )/2. Each of (k) (j) (k) ∗ ( η = x , η = xi + 0), Wt [θ (x∗ , F o)], the functions ∂θ (x, F o) ∂F o j

x=η

j

j

± Fj+1 [θj+1 (xj , F o)], θc0 (F o), is approximated by linear spline of the form (k)

(1)

(0)

fj (F o) = sj1 F o+sj1 +

K τ −1

(1)

(0)

(1)

(0)

(sj,i+1 F o+sj,i+1 − sji F o−sji )S(F o−F oi ), (1.18)

i=1

where (1)

sji =

fj (F oi ) − fj (F oi−1 ) −fj (F oi )F oi−1 + fj (F oi−1 )F oi (0) , sji = , ΔF oi ΔF oi

ΔF oi = F oi − F oi−1 , i = 1, Kτ , 0 = F o0 < F o1 < F o2 < · · · < F oKτ = F o. Using (1.17), approximations (1.18), formulas [7]–[9] 2y (k) (x, ρ)

∞ (k) (k) Φ(k) (μm , x)Φ(k) (μm , ρ) m=1

(k) (k) 2q+2 N (k) (μm )μm

(k)

= gq+1 (x, ρ, q = 0, 1,

(1.19)


149

relations (1.15), and collocation method, we obtain the recurrent systems of nonlin (k) ∂θ (x, F o) ear algebraic equations to find the values in the spline nodes. ∂F o x=η

Having solved these systems, we obtain the expressions for the Kirchhoff variables by substitution of the found values into (1.15). In formulas (1.19): (k)

(k)

k (k) g1 (x, ρ) = κ∗ (x)[κ(k) (ρ)] − [f (k) (x) − f (k) (ρ)]S(x − ρ), n − xn Bin f (1) λ0 (k) (k) (k) (k) (k) k g2 (x, ρ) = κ∗ (x) (n) g11 (xn , ρ) + xn Bin g12 (xn , ρ) − g12 (x, ρ), λ0 6 (k) (k) (k) (k) κ∗ (x) = [κ0 + xk0 Bi0 f (k) (x)] D(k) , D(k) = xkn Bin κ0 + xk0 Bi0 κn ;

f (0) (x) = x +

n−1

(i)

Hλ (x − xi )S(x − xi ),

i=1 n−1

f (1) (x) = ln x + f (2) (x) = −

1 − x

n−1

(i)

Hλ

i=1

(0)

i=1

1 1 − x xi

(i)

Hλ ln

x S(x − xi ), xi

(1)

λ0

(i)

S(x − xi ), Hλ =

(1)

(i+1)

λ0

(1)

−

λ0

(i)

,

λ0

(2)

κ0 = 1, κ0 = 1 − x0 Bi0 ln x0 , κ0 = 1 + x0 Bi0 , (1)

κ(k) n = (k) g11 (x, ρ)

x = x0

λ0

(n) λ0

+ xkn Bin f (k) (xn ),

c0 (ς)

(k) ς k g1 (ς, ρ)dς, (1) c0

(k) g12 (x, ρ)

x = x0

(1)

λ0 (k) g (ς, ρ)dς. k ζ λ0 (ς) 11

It is worthy to note that formulas (1.19) give the possibility to explain the “Gibbs phenomenon”, which is well known from mathematical analysis. To find the Kirchhoff variables, the iterative methods can be also employed. As the first approximation for the Kirchhoff variables, therewith, the expressions can be accepted, which have been obtained from solving the heat conduction problems for non-thermosensitive material. Having determined the Kirchhoff variables, the temperature field in layered bodies can be described by expression (k)

t(k) (x, F o) = t1 (x, F o) +

n−1

(k)

(k)

[tj+1 (x, F o) − tj (x, F o)]S(x − xj )

(1.20)

j=1

by taking (1.5) into account. Henceforth, we assume xj = zj for a plate, and xj = rj for cylinder and sphere.

150


2. Thermal stresses in the multilayer circular plate The temperature, t(z, F o) = t(0) (z, F o), vary with respect to the thickness only. Thus, the corresponding stresses are [2] σrr = σφφ =

E ∗ (z, F o) E ∗ (z, F o) [C1 (F o) + zC2 (F o)] − Φ∗ (z, F o), ∗ 1 − ν (z, F o) 1 − ν ∗ (z, F o)

(2.1) σzz = σrz = σφz = σrφ = 0, ∗ where ν (z, F o), E (z, F o), and Φ (z, F o) are expressed in the form (1.13), νi∗ (z, F o) = νi (ti ) are the Poisson’s ratios, Ei∗ (z, F o) = Ei (ti ) denote elasticity ti (z,F 3 o) (i) αt (ζ)dζ are the total thermal strains; t∗0 is the temmoduli, Φ∗i (z, F o) = ∗

∗

t∗ 0

perature of plate in ordinary state;

−1 , C1 (F o) = [d1 (F o)a22 (F o) − d2 (F o)a12 (F o)] a11 (F o)a22 (F o) − a212 (F o) −1 , C2 (F o) = [d2 (F o)a11 (F o) − d1 (F o)a12 (F o)] a11 (F o)a22 (F o) − a212 (F o) z z i i n n Ei∗ (z, F o) zEi∗ (z, F o) dz, a dz, a11 (F o) = (F o) = 12 1 − νi∗ (z, F o) 1 − νi∗ (z, F o) i=1 i=1 zi−1

zi−1

a22 (F o) =

n i=1 z

d1 (F o) =

n i=1 z

zi

zi

z 2 Ei∗ (z, F o) dz, 1 − νi∗ (z, F o)

i−1

Ei∗ (z, F o)Φ∗i (z, F o) dz, 1 − νi∗ (z, F o)

i−1

n zi zEi∗ (z, F o)Φ∗i (z, F o) dz. d2 (F o) = 1 − νi∗ (z, F o) i=1 zi−1

3. Thermal stresses in a multilayer cylinder and sphere To determine the thermal stresses, we use the formulas (k) ν (k) (r, F o) u(k) E (k) (r, F o) du +k − Φ∗(k) (r, F o), σr(k) = c(k) (r, F o) dr 1 − ν (k) (r, F o) r 1 − 2ν (k) (r, F o) k−1 (k) u 1 ν (k) (r, F o) du(k) (k) (k) + σφ = c (r, F o) r 1 − ν (k) (r, F o) dr 1 − ν (k) (r, F o) −

E (k) (r, F o)Φ∗(k) (r, F o) , 1 − 2ν (k) (r, F o)

σz(1) = ν (1) (r, F o)(σr(1) + σφ ) − E (1) (r, F o)Φ∗(1) (r, F o), (1)

(3.1)


151

where the displacement u(k) = u(k) (r, F o) satisfies the equilibrium equation for inhomogeneous bodies with piecewise-continuous coefficients, d (k) d (k) du(k) ν (k) (r, F o) u(k) c (r, F o) +k c (r, F o) dr dr dr 1 − ν (k) (r, F o) r +kc(k) (r, F o)

=

1 − 2ν (k) (r, F o) 1 1 − ν (k) (r, F o) r

du(k) u(k) − dr r

E (k) (r, F o) d ∗(k) Φ (r, F o) , dr 1 − 2ν (k) (r, F o)

(3.2)

under the boundary conditions σr(k) (r0 , F o) = σr(k) (rn , F o) = 0. Here c(k) (r, F o) =

(3.3)

E (k) (r, F o)[1 − ν (k) (r, F o)] , E (k) (r, F o), ν (k) (r, F o), [1 + ν (k) (r, F o)][1 − 2ν (k) (r, F o)]

Φ∗(k) (r, F o) are of the form (1.13); (k)

(k) νi (r, F o)

=

(k) νi (ti ),

(k) Ei (r, F o)

=

(k) Ei (ti ),

∗(k) Φi (r, F o)

ti

=

(r,F 3 o)

t∗ 0

(i)

αt (ζ)dζ.

The problems (3.2), (3.3) are solved approximately. For each time-moment, the coefficients E (k) (r, F o) and ν (k) (r, F o) are approximated by piecewise-constant functions of radial coordinate. As a result, we get the thermoelasticity problems n 0 for the multilayered cylinder and sphere, consisting of N = nj layers. Here nj j=1

denotes the number of parts, into which the j -layer of the n-layer-body is divided. By means of the Green’s functions for a multilayered isotropic cylinder and sphere [9], the solutions of the mentioned problems are obtained in the form u(k) p =

r1/k (k) 2qk Qn r

# $ (k) (k) (k) (k) φ2p (r, F o)S1p (F o) + φ1p (r, F o)S2p (F o) (k)

+

1 + ν˜p 1−

(k) ν˜p

1 (k) V (r, F o). r p

Substituting (3.4) into the two first relations (3.1), we obtain (k) = σrp

(k) $ c˜p r1/k # (k1) (k) (k1) (k) g2p (r, F o)S1p (F o) + g1p (r, F o)S2p (F o) (k) 2 2qk Qn r (k)

−

˜p kE 1−

(k) ν˜p

1 (k) V (r, F o), r2 p

(3.4)

152

(k) σφp

R. Kushnir and B. Protsiuk ⎤ ⎡ k−1 (k) (k) c˜p r1/k ⎣ ν˜p (k2) (k) (k2) (k) = g2p (r, F o)S1p (F o) + g1p (r, F o)S2p (F o)⎦ (k) (k) 2qk Qn r2 1 − ν˜p ˜p(k) 1 E (k) ∗(k) V (r, F o) − Φp (r, F o) , p = 1, N. (3.5) + (k) r2 p 1 − ν˜p

In formulas (3.4), (3.5):

qk 2 rp−1 = + , rrp q qk rrp−1 k (k) (k)+ rp−1 (k)− − M2p , φ2p (r, F o) = M2p r rp2 q q qk rp−1 k r k (km) (k)+ rp−1 (k)− (k)+ g2p (r, F o) = −d(k)− M − d M , mp mp 2p 2p r rp rp qk q r rp−1 k rp−1 qk (km) (k)+ (k)− (k)− M − d M , m = 1, 2; g1p (r, F o) = d(k)+ mp mp 1p 1p rp rp r r (k) 1−k ρk Φ∗(k) (ρ, F o)dρ, q1 = 1, q2 = 3/2; Vp (r, F o) = r p

(k) φ1p (r, F o)

(k)+ M1p

r rp

qk

(k)− M1p

rp−1 (k)±

M11

(k)±

= 2qk (qk ∓ βk1 ); M1p

(k0) = Φp−1

r=rp−1

(k) (k1) (qk ± Kp ) ± Φp−1

(k)

r=rp−1

p = 2, n; (k)±

(k1) (k4) (k3) = κ(k2) n,p + βkn κn,p ± qk (κn,p + βkn κn,p ); (k1) Q(k) + Φ(k0) βkn ; n = Φn n r=rn r=rn r 2qk 0 (km) m qkm , Φ1 = qk − βk1 + (−1) (qk + βk1 ) r (k0) (km) (k1) (km) Φ(km) = Φp−1 fp1 + Φp−1 fp2 , p r=rp−1 r=rp−1 2qk r 1 p−1 (km) m−1 (k) m (k) qk + Kp + (−1) (qk − Kp ) fp1 = qk , 2 r 2qk r p−1 (km) (k) 1 1 − (−1)m , m = 0, 1; fp2 = qkm−1 Kcp 2 r (k0) (k1) (k0) (k2) κ(k1) κn−1,p + fn2 κn−1,p , n,p = fn1 r=rn r=rn (k1) (k1) (k1) (k2) = f κ + f κn−1,p , κ(k2) n,p n1 n−1,p n2 r=rn r=rn (k0) (k3) (k0) (k4) κn−1,p + fn2 κn−1,p , κ(k3) n,p = fn1 r=rn r=rn (k1) (k3) (k1) (k4) κn−1,p + fn2 κn−1,p , κ(k4) n,p = fn1

M2p

r=rn

r=rn

Kcp ,

A Method of the Green’s Functions. . . (k1) (k0) κp+1,p = fp+1,1

r=rp+1

(k4) (k1) κp+1,p = fp+1,2

(k2) (k1) , κp+1,p = fp+1,1

r=rp+1

(k)

(k)

n

(k)−

M2j

j=p+1

(k) Pjp

=

(k3) (k0) , κp+1,p = fp+1,2

, r=rp+1

(k4) (k2) (k3) , p < n; κ(k1) n,n = κn,n = 1, κn,n = κn,n = 0;

S1p (F o) =

S2p (F o) = −

r=rp+1

153

p−1

(k)

c˜j

(k) ˜p j=1 c

rj−1 rj

(k)+

M1j

qk

(k) (k)

Pjp Jj (rj , F o),

(k) (k)

(k)−

Ppj Jj (rj−1 , F o) − M2p

Jp(k) (rp , F o);

qk q p (k) ri−1 k 1 + ν˜p −1/k (k) ; Jp(k) (r, F o) = r Vp (r, F o); (k) rp−1 ri 1 − ν˜p i=j+1 rp

(k)

(k) (k) Kp(k) = Kcp βk,p−1 − βkp , Kcp =

c˜p−1 (k)

c˜p

, c˜(k) p =

(k) (k) E˜p (1 − ν˜p ) (k)

(k)

(1 + ν˜p )(1 − 2˜ νp )

;

6 6 (2)− νp(2) − 1) (1 − ν˜p(2) ), d1p = 2(1 − 2˜ νp(2) ) (1 − ν˜p(2) ), 2β2p = (5˜ (2)+

d1p

6 6 6 (2)− (2)+ = (1 + ν˜p(2) ) (1 − ν˜p(2) ), d2p = (2˜ νp(2) − 1) ν˜p(2) , d2p = (1 + ν˜p(2) ) ν˜p(2) ; 6 6 ˜p(1) ), β1p = ν˜p(1) (1 − ν˜p(1) ), d(1)+ mp = 1 (1 − ν 6 m+1 (1 − 2˜ νp(1) ) (1 − ν˜p(1) ); d(1)− mp = (−1) ˜ (k) = E (k) (r∗ , F o), ν˜(k) = ν (k) (r∗ , F o), rp−1 < r∗ < rp . E p p p p p

4. Concluding remarks Special cases of the above-mentioned problems for the layered plates and cylinders were studied in [2, 3, 4]. Particularly in [2, 3], the numerical studies of the temperature and thermal stresses in the specified bodies subjected to the heat flux, are presented for the case n = 3 and constant thermal conductivity coefficients. The temperature field and thermoelastic state have been studied in [4] for the five-layer-plate subjected to the convective-radial heat exchange with neglecting, however, a temperature-dependence of heat and thermal conductivity.

154


References [1] B.V. Protsiuk, V.M. Syniuta, Exact solution of one non-linear heat conduction problem for a multi-layer plate (in Ukrainian). Proceedings of International Conference: Modern problems of mathematics, Kyiv, Instytut mathematyky NAN Ukraine 2 (1998), 247–249. [2] B.V. Protsiuk, Quasi-static temperature stresses in a multi-layer plate under heating by heat flux (in Russian). Theor. i prikl. mehanika 38 (2003), 63–69. [3] R.M. Kushnir, B.V. Protsiuk, V.M. Syniuta, Quasistatic Temperature Stresses in a multilayer thermally sensitive cylinder. Materials Science 40, 4 (2004), 433–445. [4] R.M. Kushnir, B.V. Protsiuk, V.M. Syniuta, Temperature stresses and displacements in a multi-layered plate with nonlinear heat exchange. Materials Science 38, 6 (2002), 798–808. [5] V.M. Judin, Method for solution of heat conduction problems with variable heat transfer coefficient (in Russian). Teplovyje napriazhenija v elementah konstruktzyj 5 (1965), 68–75. [6] B.V. Protsiuk, Method of Green’s function in axially symmetric elasticity and thermoelasticity problems of piecewise homogeneous orthotropic cylindrical bodies (in Ukrainian). Mathematical Methods and Physicomechanical Fields 40, 4 (2000), 94– 101. [7] B.V. Protsiuk, V.M. Suniuta, Green’s Function Method in One-Dimensional NonStationary Heat Conduction Problems for Multilayered Plates. (in Ukrainian) Visnyk Lvivskoho Universytetu. Mathematics and Mechanics 51 (1998), 76–84. [8] B.V. Protsiuk, V.M. Suniuta, The temperature field of a multilayer cylinder in asymptotic thermal mode. Journal of Mathematical Sciences 96, 2 (1999), 3077–3083. [9] B.V. Protsiuk, Application of Green’s function method to determination of thermostressed static of layer transversally-isotropic spherical Bodies (in Ukrainian). Mathematical Methods and Physicomechanical Fields 47, 3 (2004), 95–109. Roman Kushnir and Borys Protsiuk Naukova Str. 3b P.O. Box 19 79060 Lviv, Ukraine e-mail: [email protected] [email protected]


On the Application of the M.G. Krein Method for the Solution of Integral Equations in Contact Problems in Elasticity Theory S.M. Mkhitaryan Abstract. In the present paper the M.G. Krein spectral method of integral equations solutions of the first kind based on his investigations on inverse problems of differential operators spectral theory, is briefly stated. A brief review of basic results on the solution of fairly wide class of integral equations, met in contact problems of elasticity theory is given. Keywords. Inverse problem, the M.G. Krein method, differential system, fundamental function, integral equation, Fourier generalized transformation, contact problem, elasticity theory.

In his investigation [1–4] on inverse problems of spectral theory of differential operators M.G. Krein, particularly developed a theory in which the connection between the integral equations with kernels, depending on the difference of the arguments, and Sturm-Liouville differential operators is explained. This theory permitted us to discover Sturm-Liouville differential equations whole classes, integrated in the final form for any value of the spectral parameter and thus to obtain new formulae of Fourier generalized transformation. Simultaneously, on its base and on the base of Fourier generalized transformation a new method of integral equations with real symmetrical kernels, depending on the difference of the arguments, was suggested. This method, based on the ideas of differential operators spectral theory and generated by them formulae of Fourier generalized transformation, is known as spectral method. The spectral method permits us to obtain (under certain conditions) the solution of such integral equation for any right part, if the solution of this equation at the right part identical to equal unit is known. Later on [5] M.G. Krein used his method of integral equations with kernels of the noted type in Fredholm general integral equations of the first and second kind. M.G. Krein formulae of the integral equations solutions are free from singular integrals, taken in the sense of the principal value according to Cauchy. Besides, the characteristic singularities of the solutions which play an important role in the

156

S.M. Mkhitaryan

applications of the integral equations into the problems of mathematical physics are distinguished in them. By the M.G. Krein method a lot of significant classes of integral equations, coming across in various problems of mathematical physics, particulary, in boundary value problems of potential theory, are effectively solved. The M.G. Krein method was especially widely applied in contact problems of elasticity theory, plasticity theory and nonlinear theory of creep. This methods allowed us to simplify and unificate the solutions of a class of the known contact problems and to get the solutions of a class of new problems, as well. It was applied in N.Kh. Aroutyunyan [6, 7], N.Kh. Aroutyunyan and M.M. Manoukyan [8], G.Ya. Popov [9, 10], I.E. Prokopovich [11], V.M. Alexandrov [12] papers. The detailed presentation of the M.G. Krein general method, outside of this method with the theories of differential operators spectral functions, is given in [13], where the table of the solutions of integral equations for a class of contour kernels is reduced as well. The kernels of this table depend on the absolute value of the arguments difference and are expressed through the elementary transcendental functions. But the domain of the concrete application of the method is not limited by these kernels. In the author’s papers [14, 15] the above-mentioned table is significantly widened and by the M.G. Krein method effective solutions of some classes of integral equations of the first kind with Hermitian kernels, being generalizations of kernels reduced in [13], are built. By these integral equations plane contact problems of elasticity theory, taking into account of the cohesion or friction forces in the contact zone, are described. At first, by contour integral methods their solutions at the right parts identically equal to unit are found. Then starting from M.G. Krein above-mentioned results on differential equations, the differential systems corresponding to these integral equations are composed, their fundamental functions are represented in explicit form and the formulae of Fourier generalized transformation are noted. The fundamental functions equivalent to their canonical systems, from two equations make up in the space quadratically integrable two-dimensional vector functions, perhaps, a new class of full orthogonal systems of functions. In [16] the Krein spectral method was applied to the solution of the integral equation of the first kind with kernel, expressed by a hypergeometrical function, which is represented as Veber-Sonin known integral. Various private cases of this equation are often met in contact problems of elasticity theory and earlier it was solved by N.I. Akhiezer and V.A. Shcerbina in [17]. Yet the application of the spectral method to the solution of the denoted equation is represented more actual and adequate. In the present paper the M.G. Krein spectral method of the linear integral equations solutions (but only for the first kind) is briefly stated. This method is very important and actual at present too and a brief review of the basic results from [14–16], obtained on the base of this methods, is reduced. M.G. Krein one general result, concerning above-denoted connections of integral equations with Hermitian kernels, depending on the difference of the arguments with differential systems, is reduced before hand.

On the Application of the M.G. Krein Method for the Solution. . .

157

Later on, we note that in the contact and mixed problems of elasticity theory the methods of the boundary value problems theory of analytical functions and singular integral equations are widely applied as well [18–21]. More over, in [22–24] by a fairly effective method of orthogonal polynomials is developed G.Ya. Popov. The solutions determining integral equations of the contact problems by M.G. Krein methods the singular integral equations and orthogonal polynomials are represented by the formulae of the various analytical structures. M.G. Krein in due course, raised a question about correlation of these formulae and, as a final result, the question about their identity. In the present paper on the examples of the integral equation with symmetric difference logarithmic kernel, corresponding to the plane contact problem and Carleman equations, it is shown that the formulae of their solution pointed above by three methods, turn one into another and, therefore, they are identical. 1. In the M.G. Krein papers [1–4], particularly, the following results are obtained. Let K(t) = K(−t) (−2T < t < 2T ) be a measurable, locally summable function, satisfying the following condition: at any r (0 ≤ r < T ) integral equation r q(t, r) + K(t − s)q(s, r)ds = 1 −r

has the only bounded, and it means continuous solution q(t, r). This condition is equivalent to the condition of the positive definiteness squared −T < t, s < T of the Hermitian kernel K(t − s) (see [13], Chap. IV §8). Then for any complex λ we put λ r χ(r, λ) = q(s, r)e−iλs ds. (1.1) 2 −r Function χ(r, λ) turn out to be the solution of differential system [3, 4] χ d 1 dχ + [λ2 + 2λl(r)] = 0, dr p(r) dr p(r) χ(0, λ) = 0; where

χ (0, λ) = λ;

(1.2)

d [Arg q(r, r)] (0 ≤ r < T ) dr This differential system can be transformed into a canonical system of two equations, after which the formulae of Fourier generalized transformation for an arbitrary two-dimensional vector-function from L2H (0, T ) are obtained. They have the form > > T > Φ (r, λ) > > > (1.3) F (λ) = f1 (r) , f2 (r) H (r) > > dr > Ψ (r, λ)> 0 > > > > ∞ > f1 (r)> > Φ (r, λ) > > > > > (1.4) F (λ) > > > dσ (λ) ; >=2 > f2 (r)> > Ψ (r, λ)> p(r) = |q(r, r)|2

l(r) = −

−∞

158

S.M. Mkhitaryan

where

> > > 1 > > + V 2 (r) p (r) V (r) p (r)> > >; p (r) H (r) = > > > > V (r) p (r) p (r) r

V (r) = −2

l (t) dt; p (t)

0

and σ(λ) (σ (λ) = σ (λ − 0) , tion, satisfying condition

> > > Φ (r, λ) > > 1 > > > >=> > > Ψ (r, λ)> > −V (r) σ (0) = 0,

∞

−∞

> > > > χ (r, λ) > 0 >> > ? > > dχ >, 1> > λp (r)> > > dr

−∞ < λ < ∞) is a nondecreasing func-

dσ(λ) −1, ν < 1) Γ(1 − ν)tμ−ν+1 0 (t2 − r2 )ν expressing any Bessel function of the first kind through the integral, containing Bessel function with less index. Then equation (2.1)–(2.2) with the help of (2.4) is represented in the form of a ∞ 1−q/2 s ϕ(s)ds Jq/2−1 (λt)Jq/2−1 (λs)λ2p−q+1 dλ = Mp,q tq/2−1 f (t) (2.5) 0

0

according to (1.25). Placing μ = p − 1, ν = 1 + p − q/2 into Sonin formula we get t Jp−1 (λr)rp dr λq/2−1 t1−q/2 Jq/2−1 (λt) = q/2−p−1 , 2 Γ(q/2 − p) 0 (t2 − r2 )1+p−q/2 where under conditions (2.3) by Abel inversion formula we find d t Jq/2−1 (λr)rq/2 dr 2q/2−p t−p q/2−p λ Jp−1 (λt) = . Γ(1 + p − q/2) dt 0 (t2 − r2 )q/2−p

(2.6)

Functions Jq/2−1 (λt) and Jq/2−1 (λs) entering the inner integral of the left part (2.5) are not orthogonal with weight λ2p−q+1 . With this weight functions √ √ λq/2−p tJp−1 (λt) λq/2−p sJp−1 (λs), are orthogonal as ∞ √ √ λq/2−p tJp−1 (λt)λq/2−p sJp−1 (λs)λ2p−q+1 dλ = δ(t − s), 0

which directly generates from Hankel integral transformation formulae. On the other hand the transformation operator which is given by formulae (2.6), is known. These considerations note that the solution of equation (2.5),

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S.M. Mkhitaryan

i.e., equations (2.1)–(2.3) may be obtained by the M.G. Krein spectral method, described in the previous item. Having applied the above-described procedure to equation (2.5), its solution will be represented by formula a 1−2p # a1−2p F (a) $ tq−1 [s F (s)] ϕ(t) = − ds . 2 2 q/2−p Γ(1 + p − q/2) (a2 − t2 )q/2−p t (s − t ) where

2Γ(p)sin[π(p − q/2)] d t f (r)rq−1 dr . πΓ(q/2) dt 0 (t2 − r2 )q/2−p This formulae after elementary transformations coincides with the corresponding formulae from [17]. In [16] important private cases of kernel (2.2) are considered, when Carleman, Hilbert-Riesz and Hertz kernels are obtained. The integral equations with such kernels are often met in contact problem of elasticity theory and creep theory. Then by the M.G. Krein general method [5, 13], in papers [14, 15] the integral equations solutions of the first kind are built a K(t − s)ϕ(s)ds = f (t), (2.7) F (t) = −

−a

where Hermitian kernel K(t−s) is generated by one of the following seven functions K(t) (−2T < t < 2T ): iπ 1 − th (πμ) sign t (T ≤ π; −∞ < μ < ∞) ; 1) ln 2 sin (|t|/2) 2 √ 1 iπ 2) ln − th (πμ) sign t T ≤ 2ln 1 + 2 ; −∞ < μ < ∞ ; 2sh (|t|/2) 2 |t| iπ − th (πμ) sign t (T ≤ π; −∞ < μ < ∞) ; 3) lnctg 4 2 |t| iπ 4) lncth − th (πμ) sign t (T ≤ ∞; −∞ < μ < ∞) ; 4 2 πh h ; T = ∞; 0 < h < 1, |μ| < th 5) (1 − iμsign t)/|t| 2 h ? |t| πh iht/2 6) (1 − iμsign t) e ; 2sin T ≤ π; 0 < h < 1, |μ| < tg 2 2 h ? |t| πh 7) (1 − iμsign t) eiht/2 . 2 sh T ≤ ∞; 0 < h < 1, |μ| < tg 2 2 Integral equations (2.7) with such kernel functions arise in contact problems of elasticity theory and nonlinear theory of established creep with power physical law, when in the contact zone the forces of cohesion and friction are taken into account. In [14, 15] corresponding to integral equations (2.7) differential systems (1.2) are composed, their fundamental functions, which are expressed by known special


165

functions, are found and Fourier generalized transformation formulae (1.3) and (1.4) are written. On this way by the unique method the formulae of several known integral transformations are again obtained, and in some cases their generalizations are obtained. For example expansion formulae of two-dimensional vector-function from L2 (0, ∞) by Whittaker functions, which in private case turn into Hankel integral transformation known formula. 3. We pass on to the question of correlation and identity of formulae of the integral equations solution, obtained by the M.G. Krein method and by other methods. Here we shall be confined to two integral equations, met in plane contact problems of elasticity theory and nonlinear creep theory or in contact problems for linearlydeformable foundation. The solution of plane contact problem of elasticity theory on compression in vertical direction of two elastic bodies with forces statically equivalent to force P and a pair of forces with moment M , in the frame-work of Hertz’s known hypothesis and in case of one locality of contact [−a; a] is reduced to the integral equation solution [18] a 1 p(s)ds = f (x) (3.1) ln |x − s| −a under the conditions a a p(x)dx = P ; xp(x)dx = M. (3.2) −a

−a

Here f (x) = [c + αx − f1 (x) − f2 (x)](ϑ1 + ϑ2 )−1 ;

ϑi = 2(1 − νi2 )/πEi

(i = 1, 2),

where Ei , νi (i = 1, 2) are the elastic constants contacting among themselves of the bodies, fi (x)(i = 1, 2) are the functions, characterizing the surfaces of compressed bodies up to deformation, α is the angle of the mutual rotation of the bodies from each other relatively subject to determination of the constant depending on the degree of mutual approach of bodies, p(x) is the unknown normal pressure in the zone of contact. Placing (3.1) into equation f (x) = f+ (x) + f− (x),

p(x) = p+ (x) + p− (x)

(x ∈ (−a; a))

p± (−x) = ±p± (x)) (f± (−x) = ±f± (x), we expand the input contact problem on symmetrical problem, described by equation a 1 p+ (s)ds = f+ (x) ln 2 (x ∈ (0; a)) (3.3) | x − s2 | 0 and on skew-symmetric problem, described by integral equation a x+s p− (s)ds = f− (x) ln (x ∈ (0; a)) (3.4) | x−s| 0

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In the given case the solution of integral equation (1.8), when K(|t − s|) = ln(1/|t − s|), and function M (r) from (1.6) has the form [1, 5, 13] q(t, r) =

1 1 √ , 2 π ln(2/r) r − t2

M (r) =

1 . 2 ln(2/r)

(3.5)

Taking into account (3.5), with the help of the M.G. Krein formula (1.22) the solution of integral equation (3.3) after not complicated transformations may be expressed in the form of (0 < x < a) u a J(a) 2 du d # d f+ (s)ds $ 1 √ √ √ p+ (x) = u − 2 π ln(2/a) a2 − x2 π x u2 − x2 du du 0 u 2 − s2 u u # f (s)ds f (s)ds $ 2 d √+ √+ J(u) = (3.6) + u ln(2/u) π 0 du 0 u 2 − s2 u 2 − s2 and the solution of equation (3.4) in the form of (0 < x < a) a u udu df (s) 2 d √ √ p− (x) = − 2 . π dx x u2 − x2 0 u 2 − s2

(3.7)

Note that formula (3.6) for the solution of equation (3.1) in symmetric case was first obtained by N.A. Rostovtsev [26]. Later assume that f (x) ∈ C1 [−a; a] (C1 [−a; a] is a class of continuous and continuously differentiable on the segment [−a; a] functions), and also, derivative f (x) in the interval (−a; a) satisfies the Hölder condition. Then having differentiated the both parts of equation (3.1), we come to a singular integral equation a p(s)ds = f (x) (|x| < a) (3.8) −a s − x with Cauchy kernel where the integral at s = x is understood in the sense of the principal value by Cauchy. The solution of equation (3.8) under the first condition (3.2) has the form [18, 20] (−a < x < a) a √ 2 1 a − s2 f (s)ds P p(x) = − √ + √ . (3.9) s−x π 2 a2 − x2 −a π a2 − x2 At last, supposing that function f (x) in the interval (−a, a) satisfies the Dirichlet ordinal condition, we place f (x) =

∞

an Tn (x/a)

(−a < x < a),

(3.10)

n=0

where Tn (x) are Chebishev polynomials of the first kind. Then using the known spectral relationships [24] ⎧ a π ln (2/a) (n = 0) ; 1 Tn (s/a) ds ⎨ √ (|x| < a) (3.11) ln = π ⎩ Tn (x/a) (n = 1, 2, . . . |x − s| a2 − s2 n −a


167

the solution of equation (3.1) with the first condition (3.2) can be represented by formula ∞ # $ 1 p(x) = √ P+ nan Tn (x/a) (|x| < a). (3.12) π a2 − x2 n=1 Note, that unknown parameter α, which may be determined from the second condition (3.2) with the help of (3.12) or (3.9), linearly enters coefficient a1 . Thus, the solution of integral equations (3.1), built with the above-mentioned three methods, is given with formulae (3.6)–(3.7), (3.9) and (3.12). Later on, assuming, that f (x) ∈ C1 [−a; a] and the second derivative f (x) in interval (−a; a) satisfies Dirichlet conditions, we shall show that in this class of functions formulae (3.6)–(3.7) and (3.9) are reduced to formula (1.17). Hence it will be shown that in spite of the difference of the analytical types of formulae (3.6)–(3.7), (3.9) and (3.12), they coincide. In fact, with the assumptions made relatively to f (x), we have an = 0(n−3 ) at n → ∞ and, therefore, series (3.10) may be differentiated term by term: f (x) = a−1

∞

nan Un−1 (x/a)

(| x |< a),

(3.13)

n=1

where Un−1 (x) (n = 1, 2, . . . ) are Chebishev polynomials of the second kind. Substituting now (3.13) into (3.9) and using the known relationship from [27] (p. 188, formula (48)) we come to formulae (3.12). In order to show that formula (3.12) arises from the M.G. Krein formula (3.6)–(3.7), at first we place f+ (x) = T2n (x) (n = 1, 2, . . . ), into (3.6), where, without loss of generality, we can admit a = 1. Then for the integral calculation u In (u) = 0

T2n (s) ds √ = u 2 − s2

1 0

T2n (ut) dt √ (s = ut; n = 1, 2, . . .) 1 − t2

the known formulae from [28] (p. 849, form. 7.349; p. 1050, form. 8.961.8) may be used, with formulae T2n (t) = Tn (2t2 − 1) (n = 0, 1, 2, . . . ) they will make π (n = 1, 2, . . .) , In (u) = Pn(−1,0) 2u2 − 1 2 (α,β)

where Pn (x) (n = 0, 1, 2, . . . ) are the Jacobi polynomials. Later, having successively calculated the rest of the necessary integrals [29, 30] from (3.6) we get p+ (x) =

2nT2n (x) √ π 1 − x2

(0 < x < 1, n = 1, 2, . . .) ,

which is equivalent to spectral relationship (3.11) at even index of n. In the same way starting from formula (3.7) and assuming a = 1, f (x) = T2n−1 (x), we obtain (3.11) at odd indexes of n. Now, if we substitute class (3.10) into (3.6)–(3.7), we come to formula (3.12) at once.

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Let us pass to the second equation. A lot of contact and mixed problems of elasticity theory, when the base elasticity module by vertical coordinate changes by the power law, and the Poisson ratio is constant [23] or in the first approach of nonlinear theory of the established creep at degree dependence between the stresses intensities and deformations velocities [7], are reduced to the Carleman integral equation [31] a ϕ(s)ds = f (t) (0 < h < 1). (3.14) |t − s|h −a Later, as above, after representation of the right part f (t) of equation (3.14) in the form of even sum and odd function, we come to the following equations: a# $ 1 1 ϕ± (s)ds = f± (t) ± (0 < t < a). (3.15) |t − s|h (t + s)h 0 Now, with the M.G. Krein formula (1.22) the solution of equation (3.15) with the even right part f+ (t) has the form [13] ,

ϕ+ (t) = C (h) a

a −t

1−h

2

2 (h−1)/2

a J (a) −

ξ 2 − t2

h−1/2

t

×

: d 1−h d ξ J (ξ) dξ ; J (ξ) = dξ dξ

ξ

ξ 2 − s2

(h−1)/2

(3.16) f+ (s) ds

0

C (h) = π −5/2 Γ (h/2) Γ [(1 − h)/2] cos2 (πh/2)

(0 < t < a) ;

and by the M.G. Krein formula (1.23) the solution of equation (3.15) with the odd right part f− (t) has the form of d ϕ− (t) = −C (h) dt

(h−1)/2 ξ a f− (s) ds t2 1− 2 dξ . (1−h)/2 2 ξ (ξ − s2 ) t

(3.17)

0

In formulae (3.16)–(3.17) again without restriction of generality we place a = 1. Later according to the well-known results [18, 20] by the method of singular integral equations the solution of equation (3.14) is reduced to the solution of the following Abelian integral equation: t

ϕ (s) ds

=

h

−1

(t − s) 1

× −1

(1−h)/2 1 1 f (t) − ctg (πh/2) 1 − t2 2 2π

2 (h−1)/2

1−s

s−t

(3.18) f (s) ds

(−1 < t < 1) .


169

Now if we place ∞

f (t) =

an Cnh/2 (t)

(−1 < t < 1)

(3.19)

n=0

and use well-known spectral relationship [22, 24] 1

h/2

Cn h

−1

(s) ds (1−h)/2

|t − s| (1 − s2 )

= λn Cnh/2 (t)

(−1 < t < 1)

(3.20) λn = πΓ (n + h) [n!Γ (h) cos (πh/2)]−1 , where are Gegenbauer polynomials, then the solution of integral equation (3.14) may be substituted by infinite series Cnλ (t)

∞ (h−1)/2 h/2 ϕ (t) = 1 − t2 an λ−1 (t) n Cn

(−1 < t < 1) .

(3.21)

n=0 h/2

h/2

on the other hand, if f+ (t) = C2n (t) is set into (3.16), f− (t) = C2n−1 (t) (n = h/2 1, 2, . . . ) into (3.17), and f (t) = Cn (t) into (3.18), then as it is shown in [32, 33] we get relationship (3.20). In the same place it is shown, that from the M.G. Krein formulae (3.16)–(3.17) with the help of (3.19) we obtain the solution of equation (3.18) and formula (3.21). In the conclusion we note, that the M.G. Krein method may be also applied to the solution of dual integral equations and dual series-equations [34–36]. Note, too, that the M.G. Krein formulae of the integral equations solution, as it is represented, contain most full information on the solution structure. Acknowledgment Many thanks to L. Ghulghazaryan, A. Khachatryan, A. Melikyan, M. Minasyan for their kindly help in design of the paper.

References [1] M.G. Krein, On one method of effective solution of inverse boundary value problem. Reports of AS USSR, v. 94, 6, 1954, 987–990. [2] M.G. Krein, On integral equations, generating differential equations of the second order. Reports of AS USSR, v. 97, 1, 1954, 21–24. [3] M.G. Krein, On the determination of the particle potential by its S-function. Reports of AS USSR, v. 105, 3, 1955, 433–436. [4] M.G. Krein, Continual analogs of the proposals on polynomials orthogonal on the unit circle. Reports of AS USSR, v. 105, 4, 1955, 637–640. [5] M.G. Krein, On one new method of linear integral equations solution of the first and second kind. Reports of AS USSR, v. 100, 3, 1955, 413–416. [6] N.Kh. Aroutjunyan, Plane contact problem of plasticity theory with power hardening material. Izvestija AS Arm SSR, Series Phys. Math. Sci., v. 12, 2, 1959, 77–105.

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[7] N.Kh. Aroutjunyan, Plane contact problem of creep theory. Applied Math. and Mech., v. 23, 2, 1959, 901–924. [8] N.Kh. Aroutjunyan, M.M. Manukyan, Contact problem of creep theory with account of friction forces. Applied Math. and Mech., v. 27, 5, 1963, 813–820. [9] G.Ya. Popov, On one method of axially symmetric contact problem solution of elasticity theory. Applied Math. and Mech., v. 25, 1, 1961, 76–85. [10] G.Ya. Popov, On one approach method of some plane contact problems solution of elasticity theory. Izvestija AS Arm. SSR, Series Phys. Math. Sci. v. 14, 3, 1961, 81–96. [11] I.E. Prokopovich, On the solution of plane contact problem with account of creep. Applied Math. and Mech., v. 20, 6, 1956, 680–687. [12] V.M. Alexandrov, To the solution of some contact problems of elasticity theory. Applied Math. and Mech., v. 27, 5, 1963, 970–974. [13] I.Ts. Gokhberg, M.G. Krein, Volterra operators theory in Hilbert space and its applications. M.: Nauka, 1967, 508p. [14] S.M. Mkhitaryan, On some classes effective solution of linear integral equations of the first kind and connected with them differential equations. Reports of AS Arm. SSR, v. 48, 2, 1969, 71–78. [15] S.M. Mkhitaryan, On some plane contact problems of elasticity theory, with account of cohesion forces and connected with them differential equations. Izvestija AS Arm. SSR, Mechanics, v. 21, 5–6, 1968, 3–20. [16] S.M. Mkhitaryan, On Akhiezer and V.A. Shcherbin inversion formulae of some singular integrals. Math. Investigations, v. 3, edition 1(7), 1968, 61–70. [17] N.I. Akhiezer, V.A. Shcherbina, On inversion of some singular integrals. Notations of the Math. Dept, Phys.-Math. Faculty and Kharkov Mathem. Society, v. 25, ser. 4, 1957, 191–198. [18] I.Ya. Shtaerman, Contact problem of elasticity theory. M.-L.-d: Gostekhteorizdat, 1949, 270p. [19] L.A. Galin, Contact problems of elasticity theory and viscoelasticity. M.: Nauka, 304. [20] F.D. Gakhov, Boundary value problems. s-M.: Nauka, 1974, 640p. [21] N.I. Mouskhelishvili, Some basic problems of mathematical elasticity theory. M.: Nauka, 1966, 708p. [22] G.Ya. Popov, Some properties of classical polynomials and their application to contact problems. Applied Math. and Mech., v. 27, 5, 1963, 821–832. [23] G.Ya. Popov, Contact problems for linear-deformable foundation. Kiev-Odessa: Vishcha shcola, 1982, 168p. [24] G.Ya. Popov, Concentration of elastic stresses near punches, cuts, thin inclusions and confirmations. M.: Nauka, 1982, 344p. [25] G.N. Watson, Theory of Bessel functions. Part 1, M.: IL, 1949. [26] N.A. Rostovtsev, To the solution of a plane contact problem. Applied Math. and Mech., v. 17, 1, 1953, 99–106. [27] H. Beiteman, A. Erdely, Higher transcendental functions. V. 2. M.: Nauka, 1974, 295p.


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[28] I.S. Gradshtein, I.M. Righik, Integrals, sums, series and products tables. M.: Nauka, 1971, 1108p. [29] S.M. Mkhitaryan, On various methods of solution of the integral equation of plane contact problem of elasticity theory. Reports of AS Arm. SSR. v. 89, 2, 1989, 69–74. [30] S.M. Mkhitaryan, M.A. Abdu, On comparison of various methods of integral equations solution of plane contact problem of elasticity theory. Reports of AS Arm. SSR, v. 90, 2, 1990, 75–80. ¨ [31] T. Carleman, Uber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Mmath. M., Bd. 15, 1922, 111–120. [32] S.M. Mkhitaryan, M.A. Abdu, On various methods of Karleman integral equations solution. Reports of AS Arm. SSR, v. 89, 3, 1989, 125–129. [33] S.M. Mkhitaryan, M.A. Abdu, On comparison of various methods of Karleman integral equation solution, met in elasticity theory. Reports of AS Arm SSR, v. 90, 1, 1990, 6–10. [34] I.A. Tseitlin, On the method of dual integral equations and dual series and on its applications to the problems of mechanics. Applied Math. and Mech., v. 30, 2, 1966, 259–270. [35] A.A. Babloyan, S.M. Mkhitaryan, To the solution of some “triple” equations with trigonometrical functions. Information of AS Arm. SSR, Mechanics, v. 22, 6, 1969. [36] S.M. Mkhitaryan, On some full orthogonal systems of functions and on their applications to the solution of two types of dual series-equations. Information of AS Arm. SSR, Mechanics, v. 29, 2, 1970, 5–21. S.M. Mkhitaryan M.Baghramyan ave., 24b, Institute of Mechanics of NAS Armenia, 375019 Yerevan, Armenia e-mail: [email protected]



The Stress Concentration in the Neighborhood of the Spherical Crack Inside the Infinite Elastic Cone Gennadiy Popov and Nataly Vaysfel’d Abstract. The aim of this work is to estimate the stress intensity factor near the spherical crack inside the infinite cone under the compressing load on the cone vertex. It is supposed that on the cone surface the conditions of the first main elasticity problem are fulfilled. Mathematics Subject Classification (2000). Primary 74B05; Secondary 14B05. Keywords. Spherical crack, cone, integro-differential equation, orthogonal polynomial method.

1. The problem statement The problems on the stress concentration in the neighborhood of the spherical form defects, situated inside the elastic conical bodies, were considered earlier [1]–[3]. They were solved for the torsion load in the dynamic statement. In this work the solution of the axisymmetric problem on stress concentration near the spherical crack, weakened the infinite elastic cone under the compressing load on the cone vertex, is proposed. The compressing load of intensity P is applied to the vertex of the infinite elastic cone (the shear module is G, Poisson coefficient is μ). The conditions of the first main elasticity problem are fulfilled on the cone surface, described in the spherical coordinates by the correspondences, 0 ≤ r ≤ ∞,

0 ≤ θ ≤ ω,

−π ≤ ϕ ≤ π,

τrθ (r, θ)θ=ω = 0, σθ (r, θ)θ=ω = 0.

(1.1)

Only the two components of the displacement vector are different from zero, ur (r, θ), and uθ (r, θ).

174

G. Popov and N. Vaysfel’d The spherical crack on the surface r = R,

0 ≤ θ ≤ ω0 (0 < ω0 < ω),

−π ≤ ϕ ≤ π

is situated on the distance R from the cone vertex. It is supposed that the crack branches are in the smooth contact, i.e., τrθ (r, θ)r=R+0 = τrθ (r, θ)r=R−0 = 0, τrθ (R, θ) = 0, θ ∈ (0; ω0 ).

(1.2)

Here and feather f (R, θ) = f (R − 0, θ)−f (R + 0, θ) . The compressing behavior of the applied load allows to write the rest of conditions on the crack (θ ∈ (0; ω0 ): ur (R − 0, θ) = ur (R + 0, θ) , ur (R, θ) = 0, uθ (R − 0, θ) − uθ (R + 0, θ) = uθ (R, θ) = X(θ),

(1.3)

σr (R − 0, θ) = σr (R + 0, θ) , σr (R, θ) = 0, where X(θ) is the unknown jump of the displacement uθ (r, θ), which required to be estimated. Also it is needed to estimate the stress τrθ (r, θ) intensity factor in the crack neighborhood.

2. The problem reduction to the solving of the integro-differential equation The stress and displacement fields are searched as the superposition of the continuous and discontinuous field vectors T (r, θ) = T M (r, θ) + T ∗ (r, θ). The components of the continuous field T M (r, θ) constructed for the case of the defect absence inside the cone, was obtained in [4] Bctg 2θ , r B B cos θ σrM = −2G 2 , σθM = 2G 2 , r r (1 + cos θ) ω 2GB sin θ M τrθ , B = −A cos2 , (r, θ) = 2 r (1 + cos θ) 2 2P (μ + 2G) A= . μ (1 − cos3 ω) + G (1 − cos ω) (1 + cos2 ω) uM r (r, θ) =

B , r

uM θ (r, θ) = −

(2.1)

For the discontinuous field construction, when it taken in consideration the presence of the crack inside the cone, let’s apply to equilibrium equations, written in the spherical coordinates [5] / • / v sin θ (u• sin θ)• (v sin θ)• 2 / μ∗ r u + + μ0 r = 0, − 2μ∗ u − (μ0 + 2) sin θ sin θ sin θ • • / v (v sin θ) r2 v / + μ∗ + + μ0 ru/• + 2μ∗ u• = 0, (2.2) sin θ sin2 θ u (r, θ) ≡ ur (r, θ) , v (r, θ) ≡ uθ (r, θ) , μ∗ = 1 + μ0 , mu0 = (1 − 2μ)−1 .

The Stress Concentration in the Neighborhood of the Spherical. . .

175

the integral transforms, obtained in work [6]. To the first equation in (2.2) let’s apply the transform ω uk (r) = u(r, θ)Pν0k (cos θ) sin θdθ, k = 0, 1, 2, . . . (2.3) 0

and to the second one – ω v(r, θ)Pν1k (cos θ) sin θdθ, k = 1, 2, . . .

vk (r) =

(2.4)

0

where νk are the roots of the transcendental equation Pν1 (cos ω) = 0, ν = νk , k = 0, 1, 2, . . . , ν0 = 0.

(2.5)

(Here and feather a stroke above a symbol defines the derivative with respect to a first variable, and a dot – the derivative with respect to a second variable.) In the transform space the equation system is obtained: / / / μ∗ r2 uk (r) − (2μ∗ + Nk )uk (r) + (μ0 + 2)vk (r) − μ0 rvk (r) = ((μ0 + 2)v(r, ω) − u• (r, ω) − μ0 rv / (r, ω))Pν0k (cos ω) sin ω, / / / r2 vk (r) + μ∗ Nk vk (r) + μ0 rNk uk (r) + 2μ∗ Nk uk (r)

(2.6)

= μ∗ v(r, ω)(Pν1k (cos ω))• sin ω Nk = νk (νk + 1) . The boundary conditions (1.1) should be fulfilled. Let’s rewrite them in the following form r2 v / (r, ω) − v (r, ω) + u• (r, ω) = 0, (2.7) / • r2 u(r, θ) (v • (r, θ) + u(r, θ)) (v(r, θ) sin θ) + + μμ0 θ=ω θ = ω = 0. r2 r sin θ r The function u• (r, ω) is expressed from the first equation (2.7) and is substituted to the right-hand part of the first equation in (2.6). So, only two functions v(r, ω), v / (r, ω) should be defined feather. Temporally we take them as known. The integral Mellin transform is applied by the generalized scheme [7] to the obtained system (2.6) ∞ fsk = fk (r)rs−1 dr (2.8) 0

with the inversion formula 1 fk (r) = 2πi

γ+i∞

fsk r−s ds.

γ−i∞

176

G. Popov and N. Vaysfel’d

As a result, we get the equation system, which is written in transform space: (μ∗ s(s − 1) − 2μ∗ − Nk ) usk + (μ0 + 2 + μ0 s) vsk @ A / = μ∗ (s − 1)Rs uk (R) − μ∗ Rs+1 uk (R) + μ0 Rs vk (R) + [(μ∗ + 2μμ0 s) vs (ω) − 2μμ0 Rs (v(R − 0, ω) − v(R + 0, ω))] Pν0k (cos ω) sin ω, (2μ∗ − μ0 s)Nk usk + (s(s − 1) − μ∗ Nk ) vsk @ A / = (s − 1)Rs vk (R) − Rs+1 vk (R) − μ0 Rs Nk uk (R) • + μ∗ vs (ω) Pν1k (cos ω) sin ω.

(2.9)

With regard of the first condition on the crack (1.3) we have uk (R) = 0, and with regard of the second one – vk (R) = Xk . From the condition of the A @ tangent / stress equality to zero on the crack branches we obtain the relation R vk (R) = Xk . With this purpose the integral transform (2.4) is applied to this condition. From@ the normal stress jump equality to zero we obtain another one relation, A / Rμ∗ uk (R) = 2μμ0 Xk , in advance the integral transform (2.3) is applied to it. So, the algebraic system of equation is written in the form (μ∗ s(s − 1) − 2μ∗ − Nk ) usk + (μ0 + 2 + μ0 s) vsk = Rs Xk + (μ∗ + 2μμ0 s) vs (ω)Pν0k (cosω) sin ω, (2μ∗ − μ0 s) Nk usk + (s(s − 1) − μ∗ Nk ) vsk • = (s − 2)Rs Xk + μ∗ vs (ω) Pν1k (cosω) sin ω. Its solution is the following: , usk = Δ−1 α (s, k) Rs Xk + β (s, k) Pν0k (cos ω) sin ωvs (ω) & • −λ (s, k) Pν1k (cos ω) sin ωvs (ω) , % ˜ (s, k) Rs Xk + β˜ (s, k) Pν0k (cos ω) sin ωvs (ω) vsk = Δ−1 α & ˜ (s, k) P 1 (cos ω) • sin ωvs (ω) , −λ νk

(2.10) (2.11)

α (s, k) = −2μμ0 s2 + (μ0 − 3) s + 4 + 2μ0 − μ∗ Nk , β (s, k) = 2μμ0 s3 + 2s2 − μ∗ (1 + 2μμ0 Nk ) s − μ2∗ Nk , λ (s, k) = μ∗ (μ0 s + μ0 + 2) , α ˜ (s, k) = μ∗ s3 − 3μ∗ s2 + 2μμ0 Nk s + 4μ∗ − 2μ0 Nk , β˜ (s, k) = Nk 2μμ2 s2 + (1 − 4μ) μ0 μ∗ s − 2μ2 , 0

∗

˜ (s, k) = μ∗ (μ∗ s(s − 1) − 2μ∗ − Nk ) , λ Δ = (s − νk ) (s − νk − 2) (s + νk + 1) (s + νk − 1) . Let’s apply to the obtained displacement transforms (2.11) the inverse Mellin transform (2.8) taking into consideration that denominator have the isolated singularities s1 = νk ; s2 = νk + 2; s3 = −νk − 1; s4 = −νk + 1. With the help of


177

the residue theorem, we get the expression for the displacement transforms (2.3), (2.4) in the form ∞ uk (r) = Xk Ak (r) +

Pν0k (cos ω) sin ω

v (ρ, ω) 0 Ak (r, ρ) dρ ρ

0

• + Pν1k (cos ω) sin ω

∞

v (ρ, ω) 1 Ak (r, ρ) dρ, ρ

0

∞ vk (r) = Xk Bk (r) +

Pν0k (cos ω) sin ω

v (ρ, ω) 0 Bk (r, ρ) dρ ρ

0

• + Pν1k (cos ω) sin ω + Ak (r) =

A0k (r, ρ) =

A1k (r, ρ) =

⎧ ⎨ ⎩ ⎧ ⎨ ⎩

∞

v (ρ, ω) 1 Bk (r, ρ) dρ, ρ 0 2 R νk a1 + a2 Rr ,R < r r , r νk r R −a , R>r − a 3R 4r R ρ νk ρ 2 a ,ρ < r + a 5 6 r r νk , r −a7 ρr − a8 ρr , ρ > r ρ ρ νk 2 a9 + a10 ρr ,ρ < r r νk , r −a11 ρr − a12 ρr , ρ > r ρ

(2.12)

here ai , i = 1, 12 are the residuals of the functions α(s, k), β(s, k), λ (s, k) in the simple poles si , i = 1, 4. The coefficients Bk (r), Bk0 (r, ρ), Bk1 (r, ρ) have the analogues structure to the coefficients Ak (r), A0k (r, ρ), A1k (r, ρ) structure, but vice the coefficients ai , i = 1, 12 we should take bi , i = 1, 12, which are the residuals of ˜ k), λ ˜ (s, k) in the simple poles si , i = 1, 4. the functions α ˜ (s, k), β(s, To the obtained displacement transforms the inverse formulas of integral transforms (2.3), (2.4) are applied u(r, θ) = v(r, θ) =

∞ k=1 ∞

> >−2 uk (r)Pν0k (cos θ) >Pν0k (cos θ)> , > >−2 v k (r)Pν1k (cos θ)−2 >Pν1k (cos θ)> .

k=1

And so, ω0 X (η) sin η

u(r, θ) = 0

∞ k=1

> >−2 Ak (r)Pν1k (cos η) Pν0k (cos θ) >Pν0k (cos θ)> dη

(2.13)

178


+

∞

Pν0k

(cos ω) sin ωPν0k

> >−2 (cos θ) >Pν0k (cos θ)>

∞

k=1

+

∞

v (ρ, ω) 0 Ak (r, ρ) dρ ρ

0

Pν1k

∞

> >−2 • (cos ω) sin ωPν0k (cos θ) >Pν0k (cos θ)>

k=1

0

ω0 X (η) sin η

v(r, θ) =

∞

> >−2 Bk (r)Pν1k (cos η) Pν1k (cos θ) >Pν1k (cos θ)> dη

∞

Pν1k

(cos θ) sin ωPν0k

> >−2 (cos ω) >Pν1k (cos θ)>

∞

k=1

+

(2.14)

k=1

0

+

v (ρ, ω) 1 Ak (r, ρ) dρ, ρ

∞

v (ρ, ω) 0 Bk (r, ρ) dρ ρ

0

Pν1k

(cos θ) sin ω

Pν1k

>−2 • > (cos ω) >Pν1k (cos θ)>

k=1

∞

v (ρ, ω) 1 Bk (r, ρ) dρ. ρ

0

During these formulas obtaining it was taken in consideration that ω0 X (η) Pν1k (cos η) sin ηdη.

Xk = 0

As we can see, in the right-hand part of these expressions there are two unknown functions X (η) and v (ρ, ω). For the expressing the last function from the previous one X (η) let’s fulfill the condition of the normal stress equality to zero on the cone surface (it is the second equality in the boundary conditions (2.7)). With this aim the Mellin transform is applied to it. As a result, the formula connecting the Mellin transforms of the displacements, is obtained vs (ω) = −

2μ0 (1 − μs) us (ω) + (μ0 + 1) vs• (ω) . 2μμ0 ctgω

(2.15)

Let’s find the transforms in the correspondence (2.15) by the following procedure. With the help of the conjugation theorem for Mellin transform the forward Mellin transform (2.8) is applied to the formulas (2.14) ω0 X (η) sin ηRs

us (θ) =

∞

> >−2 Ask Pν1k (cos η) Pν0k (cos θ) >Pν0k (cos θ)> dη

k=1

0

+ Vs (ω) sin ω × Ask =

#

∞

> >−2 Pν0k (cos θ) >Pν0k (cos θ)>

k=1

Pν0k

$ • (cos ω) A0sk + Pν1k (cos ω) A1sk ,

−a2 −a3 −a4 −a1 + + + , −νk + s −νk + s − 2 νk + s + 1 νk + s − 1

(2.16)

The Stress Concentration in the Neighborhood of the Spherical. . . A0,1 sk =

179

−a6,10 −a7,11 −a8,12 −a5,9 + + + , −νk + s −νk + s − 2 νk + s + 1 νk + s − 1 ∞ Vs (ω) = v(ρ, ω)ρs−1 dρ. 0

The analogical formula is obtained for the displacement transform vs (ω), and then passage to the limit θ = ω was done. As a result, with regard of formula (2.5), the following equality have place: ω0 ∞ >−2 • > • vs (ω) = X (η) sin ηRs Bsk Pν1k (cos η) Pν1k (cos ω) >Pν0k (cos θ)> dη k=1

0

+ Vs (ω) sin ω

∞

>−2 • > Pν1k (cos ω) >Pν1k (cos θ)>

k=1

# • 1 $ 0 × Pν0k (cos ω) Bsk + Pν1k (cos ω) Bsk

(2.17)

0,1 where coefficients Bsk, Bsk have the analogical structure to coefficients Ask, A0,1 sk , where vice ai , i = 1, 12 it is necessary to take bi , i = 1, 12. Let’s substitute the equalities (2.16) and (2.17) in the equality (2.15), from the last one the transform Vs (ω) will be found ω ∞ 30 0 Pν1k (cos η) Lsk dη Rs X (η) sin η k=1 0 , (2.18) Vs (ω) = ∞ 0 sin ω Msk k=1

where

& > >−2 % 0 • Pνk (cosω)A0sk + Pν1k (cosω) A1sk Msk = 2μ0 (1 − μs)Pν0k (cosω) >Pν0k (cosθ)> > >−2 1 • , 0 0 0 +μ∗ >Pν1k (cos θ)> Pνk (cos ω) Pνk (cos ω) Bsk , + Pν1k (cos ω) Bsk % > > −2 Lsk = − 2μ0 (1 − μs) Ask Pν0k (cos ω) >Pν0k (cos θ)> > >−2 1 • & . Pν (cos ω) +μ∗ Bsk >Pν1 (cos θ)> k

k

After inversion of function (2.18), the expression for the original calculation is obtained γ+i∞ 1 v (ρ, ω) = Vs (ω) ρ−s ds (2.19) 2πi γ−i∞

Here the γ value should be chosen from the interval (1 − νk ; νk ) [8]. With regard of the equation (2.5) root values νk , obtained in [6], it should be executed γ ∈ (−1; 2). So, let’s chose γ = 0. After it, in the integral (2.19) the change of the variable ϕ = −is was done. It lead to the integration in the line of the real axis. The

180


obtained integral is divided on the two integration intervals (−∞; 0) ∪ (0; ∞) with the corresponding change of variables s = −ϕ and s = ϕ. After the substitution of the equality (2.19) into the correspondence (2.14) the both displacements u(r, θ), v(r, θ) will be expressed through the one unknown function X (η): ω0 ∞ > >−2 u (r, θ) = X (η) sin η Ak (r)Pν1k (cos η) Pν0k (cos θ) >Pν0k (cos θ)> dη (2.20) k=1

0

1 + 2π

ω0 X (η) sin ηΦ0 (r, θ, η) dη, 0

ω0 X (η) sin η

v (r, θ) =

∞

> >−2 Bk (r)Pν1k (cos η) Pν1k (cos θ) >Pν1k (cos θ)> dη (2.21)

k=1

0

1 + 2π

ω0 X (η) sin ηΦ1 (r, θ, η) dη, 0

Φ0 (r, θ, η) ∞ ∞ • (cos θ) Pν0k (cos ω) A0k (r, ρ) + Pν1k (cos ω) A1k (r, ρ) = > >2 > 0 > k=1 Pνk (cos θ) 0 0 ⎞ ⎛ 0 ∞ ∞ 0 1 Pνm (cos η) L−is,m is Pν1m (cos η) Lis,m is ⎟ ⎜ m=1 R ρ ⎟ dsdρ, + m=1 0 ×⎜ ∞ ∞ ⎠ ⎝ 0 R ρ M−is,m Mis,m ∞

Pν0k

m=1

m=1

Φ1 (r, θ, η) =

∞ k=1

Pν1 (cos θ) > k > >P 1 (cos θ)>2 νk

∞ ∞ • Pν0k (cos ω) Bk0 (r, ρ) + Pν1k (cos ω) Bk1 (r, ρ) 0

0

⎞ ⎛ 0 ∞ ∞ 0 Pν1m (cos η) L−is,m is Pν1m (cos η) Lis,m is ⎟ ⎜ m=1 R ρ ⎟ dsdρ. + m=1 0 ×⎜ ∞ ∞ ⎠ ⎝ 0 R ρ M−is,m Mis,m m=1

m=1

It is remain to satisfy the first condition from (1.2) on the one of the crack branches, for example, on the branch r = R − 0, to find the unknown function X (η). Taking in consideration the form of the solution as the superposition of the continuous and discontinuous fields, we get the equality M Rv / (R − 0, θ) − v (R − 0, θ) + u• (R − 0, θ) = −2Rτrθ (R, θ) .

(2.22)

Before the differentiation of the correspondence (2.21) and passing to the limit with r = R − 0, it is necessary to sum the series, including in this expression (if not


181

– all operations will be illegitimate because of the series conditional convergence). With this aim let’s investigate the convergence of these series with the formulas [9] for the asymptotic behavior of the functions Pνμ (cos ϕ) when ν → ∞, for the , k → ∞ and for the norm eigenvalues νk asymptotic [6] when k → ∞, νk ∼ kπ >−2 > >−2 ω 1 > 0 kπ 2 > 1 > > > ∼ ω2 , Pνk (cos θ) ∼ k . All these correspondences asymptotic Pνk (cos θ) allow to get the asymptotical behavior of the pairwise products: 1 1 0 1 (θ − η) + cos νk + (θ + η) , Pνk (cos θ) Pνk (cos η) ∼ Θ (θ, η) sin νk + 2 2 1 1 (θ − η) + sin νk + (θ + η) , Pν1k (cos θ) Pν1k (cos η) ∼ Θ (θ, η) cos νk + 2 2 1 Θ (θ, η) = √ , (2.23) π sin θ sin η 0,1 and of the coefficients Ask, A0,1 sk , Bsk, Bsk when k → ∞ μ0 k 1 − R2 r−2 , R < r q (r) Ak (r) ∼ Rr−1 − R−1 r, R > r 4μ∗ 2 −2 R R r − 1, R < r μ0 k r, R r, R , r < R. the series summation lead to the formula [10] using : + ∞ r sin x arctg 1−r rk sin kx cos x = , |r| < 1. (2.24) 1 k cos kx − ln(1 − 2r cos x + r2 ) 2 k=1 ∞ 0 k=1

The common technology of the series summation is following: the series fk (r) Pνik (cos θ)Pνik (cos η) is divided on the two summand sum

N

+

k=1

After this, the sum

N 0

∞

fk (r)Pνik (cos θ) Pνik (cos η) .

k=N +1

f˜k is added and deducted to this expression, here f˜k is the

k=1

asymptotic expression of the generic term when k → ∞ f˜k ≈ fk (r)Pνik (cos θ) Pνik (cos η) ; with the help of this asymptotic expression the generic term of the second addend is changed also ∞

fk (r)Pνik (cos θ) Pνik (cos η)

k=1

=

∞ k=1

f˜k (r) +

N k=1

fk (r)Pνik (cos θ) Pνik (cos η) −

N k=1

f˜k (r).

182


The series obtained in this equality is summed by formulas (2.24). Let’s substitute the summated with the help of this procedure series in the equality (2.22), where we should find the derivative of function v (r, θ) with respect of r and pass to limit when r = R − 0. Taking into consideration that function g(r, θ) = − 21 ln(1 − 2r cos θ + r2 ) in the function v (r, θ) kernel is the harmonic one, it is possible to express the derivative with respect to first variable through the derivative with 3 3 ∂ ∂2 respect to variable θ: r2 v / (r, θ) = −ctgθ ∂θ v (r, θ) dr− ∂θ v (r, θ) dr. 2 As a result, the integro-differential equation relatively to the unknown function X (η) is obtained: ∂2 ∂θ2

ω0

X (η) sin η ln

0

G (θ, η) =

∂S1 (θ, η) + ∂θ

1 dη + A |η − θ|

ω0 X (η)

sin ηG (θ, η) dη = f (θ),

(2.25)

0

√ 2

sin η ∂ (RΦ2 (θ, η) − Φ1 (θ, η)) − sin η Qj (θ, η) 2π ∂θ j=1

√ : N > >−2 sin η∂Φ0 (θ, η) a3 + a4 , Qj = + Pν1k (cos η)Pν0k (cos θ) >Pνj−1 (cos θ)> , k b3 + b4 2π∂η k=1 π (θ + η) θ+η π (η − θ) 1 ∂ 2arctg cos + 2arctg S1 (θ, η) = √ 2ω 2 2ω sin θ ∂η N 1 (η + θ) (η + θ) sin νk (η + θ) cos + cos νk (η + θ) sin + k 2 2 k=1 (η − θ) (η − θ) + − sin νk (η − θ) sin + cos νk (η − θ) cos 2 2 √ ∂Φ (r,θ,η) M j = 1, 2, A = 2μμ∗0πR , Φ2 (θ, η) = 1 ∂r , f (θ) = −2RA sin θτrθ (R, θ) . r=R−0

3. The obtained integro-differential solving The equation (2.25) solving will be done approximately [7]. With regard of spectral correspondence [6] existence d2 1 dx2 π

1 ln −1

1

1 − y 2 Un (y)dy = − (n + 1) Un (x) , |y − x|

|x| ≤ 1, n = 1, ∞

(3.1) we will use the scheme of the orthogonal polynomial method [7]. With this aim in the (2.25) the change of the variables was done. It allows to reduce the integration interval to the standard one – [−1; 1] y=

2η − ω0 , ω0

x=

2θ − ω0 , ω0

(3.2)


183

y+1 x+1 and take the new denotations Ω(y) = X y+1 , J (y, x) = 2 2 , F (x) = f 2 y+1 x+1 G 2 , 2 . The equation (2.25) is written after it in the form d2 dx2

1 −1

1 dy + A Ω (y) ln |x − y|

1 Ω (y) J (y, x) dy = F (x), x ∈ [−1; 1] .

(3.3)

−1

The searched function is the jump of the displacement on the crack, i.e., on the ends of the integration interval it have the power singularity 12 , so the solution is searched as the series ∞

Ω (y) = 1 − y 2 Uk (y)Ωk , (3.4) k=0

where Uk (y) – are the Chebyshev polynomials of the second kind. After the realization of the orthogonal polynomial method standard scheme, the infinity system of the linear algebraic equation relatively to the unknown coefficients of the expansion (3.4) is obtained Ωl + A

∞

Ωk Dkl = Fl ,

l = 0, 1, 2, . . .

(3.5)

k=0

Dkl =

Nl−1

1 1 J (y, x)

1 − y2

1 − x2 Uk (y)Ul (x)dydx,

−1 −1

1 Fl =

F (x)Ul (x) 1 − x2 dx.

−1

The system (3.5) is solved approximately with the help of the reduction method. The validity of its applying is proved by the scheme proposed in [7].

4. The stress intensity factor calculation The most important mechanical criteria of the body fracture is the stress intensity factor (SIF)

KII = lim 2πR (θ − ω0 )τrθ (r, θ) . (4.1) θ→ω0 +0

The stresses correspondingly to the superposition principe are the sum of the continuous and the discontinuous fields, where the first addend is defined by the formula (2.1), and the second addend – by the left-hand part of the integro-differential equation (3.3). Let’s change the variable by formulas (3.2) in the equality (4.1) and substitute in it the mentioned expression for the stresses. After the substitution and passing to the limit the second addend in the left-hand part of the

184


equation (3.3) and the addend with Mitchell solution are going to zero because of the continuity. The formula for SIF calculation take the form 1

d2 1 KII = lim dy. (4.2) 2πRω0 (x − 1) 2 Ω (y) ln x→1+0 dx |x − y| −1

Let’s substitute the expression (3.4) in the equality (4.2), and then let’s use the analytical extension of the spectral correspondence (3.1) on the values |x| > 1, obtained in [6]: 1

1 Ul (y) 1 − y 2 dy |x − y| −1 4 · 2l−1 Γ2 l + 32 1−x 3 . F l + , l + 2, 2l + 3; = l!(l + 1)!(x − 1)l+2 2 2

d2 dx2

ln

Then the formula (9.131.(1), [11]) of Gauss function analytical extension in the neighborhood of x = 1 is used. Taking it in consideration, after the passing to limit in the correspondence (4.2), the formula for SIF calculation is obtained ∞

(−1)l+1 (l + 1) KII = πRω0 Ωk . (4.3) l! k=1

5. The numerical result consideration The SIF calculation was done for steel cone with angel ω = 600 . In Fig. 1 the graphics of SIF value dependence on the crack angle ω0 are shown. The crack is situated on the distance R from the cone vertex. The curvature 1 in Fig. 1 corresponds to the load with intensity P , the curvature 2 to the intensity P1 = 2P . As it shown in the graphics, with the crack growth the SIF value increasing is assigned, and by the increasing of load the SIF values significantly increase also. In Fig. 2 the graphics of the same value dependence are shown, but for the case when crack is situated on the distance 3R from the cone vertex. At conservation of a general picture, nevertheless, is observed not only essential decreasing of SIF values, but also appreciable decreasing of SIF value difference at increase of load intensity twice. In Fig. 3 SIF values for the crack angle ω0 = 300 are shown. The curvature 1 correspond to the load intensity P , curvature 2 – to the intensity P1 = 2P , curvature 3 – to the intensity – P2 = 3P. As the graphics show, with increasing of the distance R from the cone vertex, where the load is applied, is observed not only the decreasing of SIF values, but is assigned also that the influence of applied load intensity on the SIF values become less essentially. All these calculations were done for cracks with angle that didn’t exceed 80% of the cone angle. If the crack angle is more than 80% from the cone angle, the calculations are fail because in this case the offered approach to the problem solving is not effective.


185

Figure 1

Figure 2 Acknowledgment This research was supported by Ukrainian Department of Science and Education under Project No.0101U008297.

References [1] N. Vaysfeld, The nonstationary problem on stress concentration near the spherical crack, which is situated inside the truncated cone. (in Russian) Mathematical Methods and Physicomechanical Fields 47, 3 (2004), 134–143.

186


Figure 3 [2] N. Vaysfeld, The nonstationary torsion problem for elastic cone with spherical crack. Materials Science 5 (2002), 75–81. [3] N. Vaysfeld, The nonstationary problem on stress concentration near the spherical crack inside the double truncated circular cone. (in Russian) Mashinoznavstvo 4(82) (2004), 17–23. [4] Mitchell, Proc. Math. Soc., London, 32 (1901), 24–29. [5] G. Popov, The axially symmetrical mixed problem of elasticity for the circular hollow cone. J. Appl. Maths. Mechs. 64, 3(2000), 431–443. [6] G. Popov, On the reduction of movement equations to the one independent and two combined solved equations. Russian Academy Reports 384, 2 (2002), 193–196. [7] G. Popov, Elastic stress concentration around Dies, Cuts, Thin inclusions and Supports. (in Russian) Nauka, Moscow, 1982. [8] I. Sneddon, Fourier transforms. New York-Toronto-London, 1st Edition, 1951. [9] H. Bateman, A. Erdelyi, Higher Transcendental Functions. N.Y., 1st Edition, McGraw-Hill, 1955. [10] A. Prudnikov,Yu. Brychkov,O. Marichev, Integrals and Series. Special functions. (in Russian) Moscow, Nauka, 1983. [11] I. Gradshtein, L. Rygik, The Tables of Integrals, Series and Products. (in Russian) Moscow, 1963. Gennadiy Popov and Nataly Vaysfel’d French bulv. 16, apt.21 P.O. Box 65044 Odessa, Ukraine e-mail: [email protected] [email protected]


Pseudospectral Functions for Canonical Differential Systems J. Rovnyak and L.A. Sakhnovich This paper is dedicated to the memory of the great mathematician M.G. Kre˘ın.

Abstract. A pseudospectral function for a canonical differential system is a nondecreasing function on the real line relative to which the eigentransform for the system is a partial isometry. Pseudospectral functions are constructed by means of eigenfunctions and resolvent operators which depend on boundary conditions for the system. Many results hold for Hamiltonians which have selfadjoint matrix values. The most complete results require the definite case, in which it is assumed that the Hamiltonian is nonnegative. Mathematics Subject Classification (2000). Primary 34L10; Secondary 47B50, 47E05, 46C20, 34B09. Keywords. Canonical differential equation, spectral function, pseudospectral function, indefinite inner product, Nevanlinna function.

1. Introduction We are concerned with the spectral theory of canonical differential systems, which we write in the form dY = izJH(x)Y, 0 ≤ x ≤ , dx (1.0.1) Y1 (0, z) = 0. We assume that H(x), the Hamiltonian, has 2m × 2m selfadjoint matrix values, Y1 (x, z) 0 Im , Y (x, z) = , (1.0.2) J= Y2 (x, z) Im 0 where Y1 (x, z) and Y2 (x, z) are m-dimensional vector-valued functions, and z is a complex parameter. It is assumed throughout that H(x) is integrable on [0, ].

188

J. Rovnyak and L.A. Sakhnovich

In a natural way, we shall define L2 (Hdx) as a Kre˘ın space of (equivalence classes of) 2m-dimensional vector-valued functions with inner product f1 , f2 H = f2∗ (t)H(t)f1 (t) dt. 0

Let W (x, z) be the unique 2m × 2m matrix-valued function satisfying dW = izJH(x)W, 0 ≤ x ≤ , dx W (0, z) = I2m .

(1.0.3)

The eigentransform for (1.0.1) is defined by V f = F , 0 Im W ∗ (x, z¯) H(x)f (x) dx, F (z) = 0

for any f in L2 (Hdx). For fixed f , F = V f is an m-dimensional vector-valued entire function. Our purpose here is to construct inner products ·, ·τ on vector-valued entire functions such that f1 , f2 H = F1 , F2 τ for suitable transform pairs F1 = V f1 , F2 = V f2 . The quantities τ used to define such inner products are constructed with the aid of boundary conditions at the right endpoint of [0, ], and they are called pseudospectral data for the system (1.0.1). In general, we allow Hamiltonians satisfying H(x) = H ∗ (x) a.e., and the inner product ·, ·τ is indefinite. In the definite case, that is, when H(x) ≥ 0 a.e., L2 (Hdx) is a Hilbert space, and the inner product identity becomes ∞ ∗ f2 (x)H(x)f1 (x) dx = F2∗ (x) dτ (x) F1 (x), (1.0.4) −∞

0

where τ (x) is a nondecreasing m × m matrix-valued function of real x. However, (1.0.4) is not asserted for all f1 , f2 in L2 (Hdx), and in general V is a partial isometry. For this reason we call τ (x) a pseudospectral function for (1.0.1). We call τ (x) a spectral function if V is an isometry. In some cases, the pseudospectral functions that we construct are spectral functions. These results appear in Section 4 and depend on properties of eigenfunctions and resolvent operators for constant boundary conditions. Basic notions of eigenfunction and resolvent operators relative to variable boundary conditions are introduced in Section 3. With variable boundary conditions, in the definite case H(x) ≥ 0 a.e., in place of (1.0.4) we have the weaker result that ∞

−∞

F ∗ (x) dτ (x) F (x) ≤

f ∗ (x)H(x)f (x) dx

(1.0.5)

0

whenever F = V f for some f in L2 (Hdx). This paper is a continuation of our study of indefinite generalizations of some results of [15]. It differs in two key ways from our previous work. Whereas operator identities and the inverse problem are central in [10, 11, 12], here operator

Pseudospectral Functions

189

identities do not appear, and we are concerned now with the direct problem. The approach using eigenfunctions is similar in spirit to Atkinson [2, Chapter 9] but is technically different. Our methods are most closely related to A.L. Sakhnovich [13]. The study of canonical differential equations is a large and old one and owes much to fundamental work of L. de Branges and M.G. Kre˘ın. For different approaches, historical remarks, and many additional references, see Arov and Dym [1], de Branges [3, 4], Gohberg and Kre˘ın [5], Kaltenb¨ ack and Woracek [8], and the second author [14, 15]. We present background information in Section 2. In Section 3 we study eigenfunctions and resolvent operators for systems with variable boundary conditions. Our main results assume constant boundary conditions and appear in Section 4. Theorem 4.1.11 describes the isometric properties of the eigentransform for the general case of selfadjoint Hamiltonians and pseudospectral data constructed from the boundary conditions. Theorems 4.2.2, 4.2.4, and 4.2.5 are stronger results that hold for pseudospectral functions in the definite case.

2. Preliminaries Assume given a system (1.0.1) where H(x) is a measurable 2m×2m matrix-valued function satisfying (i) H(x) = H ∗ (x) a.e. on [0, ]; 3 (ii) 0 H(x) dx < ∞;

0 = 0 a.e. on [0, ] is g = 0. (iii) the only g in Cm such that H(x) g In the definite case, that is, when H(x) ≥ 0 a.e., (iii) is equivalent to: 3 H11 (x) H12 (x) (iii ) 0 H22 (t) dt ≥ δIm for some δ > 0, where H(x) = . ∗ H12 (x) H22 (x) In fact, if (iii ) is false, we can find g = 0 in Cm such that H22 (x)g = 0 a.e. on [0, ]. Hence for any x such that H(x) ≥ 0 and any u ∈ Cm and z ∈ C, ∗ zu zu ¯ = r2 A + re−iθ B + reiθ B, H(x) 0≤ g g where z = reiθ , A = u∗ H11 (x)u, and B = u∗ H12 (x)g. This is only possible if B = 0. Since u is arbitrary, H12 (x)g = 0, and hence (iii) is false. Thus (iii) implies (iii ). The reverse implication is easy and omitted. 2.1. Fundamental solution Let W (x, z) be the unique solution of (1.0.3). In standard terminology, this is the fundamental solution of (1.0.1) whose value for x = 0 is the identity matrix. For fixed x in [0, ], W (x, z) is an entire function of z satisfying W ∗ (x, z¯)JW (x, z) = W (x, z)JW ∗ (x, z¯) = J

(2.1.1)

190


for all complex z. The Lagrange identity x W ∗ (x, w)JW (x, z) − J W ∗ (u, w)H(u)W (u, z) du = i(z − w) ¯ 0

(2.1.2)

holds for all x in [0, ] and all complex z and w. When w = z¯, (2.1.2) becomes x W ∗ (u, z¯)H(u)W (u, z) du = i W1 (x, z¯)∗ JW (x, z) 0

= −i W (x, z¯)∗ JW1 (x, z), (2.1.3)

where W1 (x, z) = d W (x, z)/dz. For by (2.1.2) and (2.1.1), x x ∗ W (u, z¯)H(u)W (u, z) du = lim W ∗ (u, w)H(u)W (u, z) du w→¯ z 0

0

W (x, w)∗ JW (x, z) − W (x, z¯)∗ JW (x, z) w→¯ z i(z − w) ¯ ∗ W (x, w) − W (x, z¯) = i lim JW (x, z) w→¯ z w − z¯

= lim

= i W1 (x, z¯)∗ JW (x, z), which is the first equality in (2.1.3). The second equality follows from the first on taking adjoints and replacing z by z¯. Throughout the paper we write

A(z) = W ∗ (, z¯) =

a(z) b(z) . c(z) d(z)

(2.1.4)

Here a(z), b(z), c(z), d(z) are m × m matrix-valued entire functions. By (2.1.1) and (2.1.3), for all complex z, a(z)b∗ (¯ z ) + b(z)a∗ (¯ z ) = 0,

a∗ (¯ z )c(z) + c∗ (¯ z )a(z) = 0,

z ) + b(z)c∗ (¯ z ) = Im , a(z)d∗ (¯

a∗ (¯ z )d(z) + c∗ (¯ z )b(z) = Im ,

∗

∗

z ) + d(z)c (¯ z ) = 0, c(z)d (¯ and 0

∗

b (¯ z )d(z) + d (¯ z )b(z) = 0,

W ∗ (u, z¯)H(u)W (u, z) du a (z)b(¯ z )∗ + b (z)a(¯ z )∗ =i c (z)b(¯ z )∗ + d (z)a(¯ z )∗ = −i

(2.1.5)

∗

a (z)d(¯ z )∗ + b (z)c(¯ z )∗

c (z)d(¯ z )∗ + d (z)c(¯ z )∗

z )∗ + b(z)a (¯ z )∗ a(z)b (¯

a(z)d (¯ z )∗ + b(z)c (¯ z )∗

c(z)b (¯ z )∗ + d(z)a (¯ z )∗

c(z)d (¯ z )∗ + d(z)c (¯ z )∗

.

(2.1.6)


191

2.2. Transform V Given a 2m-dimensional vector-valued function f , we define its transform F = V f as the m-dimensional vector-valued function 0 Im W ∗ (x, z¯) H(x)f (x) dx. F (z) = (2.2.1) 0

The functions f which we consider here are assumed to belong to the Kre˘ın space L2 (Hdx) which is defined below. For each f in L2 (Hdx), the transform F = V f is an entire function with values in Cm . In the definite case, L2 (Hdx) is the well-known Hilbert space of (equivalence classes) of 2m-dimensional vector-valued functions f on [0, ] with f 2H = f ∗ (t)H(t)f (t) dt < ∞. 0

To define L2 (Hdx) in the general case, we write H(x) = H+ (x) − H− (x), where H± (x) are measurable functions on [0, ] such that H± (x) ≥ 0 and H+ (x)H− (x) = 0 a.e. As a linear space L2 (Hdx) is defined to be L2 ((H+ +H− )dx). This is a Kre˘ın space in the inner product f2∗ (x)H(x)f1 (x) dx, f1 , f2 ∈ L2 (Hdx). f1 , f2 H = 0

We have L (Hdx) = L (H+ dx) ⊕ L2 (H− dx), and this direct sum is a fundamental decomposition. Two elements f1 and f2 of the space are considered identical if H(x)[f1 (x) − f1 (x)] = 0 a.e. The elements of L2 (Hdx) are thus cosets, but in the usual abuse of terminology we treat L2 (Hdx) as a space of functions. We use standard notions of orthogonality, continuity, and boundedness for operators on a Kre˘ın space. 2

2

Proposition 2.2.1. Let G(z) = [F (z) − F (z0 )]/(z − z0 ) where F = V f for some f ∈ L2 (Hdx) and some z0 ∈ C. Then G = iV g, where g(x) = g(x, z0 ) is the unique solution of dg = iz0 JH(x)g + JH(x)f, dx g() = 0.

0 ≤ x ≤ ,

(2.2.2)

The equation (2.2.2) is solved by setting g(x) = W (x, z0 )U (x). We get W (t, z0 )−1 JH(t)f (t) dt. (2.2.3) g(x, z0 ) = −W (x, z0 ) x

Proof. The Lagrange identity (2.1.2) can be used to show that x W ∗ (x, z¯) − W ∗ (x, z¯0 ) = −i W ∗ (u, z¯)H(u)W (u, z0 ) du JW ∗ (x, z¯0 ). z − z0 0

192


Thus by (2.2.1), G(z) = [F (z) − F (z0 )]/(z − z0 ) is given by W ∗ (t, z¯) − W ∗ (t, z¯0 ) 0 Im H(t)f (t) dt G(z) = z − z0 0 t ∗ W (u, z¯)H(u)W (u, z0 ) du JW ∗ (t, z¯0 ) H(t)f (t) dt. = −i 0 Im 0

0

On interchanging the order of integration and using the second equality in (2.1.1), we obtain 0 Im W ∗ (u, z¯)H(u)W (u, z0 ) G(z) = −i W (t, z0 )−1 JH(t)f (t) dt du. 0

3 By (2.2.3), G(z) = i 0 0

u

∗

Im W (u, z¯)H(u)g(u, z0 ) du, that is, G = iV g.

2.3. Nevanlinna pairs By a Nevanlinna pair we mean a pair R(z), Q(z) of m × m matrix-valued functions which are analytic on a region ΩR,Q containing C+ ∪ C− such that ∗ (i) R∗ (¯ z )Q(z) + z )R(z) ≡ 0 on ΩR,Q; Q∗ (¯ ¯ is nonnegative on ΩR,Q . (ii) the kernel i R (ζ)Q(z) + Q∗ (ζ)R(z) /(z − ζ)

When R(z) ≡ R and Q(z) ≡ Q are constant, ΩR,Q = C is the complex plane. Proposition 2.3.1. Let R(z), Q(z) be a Nevanlinna pair of functions analytic on ΩR,Q such that c(z)R(z) + d(z)Q(z) is invertible except at isolated points. Then the meromorphic function v(z) = i [a(z)R(z) + b(z)Q(z)] [c(z)R(z) + d(z)Q(z)]

−1

(2.3.1)

∗

satisfies v(z) = v (¯ z ) at all points of analyticity. If K(z) = c(z)R(z) + d(z)Q(z), then v(z) − v ∗ (ζ) R(ζ)∗ Q(z) + Q(ζ)∗ R(z) = K ∗ (ζ)−1 K(z)−1 z − ζ¯ i(ζ¯ − z) I I iv ∗ (ζ) W ∗ (t, ζ)H(t)W (t, z) dt (2.3.2) + −iv(z) 0 for z, ζ ∈ ΩR,Q such that z = ζ¯ and K(z) and K(ζ) are invertible. Proof. By (2.1.2) and (2.1.4), ∗ ¯ (¯ z) − J A(ζ)JA W ∗ (t, ζ)H(t)W (t, z) dt = . ¯ i(z − ζ) 0 Hence ∗ ¯ A(ζ)JA (¯ z) − J I ∗ I iv (ζ) −iv(z) ¯ i(z − ζ) I iv ∗ (ζ) W ∗ (t, ζ)H(t)W (t, z) = 0

I dt. −iv(z)


193

Breaking the left side into two parts and rearranging terms, we obtain I ∗ I iv (ζ) J −iv(z) v(z) − v ∗ (ζ) =− ¯ ¯ z−ζ i(z − ζ) I ∗ ¯ I iv ∗ (ζ) A(ζ)JA (¯ z) −iv(z) = ¯ i(ζ − z) ∗ I ∗ I iv (ζ) W (t, ζ)H(t)W (t, z) dt . + −iv(z) 0

(2.3.3)

Recall that K(z) = c(z)R(z) + d(z)Q(z). In addition, set H(z) = a(z)R(z) + b(z)Q(z). Then by (2.3.1), v(z) = iH(z)K(z)−1. By (2.1.4), R(z) a(z)R(z) + b(z)Q(z) H(z) , = A(z) = Q(z) c(z)R(z) + d(z)Q(z) K(z) and therefore

H(z)K(z)−1 R(z) −iv(z) = A(z) K(z). K(z) = Q(z) I I

By (2.1.1), A(z)JA∗ (¯ z ) = J. Hence A(z)−1 = JA∗ (¯ z )J and R(z) −iv(z) I K(z)−1 = A(z)−1 = JA∗ (¯ . z) Q(z) I −iv(z) Thus by (2.3.3), v(z) − v ∗ (ζ) = z − ζ¯

∗ ¯ iv ∗ (ζ) A(ζ)JJJA (¯ z)

I

+

I −iv(z)

i(ζ¯ − z) ∗ ∗ I iv (ζ) W (t, ζ)H(t)W (t, z)

0 ∗

−1

K (ζ)

∗ R (ζ)

= +

I

R(z) Q (ζ) J K(z)−1 Q(z) i(ζ¯ − z) ∗

iv (ζ) W (t, ζ)H(t)W (t, z) ∗

I dt −iv(z)

∗

0

R(ζ)∗ Q(z) + Q(ζ)∗ R(z) K(z)−1 i(ζ¯ − z) ∗ ∗ I iv (ζ) W (t, ζ)H(t)W (t, z) +

I dt −iv(z)

= K ∗ (ζ)−1

0

I dt , −iv(z)

which is (2.3.2). The identity v(z) = v ∗ (¯ z ) is easily deduced from (2.3.2).

194


3. Systems with boundary conditions Let R(z), Q(z) be a Nevanlinna pair of functions analytic on ΩR,Q such that c(z)R(z) + d(z)Q(z) is invertible except at isolated points. We study the eigenfunctions and resolvent operators for the system dY = izJH(x)Y, 0 ≤ x ≤ , dx Y1 (0, z) = 0, R∗ (¯ z )Y1 (, z) + Q∗ (¯ z )Y2 (, z) = 0,

(3.0.1)

where z ∈ ΩR,Q and the Hamiltonian H(x) satisfies the conditions in Section 2. 3.1. Eigenfunctions and resolvents Consider a system (3.0.1) with Hamiltonian H(x) = H ∗ (x). Definition 3.1.1. For every ζ ∈ ΩR,Q , let Lζ be the linear subspace of L2 (Hdx) consisting of all solutions of (3.0.1) with z = ζ. We call a point ζ ∈ ΩR,Q an eigenvalue for (3.0.1) if Lζ contains a function Y (x, ζ) such that Y = 0 as an element of L2 (Hdx). In this case, we call any such Y an eigenfunction and Lζ the eigenspace for the eigenvalue ζ. Proposition 3.1.2. For any ζ ∈ ΩR,Q , Lζ is the set of functions of the form 0 , g ∈ Cm , Y (x, ζ) = W (x, ζ) g (3.1.1) ∗ ¯ ∗ ¯ ∗ ¯ ∗ ¯ [R (ζ)c (ζ) + Q (ζ)d (ζ)]g = 0 . Hence Lζ = {0} except at isolated points of ΩR,Q . Proof. Let Y = Y (x, ζ) ∈ Lζ , and set g˜ Y1 (0, ζ) . = Y (0, ζ) = g Y2 (0, ζ) Since dY /dx = iζJH(x)Y on [0, ] and W (0, ζ) = I2m , g˜ Y (x, ζ) = W (x, ζ) . g The condition Y1 (0, ζ) = 0 says that g˜ = 0, so ∗ ∗ ¯ c∗ (ζ) ¯ 0 ¯ c (ζ)g a (ζ) Y1 (, ζ) = ∗ ¯ ¯ g = d∗ (ζ)g ¯ . Y2 (, ζ) b (ζ) d∗ (ζ) ¯ 1 (, ζ) + Q∗ (ζ)Y ¯ 2 (, ζ) = 0, we get [R∗ (ζ)c ¯ ∗ (ζ) ¯ + Q∗ (ζ)d ¯ ∗ (ζ)]g ¯ = 0. From R∗ (ζ)Y Thus Y has the form (3.1.1). These steps are reversible.


195

¯ ζ)+ ¯ Lemma 3.1.3. Let ζ ∈ ΩR,Q . Assume that R∗ (ζ)R(ζ)+Q∗ (ζ)Q(ζ) and R∗ (ζ)R( ∗ ¯ ¯ are invertible. Set Q (ζ)Q(ζ) ¯ R(ζ) R(ζ) and Mζ¯ = ran (3.1.2) Mζ = ran ¯ . Q(ζ) Q(ζ) Then dim Mζ = dim Mζ¯ = m, Mζ⊥ = JMζ¯, and Mζ¯⊥ = JMζ . Proof. Our assumptions imply that ¯ R(ζ) R(ζ) = ker ker ¯ = {0}, Q(ζ) Q(ζ) and this implies dim Mζ = dim Mζ¯ = m. By the definition of a Nevanlinna pair, ¯ ¯ + Q∗ (ζ)R(ζ) = 0, and hence JMζ¯ ⊆ Mζ⊥ . Equality holds because Mζ R∗ (ζ)Q(ζ) and Mζ¯ have dimension m. Similarly, Mζ¯⊥ = JMζ . ¯ ζ)+ ¯ Lemma 3.1.4. Let ζ ∈ ΩR,Q . Assume that R∗ (ζ)R(ζ)+Q∗ (ζ)Q(ζ) and R∗ (ζ)R( ∗ ¯ ¯ Q (ζ)Q(ζ) are invertible. Then c(ζ)R(ζ) + d(ζ)Q(ζ) is invertible if and only if ¯ ζ) ¯ + d(ζ)Q( ¯ ζ) ¯ is invertible. c(ζ)R( ¯ ζ) ¯ + d(ζ)Q( ¯ ¯ is not invertible. Then there is a nonzero Proof. Suppose that c(ζ)R( ζ) m ∗ ¯ ∗ ¯ ¯ ∗ (ζ)]g ¯ = 0. Therefore vector g in C such that [R (ζ)c (ζ) + Q∗ (ζ)d ∗ ¯ ∗ ¯ ∗ c (ζ)g ¯ Q∗ (ζ) ¯ c (ζ)g = 0 and R (ζ) ∈ Mζ¯⊥ = JMζ , ¯ ¯ d∗ (ζ)g d∗ (ζ)g where Mζ and Mζ¯ are as in Lemma 3.1.3. Hence there is g1 in Cm such that ∗ ¯ c (ζ)g R(ζ)g1 Q(ζ)g1 = J = . ¯ d∗ (ζ)g Q(ζ)g1 R(ζ)g1 Thus by (2.1.5), ¯ + d(ζ)c∗ (ζ)g ¯ = 0. [c(ζ)R(ζ) + d(ζ)Q(ζ)]g1 = c(ζ)d∗ (ζ)g ¯ = c∗ (ζ)g ¯ = 0, which We show that g1 = 0. In fact, if g1 = 0, then d∗ (ζ)g ∗ ¯ ∗ ¯ by (2.1.5) implies that g = [a(ζ)d (ζ) + b(ζ)c (ζ)]g = 0, a contradiction. Since c(ζ)R(ζ) + d(ζ)Q(ζ) has a nontrivial kernel, it is not invertible. The result follows ¯ on interchanging the roles of ζ and ζ. The next result prepares the way for the definition of a resolvent operator in Definition 3.1.6. Proposition 3.1.5. Suppose that z ∈ ΩR,Q and c(z)R(z) + d(z)Q(z) and c(¯ z )R(¯ z) + d(¯ z )Q(¯ z ) are invertible. Then for any f ∈ L2 (Hdx), the system dV = izJH(x)V + JH(x)f (x), 0 ≤ x ≤ , dx V1 (0, z) = 0, R∗ (¯ z )V1 (, z) + Q∗ (¯ z )V2 (, z) = 0,

(3.1.3)

196


has a unique solution given by x 0 Im V (x, z) = W (x, z) W ∗ (t, z¯)H(t)f (t) dt 0 −iv(z) 0 : 0 0 ∗ + W (t, z¯)H(t)f (t) dt , −Im −iv(z) x

(3.1.4)

where v(z) is defined by (2.3.1). Proof. If a solution V (x, z) exists and V (x, z) = W (x, z)U (x, z), then by (1.0.3) and (2.1.1), dU = W (x, z)−1 JH(x)f (x) = JW ∗ (x, z¯)JH(x)f (x), dx and so

x

U (x, z) = U (0, z) +

JW ∗ (t, z¯)H(t)f (t) dt.

(3.1.5)

0

The boundary condition V1 (0, z) = 0 and relation V (0, z) = W (0, z)U (0, z) = U (0, z) imply that U1 (0, z) = 0, and hence 0 . (3.1.6) U (0, z) = U2 (0, z) The boundary condition R∗ (¯ z )V1 (, z) + Q∗ (¯ z )V2 (, z) = 0 can be written in the ∗ ∗ z ) Q (¯ z ) V (, z) = 0. Here V (, z) = W (, z)U (, z), and so by (2.1.4), form R (¯ (3.1.5), and (3.1.6), a∗ (¯ z ) c∗ (¯ z) z ) Q∗ (¯ z) ∗ 0 = R∗ (¯ U (, z) b (¯ z ) d∗ (¯ z) z )a∗ (¯ z ) + Q∗ (¯ z )b∗ (¯ z ) R∗ (¯ z )c∗ (¯ z ) + Q∗ (¯ z )d∗ (¯ z) · = R∗ (¯ : 0 + · W (t, z)−1 JH(t)f (t) dt . U2 (0, z) 0 By (2.1.1), z )a∗ (¯ z ) + Q∗ (¯ z )b∗ (¯ z ) R∗ (¯ z )c∗ (¯ z ) + Q∗ (¯ z )d∗ (¯ z) · 0 = R∗ (¯ : 0 + · JW ∗ (t, z¯)H(t)f (t) dt U2 (0, z) 0 z )c∗ (¯ z ) + Q∗ (¯ z )d∗ (¯ z )]U2 (0, z) = [R∗ (¯ ∗ z )a∗ (¯ z ) + Q∗ (¯ z )b∗ (¯ z ) R∗ (¯ z )c∗ (¯ z ) + Q∗ (¯ z )d∗ (¯ z) · + R (¯ · W ∗ (t, z¯)H(t)f (t) dt. 0

Pseudospectral Functions Solving for U2 (0, z), we get U2 (0, z) = − Im

ϕ(z)

197

W ∗ (t, z¯)H(t)f (t) dt,

0

where ϕ(z) = i(−i)[R∗ (¯ z )c∗ (¯ z )+Q∗ (¯ z )d∗ (¯ z )]−1 [R∗ (¯ z )a∗ (¯ z )+Q∗ (¯ z )b∗ (¯ z )] = iv ∗ (¯ z ), that is, ϕ(z) = iv(z). Thus 0 0 U (0, z) = J W ∗ (t, z¯)H(t)f (t) dt. (3.1.7) −iv(z) −Im 0 Then by (3.1.5) and (3.1.7), V (x, z) = W (x, z)

0 0 J W ∗ (t, z¯)H(t)f (t) dt −iv(z) −Im 0 x : Im 0 ∗ J W (t, z¯)H(t)f (t) dt + 0 Im 0

x 0 Im J W ∗ (t, z¯)H(t)f (t) dt = W (x, z) −iv(z) 0 0 : 0 0 J W ∗ (t, z¯)H(t)f (t) dt + −iv(z) −Im x x 0 Im W ∗ (t, z¯)H(t)f (t) dt = W (x, z) 0 −iv(z) 0 : 0 0 ∗ W (t, z¯)H(t)f (t) dt , + −Im −iv(z) x which is one direction of the theorem. The other direction follows on reversing the steps. Definition 3.1.6. Let Ωv be the maximum domain of analyticity of the function v(z) defined by (2.3.1). For each z in Ωv , define a resolvent operator B(z) on L2 (Hdx) by B(z)f = V (x, z), f ∈ L2 (Hdx), (3.1.8) where V (x, z) is given by (3.1.4). The domain Ωv contains all removable singularities of (2.3.1) as well as real intervals across which this function has an analytic extension. Proposition 3.1.7. The resolvent B(z) is analytic as a function of z, has compact values, and satisfies B ∗ (¯ z ) = −B(z) on Ωv . Proof. The fact that B(z) is analytic and has compact values follows from (3.1.4). We show that B ∗ (¯ z ) = −B(z). First use (3.1.8) and (3.1.4) to write

198


B(z) = B1 (z) + B2 (z), where x 0 Im B1 (z)f = W (x, z) W ∗ (t, z¯)H(t)f (t) dt, 0 −iv(z) 0 0 0 W ∗ (t, z¯)H(t)g(t) dt, B2 (z)g = W (x, z) −Im −iv(z) x

(3.1.9) (3.1.10)

for any f, g ∈ L2 (Hdx). By (3.1.9), x 0 Im ∗ g (x)H(x)W (x, z) W ∗ (t, z¯)H(t)f (t) dt dx B1 (z)f, gH = 0 −iv(z) 0 0

=

h∗ (t)H(t)f (t) dt,

0

where

0 W ∗ (u, z)H(u)g(u) du. iv ∗ (z) x

0 h(x) = W (x, z¯) Im

By (3.1.10), 0 h(x) = −W (x, z¯) −Im

0 W ∗ (u, z)H(u)g(u) du = −B2 (¯ z )g. −iv(¯ z) x

Therefore B1∗ (z) = −B2 (¯ z ), and the assertion follows.

Proposition 3.1.8. For each f ∈ L2 (Hdx) and z in Ωv , iB(z)f, f H = F ∗ (¯ z )v(z)F (z) + iΓf (z),

(3.1.11)

where F (z) is given by (2.2.1), and z) = f ∗ (x)M (x, t, z)H(t)f (t) dt dx, Γf (z) = −Γf (¯ 0

0

⎧ 0 Im ⎪ ∗ ⎪ x > t, ⎪ ⎨ W (x, z) 0 0 W (t, z¯), M (x, t, z) = ⎪ ⎪ 0 0 ⎪ ⎩ W (x, z) W ∗ (t, z¯), x < t. −Im 0 Proof. By (3.1.8) and (3.1.4), x 0 Im f ∗ (x)H(x)W (x, z) W ∗ (t, z¯)H(t)f (t) dt B(z)f, f H = 0 −iv(z) 0 0 : 0 0 W ∗ (t, z¯)H(t)f (t) dt dx. + −Im −iv(z) x


199

The parts of the two integrals on the right containing −iv(z) combine to give 0 0 B(z)f, f H = f ∗ (x)H(x)W (x, z) W ∗ (t, z¯)H(t)f (t) dt 0 −iv(z) 0 0 x 0 Im W ∗ (t, z¯)H(t)f (t) dt + 0 0 0 : 0 0 W ∗ (t, z¯)H(t)f (t) dt dx + −Im 0 x ∗ z )v(z)F (z) + f ∗ (x)H(x)M (x, t, z)H(t)f (t) dt dx. = −iF (¯ 0

0

This yields the formula for Γf (z) in (3.1.11). The equality Γf (z) = −Γf (¯ z ) follows from the identities B ∗ (¯ z ) = −B(z) and v ∗ (¯ z ) = v(z). 3.2. Definite case: V as a contraction operator We again assume given a system (3.0.1), but now in addition we assume that H(x) ≥ 0 a.e. Then L2 (Hdx) is a Hilbert space. There are no nonreal eigenvalues in this case (Proposition 3.2.1). We derive a Cauchy representation for the resolvent and show its consequence for the transform V (Theorem 3.2.4). Proposition 3.2.1. If R∗ (z)R(z) + Q∗(z)Q(z) is invertible for every nonreal z, then (3.0.1) has no nonreal eigenvalues. ¯ and suppose that Y (t, ζ) ∈ Lζ . We show that Y = 0 in L2 (Hdx). Proof. Fix ζ = ζ, We borrow a formula from the proof of Proposition 4.1.1, which is valid under the present assumptions as well: ¯ i(ζ − ζ) Y ∗ (t, ζ)H(t)Y (t, ζ) dt = Y ∗ (, ζ)JY (, ζ). (3.2.1) 0

Define Mζ and Mζ¯ by (3.1.2). The boundary condition at in (3.0.1) implies that Y (, ζ) is orthogonal to Mζ¯. Since we assume that R∗ (ζ)R(ζ) + Q∗ (ζ)Q(ζ) and ¯ ζ) ¯ + Q∗ (ζ)Q( ¯ ¯ are invertible, it follows from Lemma 3.1.3 that Y (, ζ) ∈ ζ) R∗ (ζ)R( ⊥ Mζ¯ = JMζ . Therefore R(ζ) g Y (, ζ) = J Q(ζ) for some g ∈ Cm . Then by (3.2.1) and the definition of a Nevanlinna pair, R∗ (ζ)Q(ζ) + Q∗ (ζ)R(ζ) Y ∗ (t, ζ)H(t)Y (t, ζ) dt = g ∗ g ¯ i(ζ − ζ) 0 R∗ (ζ)Q(ζ) + Q∗ (ζ)R(ζ) = −g ∗ i g ≤ 0. ζ − ζ¯ Since H(x) ≥ 0 a.e., the left side is nonnegative and hence equal to zero.

200

J. Rovnyak and L.A. Sakhnovich In the definite case, it follows from (2.3.2) that v(z) = i [a(z)R(z) + b(z)Q(z)] [c(z)R(z) + d(z)Q(z)]

−1

(3.2.2)

is a Nevanlinna function. This means that all nonreal singularities of v(z) are removable, v(z) = v ∗ (¯ z ) for all nonreal z, and v(z) has nonnegative imaginary part on C+ . Hence v(z) has a Nevanlinna representation ∞ t 1 − dτ (t). (3.2.3) v(z) = α + βz + 1 + t2 −∞ t − z Here α and β are m × m matrices such that α =3α∗ , β ≥ 0, and τ (t) is a nonde∞ creasing m × m matrix-valued function such that −∞ dτ (t)/(1 + t2 ) is convergent. In particular, the resolvent B(z) is analytic on Ωv ⊇ C+ ∪ C− . Lemma 3.2.2. For each z ∈ C+ and f ∈ L2 (Hdx), Re B(z)f, f H ≥ Im z B(z)f, B(z)f H ≥ 0.

(3.2.4)

Proof. Without loss of generality, we can assume that c(z)R(z) + d(z)Q(z) and c(¯ z )R(¯ z ) + d(¯ z )Q(¯ z ) are invertible. Then B(z)f = V is the unique solution of (3.1.3), and thus JV , V L2

2m (0, )

= izH(x)V, V L2

2m (0, )

+ H(x)f, V L2

2m (0, )

= izV, V H + f, V H . Hence

2 Re f, V H = 2 Im z V, V H +

∗ ∗ V (t, z)JV (t, z) + V (t, z)JV (t, z) dt

0

= 2 Im z V, V H + V ∗ (, z)JV (, z) − V ∗ (0, z)JV (0, z) = 2 Im z V, V H + V ∗ (, z)JV (, z), since V ∗ (0, z)JV (0, z) = 0 by the boundary condition at 0 in (3.1.3). Let Mz and Mz¯ be as in Lemma 3.1.3, so JMz = Mz¯⊥ . By the boundary condition at , C B R(¯ z) g V (, z), = 0, g ∈ Cm . Q(¯ z) 2m C Q(z) ⊥ g for some gz ∈ Cm . Thus Hence V (, z) ∈ Mz¯ = JMz , so V (, z) = R(z) z V ∗ (, z)JV (, z) = 2 Im z gz∗

Q∗ (z)R(z) + R∗ (z)Q(z) gz ≥ 0 i(¯ z − z)

for Im z ≥ 0 by the definition of a Nevanlinna pair (see Section 2.3). Recalling that V = B(z)f , we deduce (3.2.4). Lemma 3.2.3. The resolvent operators have a representation ∞ dG(t) iB(z) = , z ∈ C + ∪ C− , −∞ t − z

(3.2.5)


201

where G(x) is a nondecreasing function of real x whose values are 3 ∞operators on L2 (Hdx) such that G(x) = 12 G(x+0)−G(x−0) for all real x and −∞ dG(t) ≤ I. z ) = −B(z), iB(z) is a Nevanlinna Proof. By Lemma 3.2.2 and the identity B ∗ (¯ function and hence has a representation ∞ t 1 − dG(t), (3.2.6) iB(z) = C1 + C2 z + 1 + t2 −∞ t − z where C1 = C1∗ , C2 ≥ 0, = and G(x) is a nondecreasing function 3 ∞ satisfying G(x) 1 2 G(x + 0) − G(x − 0) for all real x such that the integral dG(t)/(1 + t ) is 2 −∞ 2 weakly convergent. By Lemma 3.2.2, if f ∈ L (Hdx), iB(iy)f 2H ≤

1 iB(iy)f H f H , y

and hence y iB(iy) ≤ 1 for y > 0. It follows that C2 = 0. Therefore ∞ t 1 − dG(t) iB(iy) = C1 + 1 + t2 −∞ t − iy ∞ y t(1 − y 2 ) +i 2 dG(t), = C1 + 2 2 2 t + y2 −∞ (t + y )(1 + t ) and so

∞

y2 dG(t) = y Im [iB(iy)]. 2 2 −∞ t + y 3∞ Since y iB(iy) ≤ 1 for y > 0, −∞ dG(t) ≤ I. The representation (3.2.6) can thus be written in the form ∞ ∞ dG(t) t − dG(t). (3.2.7) iB(z) = C1 + 2 −∞ t − z −∞ 1 + t 3∞ Since y iB(iy) is bounded for y > 0, C1 = −∞ t(1 + t2 )−1 dG(t) and so (3.2.7) reduces to (3.2.5). Theorem 3.2.4. For any f ∈ L2 (Hdx), ∞ ∗ F (t)dτ (t)F (t) , iB(z)f, f H = t−z −∞

z ∈ C+ ∪ C − ,

where τ (x) is as in (3.2.3) and F (z) is given by (2.2.1). Moreover, ∞ F ∗ (t)dτ (t)F (t) ≤ f ∗ (t)H(t)f (t) dt. −∞

0

That is, the transform V acts as a contraction from L2 (H) into L2 (dτ ). Proof. Apply Lemma 3.2.3 and Proposition 3.1.8 to write ∞ dG(t)f, f H = F ∗ (¯ z )v(z)F (z) + iΓf (z). iB(z)f, f H = t − z −∞

(3.2.8)

(3.2.9)

202


By the Livˇsic-Stieltjes inversion formula, for any a, b, −∞ < a < b < ∞, 1 b [G(b) − G(a)]f, f H = lim Im F ∗ (t − iy)v(t + iy)F (t + iy) dt y↓0 π a b 1 b iΓf (t + iy) dt = F ∗ (t)dτ (t)F (t). + lim y↓0 π a a In the last equality the term involving Γf (z) makes no contribution because Γf (z) is continuous and 3 ∞real on the real axis. This proves (3.2.8). We deduce (3.2.9) from the inequality −∞ dG(t) ≤ I in Lemma 3.2.3.

4. Constant boundary conditions and main results Our main results construct pseudospectral data and pseudospectral functions for a system (1.0.1) by considering boundary conditions as in Section 3. For this purpose it is necessary to assume that the Nevanlinna pair R(z) ≡ R and Q(z) ≡ Q in (3.0.1) is constant. In this case, the domain of analyticity of the pair is ΩR,Q = C. Thus throughout this section we assume given a system dY = izJH(x)Y, 0 ≤ x ≤ , dx (4.0.1) Y1 (0, z) = 0, R∗ Y1 (, z) + Q∗ Y2 (, z) = 0, where R, Q are m × m matrices such that R∗ Q + Q∗ R = 0, the entire function c(z)R + d(z)Q is invertible except at isolated points, and z is any complex number. As usual, we assume that the Hamiltonian H(x) satisfies the conditions in Section 2. Notice that R∗ R + Q∗ Q is invertible, since otherwise c(z)R + d(z)Q cannot be invertible at any point. 4.1. Construction of pseudospectral data Assume given a system (4.0.1) with Hamiltonian H(x) = H ∗ (x). The goal of this section is Theorem 4.1.11, which is the basis for the notion of pseudospectral data. We also derive additional properties of eigenfunctions and resolvents that will be important for later results. By Proposition 3.1.2, for each ζ ∈ C, Lζ is the set of functions 0 , g ∈ Cm , Y (x, ζ) = W (x, ζ) g (4.1.1) ¯ + Q∗ d∗ (ζ)]g ¯ = 0. [R∗ c∗ (ζ) Proposition 4.1.1. For any ζ1 , ζ2 ∈ C and Y (x, ζ1 ) ∈ Lζ1 and Y (x, ζ2 ) ∈ Lζ2 , i(ζ1 − ζ¯2 ) Y ∗ (t, ζ2 )H(t)Y (t, ζ1 ) dt = 0. (4.1.2) 0

Hence Lζ1 ⊥ Lζ2

¯ if ζ1 = ζ¯2 , and Lζ is a neutral subspace of L2 (Hdx) if ζ = ζ.


203

Proof. By the differential equation in (4.0.1), i(ζ1 −ζ¯2 ) Y ∗ (t, ζ2 )H(t)Y (t, ζ1 ) dt

0

= 0

=

Y ∗ (t, ζ2 )J[iζ1 JH(t)Y (t, ζ1 )] dt +

[iζ2 JH(t)Y (t, ζ2 )]∗ JY (t, ζ1 ) dt

0

∗ Y (t, ζ2 )JY (t, ζ1 ) + Y ∗ (t, ζ2 )JY (t, ζ1 ) dt

0

= Y ∗ (, ζ2 )JY (, ζ1 ) − Y ∗ (0, ζ2 )JY (0, ζ1 ) = Y ∗ (, ζ2 )JY (, ζ1 ), where at the last stage we used the initial conditions Y1 (0, ζ1 ) = Y1 (0, ζ2 ) = 0. Applying Lemma 3.1.3 to the subspace R ⊆ C2m , M = ran (4.1.3) Q we see that dim M = m and M ⊥ = JM . Since R∗ Y1 (, ζ1 ) + Q∗ Y2 (, ζ1 ) = 0 and R∗ Y1 (, ζ2 ) + Q∗ Y2 (, ζ2 ) = 0, Y (, ζ1 ) and Y (, ζ2 ) are orthogonal to M . Therefore Y (, ζ1 ) ∈ M ⊥ = JM and JY (, ζ1 ) ∈ M . Since Y (, ζ2 ) ∈ M ⊥ , Y ∗ (, ζ2 )JY (, ζ1 ) = 0. This proves (4.1.2). The last part of the lemma follows in a straightforward way from (4.1.2), provided that whenever (4.1.2) is applied with ζ1 = ζ2 = ζ then Y (x, ζ1 ) and Y (x, ζ2 ) are understood to be possibly different elements of Lζ . Given a linear operator T on some linear space and an eigenvalue γ for T , let R0 (T, γ) = ker(T − γI). If Rj (T, γ) has been defined for j = 0, . . . , k, let Rk+1 (T, γ) be the set of all vectors f such that (T − γI)f ∈ Rk (T, γ). We call R0 (T, γ), R1 (T, γ), . . . the root subspaces for T for the eigenvalue γ. Lemma 4.1.2. Given a linear operator T with eigenvalue γ, (i) the subspaces R0 (T, γ), R1 (T, γ), . . . are invariant for T ; (ii) {0} ⊆ R0 (T, γ) ⊆ R1 (T, γ) ⊆ · · · , and if equality holds at one stage, it holds at all later stages; (iii) if R0 (T, γ) is finite dimensional, so is Rk (T, γ) for every k = 0, 1, 2, . . . ; (iv) if R0 (T, γ) is finite dimensional and γ = 0, then T is a one-to-one mapping from Rk (T, γ) onto itself for each k = 0, 1, 2, . . . . The details are elementary and omitted. We introduce an analogous notion for canonical systems. (0)

Definition 4.1.3. Assume given a system (4.0.1) with eigenvalue ζ. Let Lζ = Lζ (0) (k) be the corresponding eigenspace. If the subspaces Lζ , . . . , Lζ have been defined, (k+1) let Lζ be the set of all functions Y which satisfy dY = iζJH(x)Y + JH(x)Y (k) , dx (4.1.4) ∗ Im 0 Y (0) = 0, R Q∗ Y () = 0,

204

J. Rovnyak and L.A. Sakhnovich (k)

(0)

(1)

for some Y (k) ∈ Lζ . We call Lζ , Lζ , . . . the root subspaces for (4.0.1) for the eigenvalue ζ. (0)

(1)

It is easy to see that for any eigenvalue ζ of (4.0.1), Lζ ⊆ Lζ ⊆ · · · , and if equality holds at one stage, it holds at all subsequent stages. Theorem 4.1.4. Let z0 ∈ C, and assume that c(z0 )R + d(z0 )Q is invertible. (i) The nonzero eigenvalues of the resolvent operator B(z0 ) coincide with the set of numbers i/(z0 − ζ) where ζ is an eigenvalue of (4.0.1). (ii) For each eigenvalue ζ of (4.0.1) and every k = 0, 1, 2, . . . , (k)

Lζ

= Rk (B(z0 ), i/(z0 − ζ)). (0)

(0)

Proof. Let ζ be an eigenvalue for (4.0.1). Thus Lζ = Lζ = {0}. If Y ∈ Lζ , then dY = iζJH(x)Y = iz0 JH(x)Y + JH(x)[i(ζ − z0 )Y ], dx Im 0 Y (0) = 0, and R∗ Q∗ Y () = 0. Therefore B(z0 )[i(ζ − z0 )Y ] = Y . It follows that i/(z0 − ζ) is a nonzero eigenvalue of B(z0 ), and (0)

Lζ ⊆ R0 (B(z0 ), i/(z0 − ζ)). On the other hand, if Y ∈ R0 (B(z0 ), i/(z0 − ζ)), we can reverse these steps to (0) show that Y ∈ Lζ . Thus i/(z0 − ζ) is a nonzero eigenvalue of B(z0 ), and (0)

Lζ = R0 (B(z0 ), i/(z0 − ζ)). Conversely, suppose that γ is a nonzero eigenvalue of B(z0 ). Consider any eigenvector Y . Then B(z0 )[γ −1 Y ] = Y . This means that dY = iz0 JH(x)Y + JH(x)[γ −1 Y ], dx ∗ Im 0 Y (0) = 0, R Q∗ Y () = 0. Define ζ by γ = i/(z0 − ζ). Then dY = iz0 JH(x)Y + JH(x)[i(ζ − z0 )Y ] = iζJH(x)Y, dx Im 0 Y (0) = 0, and R∗ Q∗ Y () = 0. Hence Y is an eigenvector for (4.0.1). Thus far we have proved (i). We have also proved (ii) for k = 0. In particular, for any eigenvalue ζ of (4.0.1), (0)

dim R0 (B(z0 ), i/(z0 − ζ)) = dim Lζ < ∞ by the representation (4.1.1) of the elements of an eigenspace. It remains to complete the proof of (ii). Suppose we know that (k)

Rk (B(z0 ), i/(z0 − ζ)) = Lζ

Pseudospectral Functions (k+1)

for some k ≥ 0. Let Y ∈ Lζ

(k)

. Choose Y (k) ∈ Lζ

205

satisfying (4.1.4). Then

dY = iz0 JH(x)Y + JH(x)[i(ζ − z0 )Y + Y (k) ], dx ∗ Im 0 Y (0) = 0, R Q∗ Y () = 0, which means that B(z0 )[i(ζ − z0 )Y + Y (k) ] = Y , or $ # i i I Y =− B(z0 )Y (k) . B(z0 ) − z0 − ζ z0 − ζ

(4.1.5)

(k)

Since Y (k) ∈ Lζ

= Rk (B(z0 ), i/(z0 − ζ)), by (4.1.5) and Lemma 4.1.2(iv), # $ i B(z0 ) − I Y ∈ Rk (B(z0 ), i/(z0 − ζ)), (4.1.6) z0 − ζ and so Y ∈ Rk+1 (B(z0 ), i/(z0 − ζ)). Thus (k+1)

Lζ

⊆ Rk+1 (B(z0 ), i/(z0 − ζ)).

(4.1.7)

To prove the reverse inclusion, consider any Y ∈ Rk+1 (B(z0 ), i/(z0 − ζ)). Then Y satisfies (4.1.6). Using Lemma 4.1.2(iv), we deduce (4.1.5) for some (k)

Y (k) ∈ Rk (B(z0 ), i/(z0 − ζ)) = Lζ . (k+1)

Now we can reverse the steps and conclude that Y ∈ Lζ holds in (4.1.7), and the proof of (ii) is complete. We say that a real interval (a, b) is H-indivisible if H(x) = ηh(x)η ∗

a.e. on (a.b),

η=

. Therefore equality

α , β

(4.1.8)

where h(x) is a measurable function with selfadjoint m × m matrix values, and α and β are m × m matrices such that η ∗ Jη = α∗ β + β ∗ α = 0. The notion of an H-indivisible interval is due to Kac [7]. See also Hassi, de Snoo, and Winkler [6] and Kaltenb¨ ack and Woracek [8, Part IV]. It should be noted that some authors use a trace-normed Hamiltonian, and their formulas have a different appearance. 2 2 (Hdx) be the subspace of L2 (Hdx) consisting of all f such Definition 4.1.5. Let L that for every H-indivisible interval (a, b), there is a c ∈ C2m satisfying H(x)f (x) = H(x)c

a.e. on (a.b).

When H(x) ≥ 0 a.e., L2 (Hdx) is a Hilbert space, and an argument in Kac 2 2 (Hdx) is closed. In the general case [7] can be used to show that the subspace L ∗ 2 H(x) = H (x), the inner product of L (Hdx) is indefinite, and we make no similar assertion. Proposition 4.1.6. (1) Let F = V f be given by (2.2.1) for some f in L2 (Hdx). If 2 2 (Hdx), then F (z) ≡ 0. f is orthogonal to the subspace L (0) (1) 2 2 (Hdx). (2) The root subspaces Lζ , Lζ , . . . of (4.0.1) are contained in L

206


Proof. (1) Let g ∈ Cm and z ∈ C. By (2.2.1), B C 0 g ∗ F (z) = f (x), W (x, z¯) . g H We show that the function

(4.1.9)

0 Y (x) = W (x, z¯) g

2 2 (Hdx). Consider an H-indivisible interval (a, b), and suppose that belongs to L H(x) is represented as in (4.1.8) on (a, b). By (1.0.3), dY /dx = i¯ z JH(x)Y a.e. By (4.1.8) and the identity η ∗ Jη = 0, d ∗ (η Y (x)) = i¯ z η ∗ Jηh(x)η ∗ Y (x) = 0 dx a.e. on (a, b). Therefore η ∗ Y (x) ≡ const. on (a, b). The constant belongs to the range of η ∗ , and so η ∗ Y (x) = η ∗ c on (a, b) for some c ∈ C2m . Then H(x)Y (x) = ηh(x)η ∗ Y (x) = ηh(x)η ∗ c = H(x)c 2 2 (Hdx), g ∗ F (z) = 0 2 2 (Hdx). Since f is orthogonal to L a.e. on (a, b). Hence Y ∈ L by (4.1.9). By the arbitrariness of g, F (z) ≡ 0. 2 2 (Hdx) by the (2) Eigenfunctions have the form (4.1.1) and hence belong to L (0) (k+1) 2 2 proof of (1). Thus Lζ ⊆ L (Hdx). Let Y = Y (x) ∈ Lζ , k ≥ 0. Then Y satisfies an equation (4.1.4). Let (a, b) be an H-indivisible interval with H(x) represented in the form (4.1.8) on (a, b). By (4.1.4) and (4.1.8), d ∗ (η Y (x)) = iζη ∗ Jηh(x)η ∗ Y (x) + η ∗ Jηh(x)η ∗ Y (k) (x) = 0 dx a.e. on (a, b) because η ∗ Jη = 0. Therefore η ∗ Y (x) ≡ const. on (a, b). As in the 2 2 (Hdx). proof of (1), we deduce that Y ∈ L Proposition 4.1.7. The identity B(z) − B(w) = i(z − w)B(z)B(w) holds at all points w, z such that B(z) and B(w) are defined. Hence the subspace K = ker B(z) is independent of z. Proposition 4.1.7 is a statement about resolvent operators for systems (4.0.1) with constant boundary conditions. The resolvent identity does not hold in general for systems (3.0.1) with variable boundary conditions. Proof. Fix f ∈ L2 (Hdx). Without loss of generality assume that c(z)R + d(z)Q and c(w)R + d(w)Q are invertible. Then according to Definition 3.1.6, B(z)f and B(w)f are determined by Proposition 3.1.5. Set h B(w)f = h = 1 . h2


207

By Proposition 3.1.5, dh = iwJH(x)h(x) + JH(x)f (x), 0 ≤ x ≤ , dx ∗ ∗ h1 (0) = 0, R h1 () + Q h2 () = 0, Hence dh = izJH(x)h(x) + JH(x) f (x) − i(z − w)h(x) , dx

0 ≤ x ≤ ,

h1 (0) = 0, R∗ h1 () + Q∗ h2 () = 0, Therefore B(z) f − i(z − w)h = h, and the result follows.

Proposition 4.1.8. For any complex number ζ, the following are equivalent: (i) ζ is an eigenvalue for (4.0.1); (ii) c(ζ)R + d(ζ)Q is not invertible. ¯ + d(ζ)Q ¯ Proof. (ii) ⇒ (i) If c(ζ)R + d(ζ)Q is not invertible, neither is c(ζ)R by ∗ ∗ ¯ ¯ Lemma 3.1.4. Hence we can choose g = 0 in Cm such that [R∗ c∗ (ζ)+Q d (ζ)]g = 0. Thus (see (4.1.1)) 0 ∈ Lζ . Y (x, ζ) = W (x, ζ) g We show that Y = 0 as an element of L2 (Hdx). Argue by contradiction. If Y is equivalent to zero in L2 (Hdx), then f ∗ (t)H(t)Y (t, ζ) dt = 0 0

for all f ∈ L2 (Hdx). This implies that H(x)Y (x, ζ) = 0 a.e., and hence dY /dx = iζJH(x)Y = 0 a.e. on [0, ]. Therefore Y is constant, and so 0 0 = 0 a.e. and H(x) Y (x, ζ) = Y (0, ζ) = g g By the nondegeneracy condition (iii) in our assumptions on H(x) in Section 2, g = 0, a contradiction. Therefore Y = 0 in L2 (Hdx) and ζ is an eigenvalue for (4.0.1). (i) ⇒ (ii) If ζ is an eigenvalue for (4.0.1), there is a function (4.1.1) which is not equivalent to zero in L2 (Hdx). The vector g in (4.1.1) can therefore not be ¯ + d(ζ)Q ¯ is not invertible. Then c(ζ)R + d(ζ)Q is not invertible zero, and so c(ζ)R by Lemma 3.1.4. Since the functions R(z) ≡ R and Q(z) ≡ Q in (2.3.1) are constant, v(z) = i [a(z)R + b(z)Q] [c(z)R + d(z)Q]

−1

(4.1.10)

is meromorphic in the complex plane. The remaining results in this section add the hypothesis that v(z) has only simple poles.

208


Proposition 4.1.9. Assume that v(z) has only simple poles. Define γ(ζ) for every ζ ∈ C by γ(ζ) v(z) = + v1 (z), (4.1.11) ζ −z where v1 (z) is holomorphic at z = ζ. Then γ(ζ) = 0 except at the poles of v(z), ¯ = γ(ζ)∗ . For each ζ ∈ C, and γ(ζ) ¯ + Q∗ d∗ (ζ)]γ(ζ) ¯ [R∗ c∗ (ζ) = 0, (4.1.12) and hence for every g ∈ Cm ,

0 ∈ Lζ . Y (x, ζ) = W (x, ζ) γ(ζ)g

(4.1.13)

z ), Proof. Clearly γ(ζ) = 0 except at the poles of v(z). Since v(z) = v ∗ (¯ v(z) =

γ(ζ)∗ γ(ζ) z ), + v1∗ (¯ + v1 (z) = ζ−z ζ¯ − z

(4.1.14)

¯ = γ(ζ)∗ . By (4.1.10) and (4.1.14), and hence γ(ζ) i(ζ¯ − z)[a(z)R + b(z)Q] = [γ(ζ)∗ + (ζ¯ − z)v ∗ (¯ z )][c(z)R + d(z)Q]. 1

¯ we deduce (4.1.12). Then (4.1.13) follows from (4.1.1). Letting z → ζ,

The eigenvalues of (4.0.1) are isolated in the complex plane and occur in conjugate pairs by Proposition 4.1.8 and Lemma 3.1.4. Assuming again that v(z) has only simple poles, we fix the following notation for these points. • Let {λj }rj=1 be the real eigenvalues of (4.0.1) (0 ≤ r ≤ ∞). For each j = 1, . . . , r, write τj v(z) = + vj (z), τj = τj∗ = γ(λj ). λj − z ¯k }sj=1 be the nonreal pairs of eigenvalues of (4.0.1) (0 ≤ s ≤ ∞). • Let {μk , μ For each k = 1, . . . , s, write v(z) =

βk βk∗ + vk (z) = z ), + vk∗ (¯ μk − z μ ¯k − z

βk = γ(μk ).

• Let τ = τ R,Q be the collection of all eigenvalues λj , μk , μ ¯k together with the matrices τj , βk , j = 1, . . . , r and k = 1, . . . , s. By Proposition 4.1.1, Lλj ⊥ Lλk if j = k, Lλj ⊥ (Lμk + Lμ¯k ), and Lμk is a neutral subspace of L2 (Hdx) for all j = 1, . . . , r and k = 1, . . . , s. Let L20 (τ ) be the space of all Cm -valued functions defined on the points λj , μk , μ ¯k having only finitely many nonzero values, in the inner product F, GL2 (τ ) =

r

0

j=1

G(λj )∗ τj F (λj ) +

s k=1

G(¯ μk )∗ βk F (μk ) + G(μk )∗ βk∗ F (¯ μk ) .


209

Equivalently, we can consider the elements of L20 (τ ) as Cm -valued functions on the complex plane in the inner product ¯ F, GL2 (τ ) = G∗ (ζ)γ(ζ)F (ζ). (4.1.15) 0

ζ∈C

Two functions F1 and F2 are identified if γ(ζ)[F1 (ζ) − F2 (ζ)] = 0,

ζ ∈ C.

(4.1.16)

The inner product in L20 (τ ) is nondegenerate and in general indefinite. We investigate the transform F = V f defined by (2.2.1) as a mapping from L2 (Hdx) into L20 (τ ). Lemma 4.1.10. Assume that v(z) has only simple poles. Let Y (x, ζ) belong to Lζ and have the form (4.1.1). Then V Y belongs to L20 (τ ) and is equivalent to the function F defined by + z = ζ, Δζ g, (4.1.17) F (z) = 0, z = ζ, where ¯ + d (ζ)c∗ (ζ)]. ¯ Δζ = Δ∗ζ¯ = i[c (ζ)d∗ (ζ)

(4.1.18)

Proof. The identity Δζ = Δ∗ζ¯ follows from (2.1.6). Let G = V Y . For z = ζ, 0 ∗ W (t, z¯) H(t)W (t, ζ) dt G(z) = 0 Im g 0 W ∗ (, z¯)JW (, ζ) − J 0 = 0 Im g i(ζ − z) ∗ ¯ ¯ 0 a(z) b(z) a (ζ) c∗ (ζ) 1 0 Im J ∗ ¯ = ¯ g c(z) d(z) b (ζ) d∗ (ζ) i(ζ − z) =

¯ + d(z)c∗ (ζ) ¯ c(z)d∗ (ζ) g. i(ζ − z)

¯ + d(ζ)c∗ (ζ) ¯ = 0, and so By (2.1.5), c(ζ)d∗ (ζ) G(ζ) = lim

z→ζ

¯ + [d(z) − d(ζ)]c∗ (ζ) ¯ [c(z) − c(ζ)]d∗ (ζ) g i(ζ − z) ¯ + d (ζ)c∗ (ζ)]g ¯ = Δζ g. = i[c (ζ)d∗ (ζ)

Thus F (ζ) = G(ζ). To show that G is equivalent to F in L20 (τ ), by (4.1.17) we must show that for all w = ζ, γ(w)[F (w) − G(w)] = 0, that is, γ(w)G(w) = 0, or γ(w)

¯ + d(w)c∗ (ζ) ¯ c(w)d∗ (ζ) g=0. i(ζ − w)

(4.1.19)

210


Fix w = ζ and consider any g˜ ∈ Cm . Write C B ∗ ¯ ∗ c (w)γ(w)∗ g˜ d (ζ)g ∗ ∗ ¯ ∗ ¯ . g˜ γ(w)[c(w)d (ζ) + d(w)c (ζ)] g = ¯ , d∗ (w)γ(w)∗ g˜ c∗ (ζ)g C2m

(4.1.20)

Define M as in (4.1.3), so M ⊥ = JM . By (4.1.12), [R∗ c∗ (w)+ Q∗ d∗ (w)]γ(w)∗ = 0, and hence ∗ c (w)γ(w)∗ g˜ ∈ M ⊥. (4.1.21) d∗ (w)γ(w)∗ g˜ ∗ ¯ ¯ + Q∗ d∗ (ζ)]g ¯ = 0, that is, c (ζ)g ∈ M ⊥ = JM . Hence By (4.1.1), [R∗ c∗ (ζ) ¯ d∗ (ζ)g ∗ ¯ ∗ ¯ c (ζ)g d (ζ)g =J ∗ ¯ ∈ M. (4.1.22) ∗ ¯ c (ζ)g d (ζ)g ¯ + d(w)c∗ (ζ)] ¯ g = 0. Since g˜ is By (4.1.20), (4.1.21), and (4.1.22), g˜∗ γ(w)[c(w)d∗ (ζ) arbitrary, this proves (4.1.19), and the result follows. Theorem 4.1.11. Assume that v(z) has only simple poles. (1) If f1 and f2 are finite linear combinations of eigenfunctions of (4.0.1), then f2∗ (t)H(t)f1 (t) dt = V f1 , V f2 L2 (τ ) . 0

0

(2) If f ∈ L2 (Hdx) and f is orthogonal to every eigenfunction of (4.0.1), then V f = 0 as an element of L20 (τ ). Definition 4.1.12. By pseudospectral data for (1.0.1) we mean a collection τ of the type considered above satisfying the properties (1) and (2) in Theorem 4.1.11. Proof. (1) By linearity, we may assume that fj (x) = Y (ζj , x), where 0 ∈ Lζj , j = 1, 2, Y (x, ζj ) = W (x, ζj ) gj

(4.1.23)

as in (4.1.1) for some ζ1 , ζ2 ∈ C. By Lemma 4.1.10, V Y (x, ζj ) is equivalent to z = ζj , Δζj gj , Fj (z) = (4.1.24) 0, z = ζj , j = 1, 2, where Δζ is given by (4.1.18). To prove (1), we must show that Y (t, ζ2 )∗ H(t)Y (t, ζ1 ) dt = F1 , F2 L2 (τ ) . (4.1.25) 0

0

Case 1: ζ2 = ζ¯1 . Then (4.1.15),

3 0

Y (t, ζ2 )∗ H(t)Y (t, ζ1 ) dt = 0 by Proposition 4.1.1. By

F1 , F2 L2 (τ ) =

0

ζ∈C

∗ ¯ ¯ F2∗ (ζ)γ(ζ)F 1 (ζ) = F2 (ζ1 )γ(ζ)F1 (ζ1 ),


211

since F1 (ζ) = 0 for ζ = ζ1 by (4.1.24). In the same way, F2 (ζ) = 0 for ζ = ζ2 , and hence F2 (ζ¯1 ) = 0 because ζ¯1 = ζ2 . Thus F1 ⊥ F2 in L20 (τ ), and (4.1.25) follows. ¯ By (2.1.6), Case 2: ζ2 = ζ¯1 . Write the two points as ζ1 = ζ and ζ2 = ζ. ¯ ∗ H(t)Y (t, ζ) dt = 0 g ∗ ¯ ∗ H(t)W (t, ζ) dt 0 Y (t, ζ) W (t, ζ) 2 g1 0 0 0 ∗ ∗ = 0 g2∗ ¯ + d (ζ)c∗ (ζ)] ¯ g1 ∗ i[c (ζ)d∗ (ζ) = g2∗ Δζ g1 . Since F1 , F2 L2 (τ ) =

0

(4.1.26)

∗ ∗ ¯ F2∗ (¯ z )γ(z)F1 (z) = F2∗ (ζ)γ(ζ)F 1 (ζ) = g2 Δζ¯ γ(ζ)Δζ g1 ,

z∈C

in order to verify (4.1.25), we must show that g2∗ Δ∗ζ¯ γ(ζ)Δζ g1 = g2∗ Δζ g1 . ∗ ¯ d (ζ)g1 R . Hence ∈ ran As in the proof of Lemma 4.1.10, ∗ ¯ Q c (ζ)g1 ¯ 1 R˜ g1 = d∗ (ζ)g

and

¯ 1 Q˜ g1 = c∗ (ζ)g

(4.1.27)

(4.1.28)

for some g˜1 ∈ Cm . Thus ¯ + d (ζ)c∗ (ζ)]g ¯ 1 = i [c (ζ)R + d (ζ)Q]˜ Δζ g1 = i [c (ζ)d∗ (ζ) g1 .

(4.1.29)

Next use (4.1.10) and (4.1.11) to write i(ζ − z)[a(z)R + b(z)Q] = [γ(ζ) + (ζ − z)v1 (z)][c(z)R + d(z)Q], where v1 (z) is holomorphic at z = ζ. On differentiating this relation with respect to z and then setting z = ζ, we obtain −i[a(ζ)R + b(ζ)Q] = −v1 (ζ)[c(ζ)R + d(ζ)Q] + γ(ζ)[c (ζ)R + d (ζ)Q].

(4.1.30)

Therefore g2∗ Δ∗ζ¯ γ(ζ)Δζ g1

(4.1.29)

g2∗ Δ∗ζ¯ γ(ζ)i [c (ζ)R + d (ζ)Q]˜ g1 % & (4.1.30) = i g2∗ Δ∗ζ¯ − i[a(ζ)R + b(ζ)Q] + v1 (ζ)[c(ζ)R + d(ζ)Q] g˜1 % (4.1.28) ∗ ∗ ¯ + b(ζ)c∗ (ζ)]g ¯ 1 = g2 Δζ¯ [a(ζ)d∗ (ζ) & ¯ + d(ζ)c∗ (ζ)]g ¯ 1 . + iv1 (ζ)[c(ζ)d∗ (ζ) =

Thus by (2.1.5) and (4.1.18), g2∗ Δ∗ζ¯ γ(ζ)Δζ g1 = g2∗ Δ∗ζ¯g1 = g2∗ Δζ g1 . This proves (4.1.27) and verifies (4.1.25) in Case 2.

212


(2) Let F = V f , where f ∈ L2 (Hdx) and f ⊥ Lζ for all ζ ∈ C. To show that F = 0 as an element of L20 (τ ), by (4.1.16) we must show that γ(ζ)F (ζ) = 0 for every ζ ∈ C. In fact, for every g ∈ Cm , 0 ¯ ∈ Lζ¯ W (x, ζ) (4.1.31) γ(ζ)∗ g by Proposition 4.1.9. By assumption, f is orthogonal to (4.1.31), and so by (2.2.1), ∗ ∗ ¯ ∗ H(t)f (t) dt = 0. 0 Im W (t, ζ) g γ(ζ)F (ζ) = g γ(ζ) 0

Since g is arbitrary, γ(ζ)F (ζ) = 0. 4.2. Definite case: pseudospectral functions

Let a system (4.0.1) be given as before, and in addition assume that H(x) ≥ 0 a.e. Then the function v(z) defined by (4.1.10) is a Nevanlinna function by (2.3.2). The main results of this section appear in Theorems 4.2.2, 4.2.4, and 4.2.5. Proposition 4.2.1. The eigenvalues of (4.0.1) are real. For any complex number ζ, the following are equivalent: (i) ζ is an eigenvalue for (4.0.1); (ii) c(ζ)R + d(ζ)Q is not invertible; (iii) ζ is a pole of v(z). Proof. Since H(x) ≥ 0, the eigenvalues of (4.0.1) are real by Proposition 4.1.1 (or Proposition 3.2.1). The equivalence of (i) and (ii) is shown in Proposition 4.1.8. (iii) =⇒ (ii) This is obvious from the definition of v(z) in (4.1.10). (i) =⇒ (iii) If ζ is an eigenvalue of (4.0.1), then there is a Y = 0 in L2 (Hdx) of the form (4.1.1). By Lemma 4.1.10, V Y is equivalent to the function F (x) given by (4.1.17). Since H(x) ≥ 0, by Theorem 4.1.11(1) and (4.1.15), ¯ 0< Y ∗ (t, ζ)H(t)Y (t, ζ) dt = F, F L2 (τ ) = F ∗ (ζ)γ(ζ)F (ζ). 0

0

So γ(ζ) = 0, and hence ζ is a pole of v(z) by (4.1.11).

Since v(z) is meromorphic in C and a Nevanlinna function, its poles are real and simple, and hence the pseudospectral data constructed in Theorem 4.1.11 take a simpler form. By Proposition 4.2.1, the poles of v(z) coincide with the eigenvalues {λj }rj=1 of (4.0.1). Thus we have τj + vj (z), v(z) = λj − z where τj ≥ 0 and vj (z) is holomorphic at λj , j = 1, . . . , r, and ∞ t 1 − dτ (t), v(z) = α + βz + 1 + t2 −∞ t − z

(4.2.1)


213

where τ (t) is a nondecreasing m × m matrix-valued step function with jumps τj at the points λj , j = 1, . . . , r. The inner product space L20 (τ ) is positive and has a Hilbert space completion to L2 (dτ ). Theorem 4.2.2. The transform F = V f , 0 Im W ∗ (x, z¯) H(x)f (x) dx, F (z) = 0 2 2 acts as a partial isometry D r from L (Hdx) into L (dτ ) with initial space N equal to the closed span N = j=0 Lλj of all eigenfunctions for the system (4.0.1).

It will be shown in Theorem 4.2.4 that the mapping in Theorem 4.2.2 is always onto. Theorem 4.2.2 constructs a family of pseudospectral functions for the system (1.0.1) in the sense of the following definition. Definition 4.2.3. A pseudospectral function for (1.0.1) is a nondecreasing function τ (t) of real t such that the transform 0 Im W ∗ (x, z¯) H(x)f (x) dx (V f )(z) = 0

acts as a partial isometry from L2 (Hdx) into L2 (dτ ). If the partial isometry is an isometry, we call τ (t) a spectral function for (1.0.1). We say that a pseudospectral function τ (t) is orthogonal if the range of the partial isometry is all of L2 (dτ ). Proof of Theorem 4.2.2. By Theorem 4.1.11(1), V acts isometrically from the linear span of all eigenfunctions into L2 (dτ ). Hence V acts isometrically from N into L2 (dτ ). Theorem 4.1.11(2) asserts that every function in L2 (Hdx) which is orthogonal to all eigenfunctions is mapped by V to the zero element of L2 (dτ ). Alternate proof of part of Theorem 4.2.2. We give another proof that V is isometric on N, using resolvent operators and Theorems 4.1.4 and 3.2.4. This argument avoids any use of Lemma 4.1.10 and Theorem 4.1.11. By (3.2.9), V is a contraction from L2 (Hdx) into L2 (dτ ). It is therefore sufficient to show that for any eigenvalues λj and λk , Y (x, λj ), Y (x, λk )H = Fj (t), Fk (t)L2 (dτ ) ,

(4.2.2)

where Y (x, λj ) and Y (x, λk ) are corresponding eigenfunctions and Fj (z) and Fk (z) are their transforms under V . By Theorem 4.1.4, B(z)Y (x, λj ) =

i Y (x, λj ) z − λj

for every z such that c(z)R + d(z)Q is invertible. For such z, B(z)Y (x, λj ), Y (x, λk )H =

i Y (x, λj ), Y (x, λk )H . z − λj

214


We deduce that lim y B(iy)Y (x, λj ), Y (x, λk )H = Y (x, λj ), Y (x, λk )H ,

y→∞

(4.2.3)

where the limit is through points such that c(iy)R + d(iy)Q is invertible. By the identity (3.2.8) in Theorem 3.2.4, ∞ −iy Fj∗ (t) dτ (t) Fk (t) lim y B(iy)Y (x, λj ), Y (x, λk )H = lim y→∞ y→∞ −∞ t − iy = Fj (t), Fk (t)L2 (dτ ) .

(4.2.4)

We obtain (4.2.2) from (4.2.3) and (4.2.4).

Theorem 4.2.4. The pseudospectral function τ (t) constructed in Theorem 4.2.2 is orthogonal. Proof. It is sufficient to show that for each j = 1, . . . , r, V Lλj = Mj , where Mj is the subspace of functions in L2 (dτ ) which are supported at λj . By Lemma 4.1.10, V Lλj ⊆ Mj . Since V |Lλj is one-to-one and dim Mj = rank τj , we only need to show that dim Lλj ≥ rank τj . For any g ∈ Cm , 0 ∈ Lλj Y (x, λj ) = W (x, λj ) (4.2.5) τj g by Proposition 4.1.9. If Y = 0 as an element of L2 (Hdx), then Y ∗ (t, λj )H(t)Y (t, λj ) dt = 0.

(4.2.6)

0

Since H(x) ≥ 0 on [0, ] we conclude that H(x)1/2 Y (x, λj ) = 0, and hence dY = izJH(x)Y = 0 dx a.e. on [0, ]. Thus Y is constant, and so Y (x, λj ) ≡

0

g ∗ τj

H(t) dt 0

0 . Then by (4.2.6), τj g

0 = 0, τj g

and so τj g = 0 by the condition (iii ) at the beginning of Section 2. Therefore we can find a linearly independent set of elements of Lλj of the form (4.2.5) containing rank τj elements. Hence dim Lλj ≥ rank τj , and the result follows. Theorem 4.2.5. The following are equivalent. (i) The function τ (t) in Theorem 4.2.2 is an orthogonal spectral function for the system (1.0.1). (ii) The eigenfunctions for (4.0.1) are complete in L2 (Hdx). (iii) For some and hence any z in the domain of the resolvent, ker B(z) = {0}.


215

Proof. (i) ⇔ (ii) This follows from Theorems 4.2.2 and 4.2.4. (ii) ⇔ (iii) By Proposition 4.1.7, K = ker B(z) is independent of z in the domain of the resolvent. Therefore in (iii) it is sufficient to consider some z = z0 such that z0 = z¯0 and c(z0 )R + d(z0 )Q is invertible. Then iB(z0 ) is a compact selfadjoint operator by Proposition 3.1.7. By the spectral theorem, the eigenfunctions for iB(z0 ) are complete in L2 (Hdx). By Theorem 4.1.4, the eigenfunctions for iB(z0 ) for its nonzero eigenvalues have the same closed span as the eigenfunctions for (4.0.1). Now assume (ii). Then L2 (Hdx) is the closed span of the eigenfunctions for iB(z0 ) for its nonzero eigenvalues. So the origin is not an eigenvalue of iB(z0 ). Thus ker B(z0 ) = {0}, and (iii) follows. The proof that (iii) implies (ii) follows on reversing these steps. Corollary 4.2.6. Conditions (i)–(iii) in Theorem 4.2.5 hold if H(x) has invertible values a.e. Proof. We verify condition (iii) in Theorem 4.2.5. Suppose f ∈ ker B(z0 ) for some real number z0 such that c(z0 )R + d(z0 )Q is invertible. If H(x) has invertible values, then a function in L2 (Hdx) is equivalent to the zero element of the space if and only if it is equal to zero a.e. Hence by Definition 3.1.6 and (3.1.4), x 0 Im J W (x, z0 ) W ∗ (t, z0 )H(t)f (t) dt −iv(z0 ) 0 0 : 0 0 ∗ J + W (t, z0 )H(t)f (t) dt ≡ 0. −iv(z0 ) −Im x Multiply by W (x, z0 )−1 , then differentiate to get W ∗ (x, z0 )H(x)f (x) = 0 a.e. Again since H(x) has invertible values a.e., it follows that f (x) = 0 a.e. This verifies the condition (iii) in Theorem 4.2.5, and so the corollary follows. Recall that by Proposition 4.1.7, the subspace K = ker B(z) is independent 2 2 (Hdx) be as in Definition 4.1.5. of z in the domain of the resolvent. Let L 2 2 (Hdx). Proposition 4.2.7. (1) The subspace K = ker B(z) contains L2 (Hdx) ( L (2) The eigenfunctions for (4.0.1) are complete in L2 (Hdx) ( K. 2 2 (Hdx). We must show that B(z)f = 0 for z in Proof. (1) Let f ∈ L2 (Hdx) ( L the domain of the resolvent. We may suppose that z ∈ C+ ∪ C− , in which case we can use the representation (3.2.8). It follows from (3.2.8) that for any g ∈ L2 (Hdx), ∞ ∗ G (t)dτ (t)F (t) , iB(z)f, gH = t−z −∞ where F and G are the transforms of f and g as in (2.2.1). Since we assume that f is 2 2 (Hdx), F ≡ 0 by Proposition 4.1.6(1). Thus iB(z)f ⊥ L2 (Hdx), orthogonal to L and hence B(z)f = 0.

216


(2) Write K = ker B(z0 ), where z0 is a real number such that c(z0 )R + d(z0 )Q is invertible. As in the proof of Theorem 4.2.5, the eigenfunctions for (4.0.1) have the same closed span as the eigenfunctions for iB(z0 ) for its nonzero eigenvalues. This closed span is L2 (Hdx) ( K because K = ker B(z0 ). In A.L. Sakhnovich [13], the term “pseudospectral function” is used in a little different way from our definition. What is called a “pseudospectral function” in [13] will be called a “strong pseudospectral function” here. Definition 4.2.8. By a strong pseudospectral function for a system (1.0.1) we mean a pseudospectral function τ (t) such that the kernel of V as an operator from L2 (Hdx) into L2 (dτ ) coincides with the set of all f in L2 (Hdx) such that (V f )(z) ≡ 0 for all z. A strong spectral function for (1.0.1) is a spectral function τ (t) such that, whenever f belongs to L2 (Hdx) and its transform F = V f is zero in L2 (dτ ), then (V f )(z) ≡ 0 for all z. The term orthogonal applied to these notions has the same meaning as in Definition 4.2.3. Thus if τ (t) is a pseudospectral function, it may occur that the subspaces K+ = {f : f ∈ L2 (Hdx) and V f = 0 in L2 (dτ )}, K− = {f : f ∈ L2 (Hdx) and (V f )(z) = 0 for all z ∈ C},

(4.2.7)

do not coincide, although in every case K− ⊆ K+ . The condition for τ (t) to be a strong pseudospectral function is that K+ = K− . The next result follows easily from Theorem 4 of A.L. Sakhnovich [13]. Set v0 (z) = i[a(z) + b(z)][c(z) + d(z)]−1 .

(4.2.8)

Since c(z) + d(z) is entire and has value Im for z = 0, it is invertible except at isolated points. It is easy to see that Im v0 (z) ≥ 0 on C+ (but v0 (z) = v0∗ (¯ z )). Proposition 4.2.9. The pseudospectral function τ (t) constructed in Theorem 4.2.2 is strongly pseudospectral if 1 ∗ [c (−iy) − d∗ (−iy)][v(iy) − v0 (iy)][c(iy) − d(iy)] = 0. (4.2.9) lim y→∞ y Dr In this case, the closed span N = j=0 Lλj of the eigenfunctions of (4.0.1) is equal to the closed span of all functions 0 (4.2.10) Y (x, z) = W (x, z) , z ∈ C, g ∈ Cm . g Proof. Define K+ and K− by (4.2.7). By Theorem 4.2.2, VD acts as a partial isomr etry from L2 (Hdx) into L2 (dτ ) with initial space K⊥ + = j=0 Lλj . By [13, Theorem 4(a)], the condition (4.2.9) D r implies that the isometric set of V coincides with ⊥ ⊥ K⊥ . Hence K = K = − − + j=0 Lλj . In particular, K− = K+ , and therefore τ (t) isDa strong pseudospectral function. The last statement follows from the equality r ( j=0 Lλj )⊥ = K− together with the observation that a function f in L2 (Hdx) belongs to K− if and only if f is orthogonal to all functions of the form (4.2.10).


217

Example 4.2.10. A simple example, adapted from Orcutt [9], illustrates some of our results. Take m = 1 and fix a number 0 < x0 < . Consider a system (1.0.1) with ⎧ ⎪ ⎪ 1 0 ⎪ , 0 ≤ t ≤ x0 , ⎪ ⎪ ⎨ 0 0 H(t) = (4.2.11) ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ x0 < t ≤ . ⎩ 0 1 , Then L2 (Hdx) is an infinite-dimensional Hilbert space. The intervals (0, x0 ) and 2 2 (Hdx) of Definition 4.1.5 (x0 , ) are H-indivisible, and therefore the subspace L is two-dimensional. Straightforward calculations show that ⎧ ⎪ 1 0 ⎪ ⎪ , 0 ≤ t ≤ x0 , ⎪ ⎪ ⎨ izt 1 (4.2.12) W (t, z) = ⎪ 2 ⎪ z (t − x ) iz(t − x ) 1 − x 0 0 0 ⎪ ⎪ ⎪ , x0 < t ≤ , ⎩ ix0 z 1 and

(V f )(z) =

f2 (t) dt, x0

f (x) =

f1 (x) ∈ L2 (Hdx). f2 (x)

The functions (2.1.4) are given by 1 − x0 z 2 ( − x0 ) a(z) b(z) = c(z) d(z) −iz( − x0 )

−ix0 z 1

(4.2.13)

.

(4.2.14)

As an illustration of Theorem 4.2.2, consider a system (4.0.1), where R and ¯ + QR ¯ = 0. Set Q are numbers not both zero such that RQ v(z) = i

a(z)R + b(z)Q [1 − x0 z 2 ( − x0 )]R − ix0 zQ =i . c(z)R + d(z)Q −iz( − x0 )R + Q

When R = 0,

τ1 + x0 z , λ1 − z where τ1 = 1/( − x0 ), λ1 = ρ/( − x0 ), and ρ = −iQ/R. Thus τ (x) is a step function with a single jump at x = λ1 . The eigenspace Lλ1 is one-dimensional and spanned by ⎧ ⎪ 0 ⎪ ⎪ , 0 ≤ t ≤ x0 , ⎪ ⎪ ⎨ 1 Y (t, λ1 ) = (4.2.15) ⎪ ⎪ iλ (t − x ) ⎪ 1 0 ⎪ , x0 ≤ t ≤ . ⎪ ⎩ 1 v(z) =

We easily check that V is a partial isometry from L2 (Hdx) onto L2 (dτ ) with initial space Lλ1 . In particular, a pseudospectral function need not be a spectral

218


function. When R = 0, v(z) = x0 z and τ (x) is constant. There are no poles and no eigenvalues, and we interpret the span of the eigenfunctions to be the zero subspace of L2 (Hdx). Trivially V is the zero operator on L2 (Hdx) to L2 (dτ ) = {0}. The question arises if Theorem 4.2.2 generalizes to systems (3.0.1) with nonconstant functions R(z) and Q(z). That is, is the function τ (x) in (3.2.3) always a pseudospectral function? An example shows that this is not necessarily the case. Choose the Nevanlinna pair Q(z) = −iqz,

R(z) = 1,

z ∈ C,

where q > 0. By (4.2.14), v(z) = i and so

a(z)R(z) + b(z)Q(z) 1 1 = x0 z − , c(z)R(z) + d(z)Q(z) − x0 + q z ⎧ ⎨

1 , − x τ (x) = 0+q ⎩ 0,

x ≥ 0, x < 0. 2

By (4.2.13), the transform F = V f of any f in L (Hdx) is constant. The orthogonal complement of ker V in L2 (Hdx) is spanned by the element ⎧ 0 ⎪ ⎪ 0 < x < x0 , ⎪ ⎨ 0 , f0 (x) = ⎪ ⎪ 0 ⎪ ⎩ , x0 < x < , 1 whose transform F0 = V f0 is given by F0 (z) = − x0 . Thus F0 2L2 (dτ ) =

( − x0 )2 < − x0 = f0 2H , − x0 + q

so V is not isometric on the orthogonal complement of its kernel. Hence τ (x) is not a pseudospectral function. We remark that the inequality F0 2L2 (dτ ) ≤ f0 2H is a special case of (3.2.9). Added in proof. The authors plan to treat the case in which v(z) has nonsimple poles in future work.

References [1] D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems. I. Foundations, Integral Equations Operator Theory, 29 (1997), no. 4, 373–454; II. The inverse monodromy problem, ibid. 36 (2000), no. 1, 11–70; III. More on the inverse monodromy problem, ibid. 36 (2000), no. 2, 127–181; IV. Direct and inverse bitangential input scattering problems, ibid. 43 (2002), no. 1, 1–67; V. The inverse input scattering problem for Wiener class and rational p × q input scattering matrices, ibid. 43 (2002), no. 1, 68–129.


219

[2] F.V. Atkinson, Discrete and continuous boundary problems. Mathematics in Science and Engineering, Vol. 8, Academic Press, New York, 1964. [3] L. de Branges, The expansion theorem for Hilbert spaces of entire functions. Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), Amer. Math. Soc., Providence, R.I. (1968), 79–148. [4] , Hilbert spaces of entire functions. Prentice-Hall Inc., Englewood Cliffs, N.J., 1968. [5] I.C. Gohberg and M.G. Kre˘ın, Theory and applications of Volterra operators in Hilbert space. American Mathematical Society, Providence, R.I., 1970. [6] S. Hassi, H. De Snoo, and H. Winkler, Boundary-value problems for two-dimensional canonical systems. Integral Equations Operator Theory 36 (2000), no. 4, 445–479. [7] I.S. Kac, Linear relations generated by a canonical differential equation of dimension 2, and eigenfunction expansions. Algebra i Analiz 14 (2002), no. 3, 86–120, Engl. transl., St. Petersburg Math. J. 14 (2003), no. 3, 429–452. [8] M. Kaltenb¨ ack and H. Woracek, Pontryagin spaces of entire functions. I. Integral Equations Operator Theory 33 (1999), no. 1, 34–97; II, ibid. 33 (1999), no. 3, 305– 380; III, Acta Sci. Math. (Szeged) 69 (2003), no. 1–2, 241–310; IV, ibid. 72 (2006), no. 3–4, 709–835. [9] B. Orcutt, Canonical differential equations. Ph.D. thesis, University of Virginia, 1969. [10] J. Rovnyak and L.A. Sakhnovich, Spectral problems for some indefinite cases of canonical differential equations. J. Operator Theory 51 (2004), 115–139. [11] , Inverse problems for canonical differential equations with singularities. Recent advances in matrix and operator theory, Oper. Theory Adv. Appl. 179 (2007), Birkh¨ auser, Basel, 257–288. [12] , On indefinite cases of operator identities which arise in interpolation theory, The extended field of operator theory, Oper. Theory Adv. Appl. 171 (2007), Birkh¨ auser, Basel, 281–322. [13] A.L. Sakhnovich, Spectral functions of a second-order canonical system, Mat. Sb. 181 (1990), no. 11, 1510–1524, Engl. transl., USSR-Sb. 71 (1992), no. 2, 355–369. [14] L.A. Sakhnovich, Interpolation theory and its applications, Kluwer, Dordrecht, 1997. [15] , Spectral theory of canonical differential systems. Method of operator identities. Oper. Theory Adv. Appl. 107, Birkh¨ auser, Basel, 1999. J. Rovnyak Department of Mathematics University of Virginia P. O. Box 400137 Charlottesville, VA 22904–4137, USA e-mail: [email protected] L.A. Sakhnovich 735 Crawford Avenue Brooklyn, NY 11223, USA e-mail: [email protected]



On Wave Hyperbolic Model for Disturbance Propagation in Magnetic Fluid Igor Selezov Abstract. A new extended model of ferrohydrodynamics is presented. The model takes into account the fluid compressibility and heat relaxation. As a result, such a model is governed by the partial differential equations of hyperbolic-elliptic type and predicts the wave propagation with a finite velocity unlike tradition models. The solvability of the corresponding problem for plane waves is investigated. Mathematics Subject Classification (2000). Primary 76W05; Secondary 78A40. Keywords. Wave hyperbolic model, magnetic fluid, disturbance propagation, finite velocity.

1. Extended equations of magnetic fluid motion Tradition model of magnetofluid dynamics includes the heat equation of parabolic type which predicts the propagation of a weak discontinuity with infinite velocity after excitation of the medium. Here this paradox is overcome by the extension of parabolic partial differential operator up to hyperbolic ones [2]. Similarly, the elliptic partial differential operator for incompressible fluid is extended up to hyperbolic ones for compressible fluid. Moreover, unlike traditional model we take into account the effect of the temperature on a magnetization and, as a result, obtain the extended equation for magnetic potential instead of Laplace equation [1]. As a result, instead of tradition equations of ferrohydrodynamics of parabolic-elliptic type, we obtain the system of equations of hyperbolic-elliptic type predicting the propagation of disturbances with finite velocities. Propagation of plane monochromatic waves is considered on the basis of this system. The ferrofluid (magnetic fluid) is a suspension of magnetite particles with sizes of (3 − 15) 10−9 m in vacuum oil. The density of particles is 1023 particles/m3 . Such a structure allows to describe it by the equations of continuum physics. The main significant property of magnetic fluid is a strong magnetization.

222

I. Selezov

We present the extended equations describing a weak disturbance propagation in the compressible inviscid magnetic fluid which include the balance and constitutive equations: the equation of momentum conservation ∂V ·∇ V = −∇ˆ p + μ0 M ·∇ H, ρˆ + V (1.1) ∂t the extended state equation ∂ pˆ ·V + βK ∂T , = −K ∇ ∂t ∂t the extended hyperbolic equation of heat propagation ∂T ∂2T ·V , = k∇2 T − τ 2 − γ ∇ ∂t ∂t the Maxwell equations

×H = 0, ∇ · H +M = 0, ∇

(1.2)

(1.3)

(1.4)

the constitutive equations [1] = H M, ρˆ = ρ0 [1 − β (T − T0 )] , M H M = M0 − Kp (T − T0 ) + χ (H − H0 ) ,

(1.5) (1.6)

and M are the vectors of magnetic intensity where V is the velocity vector, H and magnetization, p and ρ are the pressure and density, T is a temperature, K is a bulk modulus, β is a coefficient of bulk temperature dilatation, k is a coefficient of thermal conductivity, τ is a thermal relaxation time, γ is a coefficient of thermoelastic diffusion, Kp is a pyromagnetic constant, χ is the susceptibility, is the Hamilton operator. ∇ Unlike tradition equations of ferrohydrodynamics of parabolic-elliptic type the system (1.1)–(1.6) includes the state equation (1.2) instead of the equation · v = 0 and equation (1.3) taking into account the relaxation effect τ ∂ 2 T2 and ∇ ∂t · v . fluid compressibility γ ∇ A total field is presented as a superposition of undisturbed field and small disturbed field pˆ (x, t) = p0 + p (x, t) , ρˆ (x, t) = ρ0 + ρ (x, t) , T (x, t) = T0 (x) + tˆ(x, t) , (x, t) = 0 + ν (x, t) , H (x, t) = H 0 (x) + h (x, t) , V

(1.7)

M (x, t) = M0 (x) + m (x, t) Substituting (1.7) into the system (1.1)–(1.6) yields the static and dynamic problems. In the first problem desired functions do not depend on time, they correspond to an undisturbed state. For the second problem, assuming that disturbed

Wave Hyperbolic Model

223

values are small in comparison with undisturbed ones and introducing the velocity potential ϕ and magnetic field potential ψ according to the representations h = ∇ψ generate the following system of linearized equations ν = ∇ϕ, ∂ψ 2 ˆ μ0 (1 + χ) ∂ ϕ 2 2 2 ∂t 0 · ∇ + , (1.8) ∇Ψ − c ∇ ϕ = −βc 0 0 ∂t2 ∂t ρ0 ∂t

where c0 =

∂ 2 tˆ 1 ∂ 2 tˆ 1 2 2 2ˆ = ∇ ϕ, − c ∇ t + t ∂t2 τ ∂t τ Kp ˆ 2 ∇t, ∇ ψ= χ

K ρ0

(1.9)

(1.10)

is the velocity of propagation of dilatation waves, ct = K τ is

the velocity of heat propagation, in (1.10) ∇ is the scalar Hamilton operator. Equation (1.8) includes at the right-hand part the term taking into account the influence of temperature and dissipative term connected with the losses in magnetic fluid. Equation (1.9) includes the term with the time relaxation τ and the term taking into account the effect of dilatational field. As is seen from equation 0 = 0. Thus, the (1.8) the last term is not equal to zero only in the case when ∇Ψ original equations for the vector field are reduced to a closed system of equations (1.8)–(1.10) for three scalar functions ϕ, tˆ and ψ.

2. Propagation of plane waves In this case the systems (1.8)–(1.10) is written for the functions ϕ (x, t) , tˆ(x, t) and ψ (x, t) as follows 2 2 ˆ μ0 (1 + χ) ∂ ∂2ϕ ∂ ψ 2∂ ϕ 2 ∂t Ψ0 (x) , (2.1) − c0 2 = −βc0 + ∂t2 ∂x ∂t ρ0 ∂x ∂x∂t 2ˆ ∂ 2 tˆ 1 ∂ tˆ 1 ∂ 2 ϕ 2∂ t = − c + , t ∂t2 ∂x2 τ ∂t τ ∂x2 ∂2ψ Kp ∂ tˆ . = 2 ∂x χ ∂x Later on we consider the case of the constant magnetic field H0 , so that

Ψ0 (x) = H0 x.

(2.2) (2.3)

(2.4)

As a result, the system (2.1)–(2.3) with taking into account (2.4) can be reduced to the following equation 2 2 2 2ˆ ∂ ∂ tˆ ∂ tˆ 2 ∂ 2∂ t τ 2 − τ ct 2 + − c0 2 ∂t2 ∂x ∂t ∂x ∂t 3 Kp ∂ tˆ = 0, (2.5) − (−qt + qm ) H0 χ ∂x2 ∂t where qt = βc20 , qm =

μ0 (1+χ) . ρ0

224

I. Selezov Equation (2.5) can be presented in the form τ

∂ 4 tˆ ∂ 4 tˆ ∂ 4 tˆ − τ c2t + c20 + τ c20 c2t 4 4 2 2 ∂t ∂t ∂x ∂x 3ˆ 3ˆ ∂ t ∂ t + 3 + (qt − qm ) = 0. ∂t ∂t∂x2

(2.6)

3. Stationary waves For investigation of stationary waves the function tˆ is presented as follows tˆ(x, t) = f (x − ct) = f (θ) ,

(3.1)

where θ is the phase. In this case the equation (2.6) can be reduced to the following equation a1 a0 (3.2) f − f − f = 0, a2 a2 where a0 = τ c2t + c20 c2 , a1 = c c2 + qt − qm , a2 = τ c4 + c20 c2t . The solution of equation (3.2) is of the form f (θ) = B1 eκ1 θ + B2 eκ2 θ , where κ1,2

1 a1 = ± 2 a2

= 2 1 a1 a0 + . 4 a2 a2

(3.3)

(3.4)

One can see from (3.4), the real roots exist at the conditions a21 > 4a0 , c2 +qt > qm and in this case the solutions (3.3) corresponding to stationary waves do not exist.

4. Travelling waves The solution is presented in the form of monochromatic waves tˆ(x, t) ∼ ei(kx−ωt) .

(4.1)

After substituting (4.1) into (2.6) we obtain the condition of solvability in the form λ λ τ c4p − τ c2t + c20 c2p + τ c20 c2t − i c3p − i (qt − qm ) cp = 0, (4.2) 2π 2π 2π where cp = ωk is the phase velocity, λ is the wavelength, k = 2π λ , ω = λ cp . In general case the equation (4.2) gives two pair of complex-conjugate roots. Later on we consider two degenerate cases: long wavelength approximation and short wavelength approximation. At λ → ∞ (or τ → 0) the slow motions are determined by the condition of existence of solutions √ cp = q m − qt , qm > qt .

Wave Hyperbolic Model At λ → 0 the equation degenerates to c4p − c2t − c20 c2p + c20 c2t = 0

225

(4.3)

and gives the following conditions of solvability: cp = ct and cp = c0 which correspond to two characteristics.

References [1] B. Berkovsky, V. Medvedev, M. Krakov, Magnetic fluids: engineering applications. Oxford: Oxford University Press, 1993. [2] I. Selezov, Nonlinear wave propagation in close to hyperbolic systems. Int. Series of Numerical Mathematics, Birkh¨ auser Verlag Basel/Switzerland 141 (2001), 851–860. Igor Selezov Institute of Hydromechanics NAS Sheliabov Str. 8/4, Kiev 03680, Ukraine e-mail: [email protected]


Part 2 Research Papers



High-accuracy Stable Difference Schemes for Well-posed NBVP Allaberen Ashyralyev Abstract. The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP) v (t) + Av(t) = f (t)(0 ≤ t ≤ 1), v(0) = v(λ) + μ,

0{τ (u −uk−1 )}N > 1 Lp,τ (E) ≤ M [ ϕ Lp,τ (E) + Aμ E ] k does not, generally speaking, hold in an arbitrary Banach space E and for the general strong positive operator A. Nevertheless, in [9], the well-posedness of the difference schemes (1.3) and (1.4) was established in Bohner spaces Lp,τ (E) = Lp ([0, 1]τ, E) with 1 < p < ∞ and Lτ,p (Eα,p )(0 < α < 1, 1 ≤ p ≤ ∞) and Lp (Eα,q )(0 < α < 1, 1 < p, q < ∞) under assumption (1.2). Finally, methods for numerical solutions of the evolution differential equations have been studied extensively by many researchers (see [13]–[33] and the references therein). In the present paper, the single step difference schemes of the high order of accuracy for the approximate solution of this problem are presented. The construction of these difference schemes is based on the Padé difference schemes for the solution of the initial-value problem for abstract parabolic equation and the high order approximation formula for v(0) = v(λ) + μ. The stability and coercive stability of the difference schemes (1.3) and (1.4) in Cτα,α (E) and , Cτ (Eα ) (0 < α < 1) spaces and almost coercive stability (with multiplier min ln τ1 , 1 + |ln A E→E | ) of these difference schemes in Cτ (E) spaces are established.

2. Padé difference schemes Applying the high order approximation formula for v(0) = v(λ) + μ and using the Padé difference schemes for the solution of the initial-value problem for abstract parabolic equation (see [10]), one can present the single step Padé difference schemes of (p + q)th-order of accuracy q uk − uk−1 p + αj (−A)j τ j−1 uk − βj (−A)j τ j−1 uk−1 = ϕk , (2.1) j=1 j=1 τ p j−1 ϕk = − αj (−A)j−i+1 f (i) (tk )τ j−1 j=1 i=0 q j−1 + βj (−A)j−i+1 f (i) (tk−1 )τ j−1 , 1 ≤ k ≤ N, j=1

i=0

232

A. Ashyralyev λ ∈ Z, τ

u0 = u λ + ϕ0 + μ, τ

u0 =

m=0

⎧ ⎨ ϕ0 =

p+q−1

⎩ 0p+q−1 m=1

where

(−A)m λ (λ − τ )m u[ λ ] + ϕ0 + μ, τ m! τ

λ ∈ / Z, τ

λ 0, τ ∈ Z 0 m−1 λ 1 m m−i+1 (i) f (t[ λ ] ), i=0 (−A) m! (λ − τ τ ) τ

⎧ ⎨αj = ⎩

βj =

(p+q−j)!p!(−1)j (p+q)!j!(p−j)! (p+q−j)!q! (p+q)!j!(q−j)!

λ τ

∈ / Z,

1 ≤ j ≤ p, (2.2) 1 ≤ j ≤ q.

The operator A is said to be strongly positive if its spectrum σ (A) lies in the interior of the sector of angle φ, 0 < 2φ < π, symmetric with respect to the real axis, and if on the edges of this sector, S1 (φ) = {ρeiφ : 0 ≤ ρ ≤ ∞ } and −1 S2 (φ) = {ρe−iφ : 0 ≤ ρ ≤ ∞}, and outside the resolvent (λ − A) is subject to the bound > > M (φ) > −1 > . (2.3) ≤ >(λ − A) > 1 + |λ| E→E The strong positivity of the operator implies the existence of a bounded operator Rq,p (τ A) = Pq,p (τ A)Q−1 q,p (τ A), defined on the entire space E. Here, Pq,p (τ A) = I +

q

βj (−τ A)j , Qq,p (τ A) = I +

j=1

p

αj (−τ A)j .

j=1

Note that the Padé difference schemes (2.1) for q = p−2, p−1, p include difference schemes of arbitrary order of accuracy. Moreover, the corresponding functions Rq,p (z), q = p − 2, p − 1, p are bounded at infinity. Unfortunately, the stability and coercive stability of the Padé difference schemes in the special cases q = p = 1 for the approximate solutions of problem (1.1) have not been established. Therefore, we will consider the Padé difference schemes (2.1) for q = p − 2, p − 1. Assume that (p + q)τ ≤ λ for λτ ∈ / Z. Initially, the following necessary lemmas will be provided. Lemma 2.1. [10] For any 1 ≤ k ≤ N, one has the estimates k k Rq,p (τ A) E−→E ≤ M (δ), kτ ARq,p (τ A) E−→E ≤ M (δ).

Lemma 2.2. If λ τ

λ τ

(2.4)

λ τ

∈ Z, then the operator I − Rq,p (τ A) has a bounded inverse

Tτ = (I − Rq,p (τ A))−1 and Tτ E→E ≤ M (λ, δ).

(2.5)

Tτ − (I − exp{−λA})−1 λ −1 τ Rq,p = Tτ (I − exp{−λA}) (τ A) − exp{−λA} .

(2.6)

Proof. We have

High-accuracy Stable Difference Schemes for Well-posed NBVP

233

Using estimates (1.2) and (2.4), we obtain λ

τ Rq,p (τ A) − exp{−λA} E→E ≤ M (λ, δ)τ p+q ,

(I − exp{−λA})−1 E→E ≤ M (λ, δ). Then, applying the triangle inequality, formula (2.6) and last two estimates, we obtain estimate (2.5). λ m [ λτ ] m 0p+q−1 Lemma 2.3. Let B = m=0 (−A) m! (λ − τ τ ) Rq,p (τ A) and operator I − B has a bounded inverse Tτ = (I − B)−1 and

λ τ

∈ / Z, then the

Tτ E→E ≤ M (λ, δ).

(2.7)

Proof. We have

λ τ A})−1 Tτ − (I − exp{− τ −1 λ λ τ A} τ A} . B − exp{− = Tτ I − exp{− τ τ

Using estimates (1.2) and (2.4), we obtain p+q−1 (−A)m λ [λ τ] (λ − τ )m Rq,p (τ A) E→E ≤ M (λ, δ)τ, m! τ m=1

λ τ A} E→E ≤ M (λ, δ)τ p+q , − exp{− τ λ τ A})−1 E→E ≤ M (λ, δ). (I − exp{− τ

[λ τ] Rq,p (τ A)

(2.8)

(2.9)

(2.10) (2.11)

Then, applying the triangle inequality, formula (2.8) and estimates (2.9)–(2.11), we obtain estimate (2.7). Lemma 2.4. For the solution of problem (2.1), the following formula holds: ⎧ k 0 ⎪ k k−j ⎪ Rq,p (τ A)u0 + Rq,p (τ A)Q−1 k = 1, . . . , N, ⎪ q,p (τ A)ϕj τ, ⎪ ⎪ j=1 ⎪ ⎪ λ ⎪ ⎪ τ λ 0 ⎪ ⎪ λ τ −j ⎪ { Rq,p (τ A)Q−1 T ⎪ τ q,p (τ A)ϕj τ + ϕ0 + μ}, τ ∈ Z, k = 0, ⎪ ⎨ j=1 (2.12) uk = λ m ⎪ 0p+q−1 (−A)m ⎪ ⎪ Tτ { m=0 ⎪ m! (λ − τ τ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ ⎪ [0 τ] ⎪ [λ ⎪ λ τ ]−j ⎪ Rq,p (τ A)Q−1 / Z, k = 0. ⎩ × q,p (τ A)ϕj τ + ϕ0 + μ}, τ ∈ j=1

234

A. Ashyralyev

Here,

⎧ ⎪ ⎪ ⎨ Tτ =

λ

τ (I − Rq,p (τ A))−1 for

⎪ ⎪ ⎩ I − 0p+q−1 m=0

(−A)m m! (λ

−

λ τ

λ τ

∈ Z,

−1 [λ τ] τ )m Rq,p (τ A) for

λ τ

∈ / Z.

Proof. By [10], [formula (0.50) in Chapter 3], k uk = Rq,p (τ A)u0 +

k

k−j Rq,p (τ A)Q−1 q,p (τ A)ϕj τ,

k = 1, . . . , N

j=1

for the solution of the Padé difference schemes for the approximate solutions of Cauchy problem

u (t) + Au(t) = f (t)(0 ≤ t ≤ 1), u(0) = u0 . If

λ τ

(2.13)

∈ Z, then from this formula and the condition u0 = u λ + μ τ

it follows that λ

λ τ

u0 = Rq,p (τ A)u0 +

τ

λ

−j

τ Rq,p (τ A)Q−1 q,p (τ A)ϕj τ + μ.

j=1 λ τ

λ

τ Recall that the operator I −Rq,p (τ A) has a bounded inverse Tτ = (I −Rq,p (τ A))−1 . Therefore, we have λ

u0 = Tτ {

τ

λ

−j

τ Rq,p (τ A)Q−1 q,p (τ A)ϕj τ + ϕ0 + μ}.

j=1

If

λ τ

∈ / Z, then from this formula and the condition p+q−1 (−A)m λ u0 = (λ − τ )m u[ λ ] + ϕ0 + μ τ m! τ m=0

it follows that u0 =

p+q−1 m=0

⎛

(−A)m λ (λ − τ )m m! τ ⎞

λ

[λ τ]

× ⎝Rq,p (τ A)u0 +

[τ ]

[λ τ ]−j

Rq,p

⎠ + ϕ0 + μ. (τ A)Q−1 q,p (τ A)ϕj τ

j=1

Recall that the operator I−

p+q−1 m=0

(−A)m λ [λ τ] (λ − τ )m Rq,p (τ A) m! τ


235

has a bounded inverse Tτ = (I −

p+q−1 m=0

(−A)m λ [λ τ] (λ − τ )m Rq,p (τ A))−1 . m! τ

Therefore, we have u0 = Tτ

p+q−1 m=0

: [λ τ] (−A)m λ [λ m τ ]−j (λ − τ) Rq,p (τ A)Q−1 (τ A)ϕ τ + ϕ + μ . j 0 q,p m! τ j=1

3. The main theorems The nonlocal boundary value problem (2.1) will be said to be stable in Fτ (E) if the following inequality holds: uτ Fτ (E) ≤ M [ μ + ϕ0 E + ϕτ Fτ (E) ]. Theorem 3.1. Let τ be a sufficiently small number. Then, the boundary value problem (2.1) is stable in Cτ (E) and Cτα,α (E). Proof. By [10], [Theorems 1.5 and 2.1 in Chapter 3], uτ Cτ (E) ≤ M [ u0 E + ϕτ Cτ (E) ],

(3.1)

uτ Cτα,α (E) ≤ M [ u0 E + ϕτ Cτα,α (E) ]

(3.2)

for the solution of the Padé difference schemes of (p + q)th-order of accuracy p q uk − uk−1 j j−1 + αj (−A) τ uk − βj (−A)j τ j−1 uk−1 = ϕk , τ j=1 j=1

ϕk = −

p j=1

+

q j=1

βj

j−1

αj

j−1

(3.3)

(−A)j−i+1 f (i) (tk )τ j−1

i=0

(−A)j−i+1 f (i) (tk−1 )τ j−1 ,

1 ≤ k ≤ N, u0 = v0

i=0

for the approximate solution of Cauchy problem

v (t) + Av(t) = f (t)(0 ≤ t ≤ 1), v(0) = v0 . Applying formulas (2.12), (2.13) and estimates (2.5), (2.7), we get > λ > > τ > > λ −j > λ −1 τ > Rq,p (τ A)Qq,p (τ A)ϕj τ + ϕ0 + μ> u0 E ≤ M (δ) > > , τ ∈ Z, >j=1 > E

(3.4)

(3.5)

236

A. Ashyralyev >p+q−1 > (−A)m λ > (λ − τ )m u0 E ≤ M (δ) > > m! τ m=0 λ

×

[τ ] j=1

[λ τ ]−j Rq,p (τ A)Q−1 q,p (τ A)ϕj τ

> > > + ϕ0 + μ> > , >

(3.6) λ ∈ / Z. τ

E

Using estimates (2.4), (3.5) and (3.6), we get $ # u0 E ≤ M1 (δ) μ + ϕ0 E + ϕτ Cτ (E) .

(3.7)

Finally, from estimate (3.7) for solutions of the boundary value problem (2.1) and estimates (3.1), (3.2) it follows the stability of the boundary value problem (2.1) in Cτ (E) and Cτα,α (E). Theorem 3.1 is proved. The nonlocal boundary value problem (2.1) is said to be coercively stable (well posed) in Fτ (E), if we have the following coercive inequality >⎧ ⎫N > > > p q >⎨ ⎬ > > > j j−1 j j−1 αj (−A) τ uk − βj (−A) τ uk−1 > > >⎩ ⎭ > j=1 > > j=1 1 Fτ (E)

$ # , τ + τ −1 (uk − uk−1 ) N 1 Fτ (E) ≤ M A(μ + ϕ0 ) E + ϕ Fτ (E) . Since the nonlocal boundary value problem (1.1) in the space C(E) of continuous functions defined on [0,1] and with values in E is not well posed for the general positive operator A and space E, then the well-posedness of the difference boundary value in Cτ (E) norm does not take place uniformly with respect to τ > 0. This means that the coercive norm , uτ Kτ (E) = τ −1 (uk − uk−1 ) N 1 Cτ (E) >⎧ ⎫N > > p > q >⎨ ⎬ > > > j j−1 j j−1 +> αj (−A) τ uk − βj (−A) τ uk−1 > >⎩ ⎭ > j=1 > j=1 > 1

Cτ (E)

tends to ∞ as τ → +0. The investigation of the difference problem (2.1) permits us to establish the order of growth of this norm to ∞. Theorem 3.2. Let τ be a sufficiently small number. Then, for the solution of the difference problem (2.1), we have the following almost coercive inequality : 1 τ u Kτ (E) ≤ M (δ) min ln , 1 + |ln A E→E | ϕτ Cτ (E) τ (3.8) $ + A(μ + ϕ0 ) E .


237

Proof. By [10], [Theorem 1.6 in Chapter 3], : 1 uτ Kτ (E) ≤ M [min ln , 1 + |ln A E→E | ϕτ Cτ (E) + Au0 E τ

(3.9)

for the solution of the Padé difference schemes (3.3). Applying formulas (2.12), (2.13) and estimates (2.5), (2.7), we get > λ > > τ > > > λ λ −1 τ −j > Au0 E ≤ M (δ) > ARq,p (τ A)Qq,p (τ A)ϕj τ + A(μ + ϕ0 )> > , τ ∈ Z, (3.10) > j=1 > E

>p+q−1 > (−A)m λ > Au0 E ≤ M (δ) > (λ − τ )m A > m! τ m=0

[λ τ]

×

j=1

[λ τ ]−j

Rq,p

> > > > (τ A)Q−1 (τ A)ϕ τ + A(μ + ϕ ) j 0 > , q,p >

(3.11) λ ∈ / Z. τ

E

Using estimates (2.4), (3.5) and (3.6), we get Au0 E ≤ M1 (δ) A(μ + ϕ0 ) E : 1 τ + min ln , 1 + |ln A E→E | ϕ Cτ (E) . τ

(3.12)

From estimate (3.12) for solutions of the boundary value problem (2.1) and estimates (3.10), (3.11) it follows estimate (3.8). Theorem 3.2 is proved. Theorem 3.3. Let τ be a sufficiently small number. Then, the following coercivity inequality holds: > p :N > q > > j j−1 j j−1 > > αj (−A) τ uk − βj (−A) τ uk−1 > > j=1

1

j=1

>, - > > + > τ −1 (uk − uk−1 ) N 1

Cτ (Eα )

(3.13)

Cτ (Eα )

≤

M (δ, ψ) ϕτ Cτ (Eα ) +M (δ, ψ) A(μ + ϕ0 ) Eα . α(1 − α)

Proof. By [10], [Theorem 4.2 in Chapter 3], ⎫N ⎧ p q ⎬ ⎨ αj (−A)j τ j−1 uk − βj (−A)j τ j−1 uk−1 Cτ (Eα ) ⎭ ⎩ j=1

j=1

>, - > > + > τ −1 (uk − uk−1 ) N 1 Cτ (Eα ) ≤

(3.14)

1

M ϕτ Cτ (Eα ) + Au0 Eα α(1 − α)

238

A. Ashyralyev

for the solution of the Padé difference schemes (3.3). Applying formulas (2.12), (2.13) and estimates (2.5), (2.7), we get > ⎡> λ > τ > > > λ −1 τ −j > Au0 Eα ≤ M (δ) ⎣> AR (τ A)Q (τ A)ϕ τ q,p j > q,p > > j=1 > (3.15) Eα λ ∈ Z, + A (ϕ0 + μ) Eα , τ Au0 Eα

>p+q−1 > (−A)m λ > τ )m ≤ M (δ) > (λ − > m! τ m=0 >

λ

×

[τ ]

> [λ τ ]−j > ARq,p (τ A)Q−1 q,p (τ A)ϕj τ > Eα j=1

+ A (ϕ0 + μ) Eα ,

λ ∈ / Z. τ (3.16)

λ

Let us estimate the norm of

τ 0

λ

−j

τ ARq,p (τ A)Q−1 q,p (τ A)ϕj τ for

j=1 λ

λ τ

∈ Z. We use the

−j+1

τ Cauchy-Riesz formula for the operator A(λ + A)−1 ARq,p (τ A)Q−1 q,p (τ A) ( see, [10]) λ λ 1 τ −1 −1 τ −j τ −j A(z−τ A)−1 ϕj , zRq,p (z)Q−1 A(λ+A) ARq,p Qq,p (τ A)ϕj = q,p (z) 2πi λτ + z

S1 ∪S2

where S1 = {ρeiψ , 0 ≤ ρ < ∞} and S2 = {ρeiψ , 0 ≤ ρ < ∞}, 0 ≤ ψ < one can write

π 2.

λ

A(λ + A)−1

τ

λ

−j

τ ARq,p (τ A)Q−1 q,p (τ A)ϕj τ

j=1

1 = 2πi

λ

τ

λ

−j

τ zRq,p (z)Q−1 q,p (z)

S1 ∪S2 j=1

τ A(z − τ A)−1 ϕj τ dz. λτ + z

By estimate (2.3), > > λ > > τ > > λ −1 τ −j > > A(λ + A)−1 AR (τ A)Q (τ A)ϕ τ q,p j > q,p > > > j=1 λ ∞ τ λ τ −j ≤M zRq,p (z)Q−1 q,p (z)

0

j=1

E

> > τ >A(|z| + τ A)−1 ϕj > τ d |z| . E λτ + |z|

Then,

High-accuracy Stable Difference Schemes for Well-posed NBVP Since z = ρe±iψ , with |ψ|
0. Hence, λ

τ

λ

−j

τ ARq,p (τ A)Q−1 q,p (τ A)ϕj τ Eα ≤

j=1

M (ψ) ϕτ Cτ (Eα ) . α(1 − α)

(3.17)

Finally, using the triangle inequality and estimates (3.15), (3.17), we obtain Au0 Eα ≤ for

λ τ

M (ψ) ϕτ Cτ (Eα ) +M1 (ψ) A (ϕ0 + μ) Eα α(1 − α)

(3.18)

∈ Z. Estimate (3.3) follows from (3.15) and (3.18). Now, let us estimate the norm of p+q−1 m=0

[λ τ] (−A)m λ [λ m τ ]−j (λ − τ) ARq,p (τ A)Q−1 q,p (τ A)ϕj τ m! τ j=1 m

/ Z. We use the Cauchy-Riesz formula for the operator A(λ+A)−1 (−A) for λ ∈ m! (λ− λ τ m [λ −1 τ ]−j (τ A)Qq,p (τ A) (see, [10]) τ τ ) ARq,p (−A)m λ [λ τ ]−j τ )m ARq,p (τ A)Q−1 A(λ + A)−1 (λ − q,p (τ A)ϕj m! τ (−z)m λ 1 τ [λ τ ]−j (λ − τ )m zRq,p A(z − τ A)−1 ϕj . = (z)Q−1 q,p (z) 2πi m! τ λτ + z S1 ∪S2

240

A. Ashyralyev

Then, one can write p+q−1

−1

A(λ + A)

m=0

1 = 2πi

[λ τ] (−A)m λ [λ m τ ]−j (λ − τ) ARq,p (τ A)Q−1 q,p (τ A)ϕj τ m! τ j=1 [λ τ] (−z)m λ [λ m τ ]−j (λ − τ) zRq,p (z)Q−1 q,p (z) m! τ j=1

p+q−1 m=0

S1 ∪S2

×

τ A(z − τ A)−1 ϕj τ dz. λτ + z

By estimate (2.3), > > > > [λ p+q−1 τ] (−A)m > > λ λ [ τ ]−j m −1 > A(λ + A)−1 > (λ − τ ) AR (τ A)Q (τ A)ϕ τ q,p j > q,p > m! τ > > m=0 j=1 ∞

p+q−1

≤M

m=0

0

×

[ λ ]−j |z|m λ τ m −1 (λ − τ) zRq,p (z)Qq,p (z) m! τ j=1

> > τ >A(|z| + τ A)−1 ϕj > τ d |z| . E λτ + |z|

Since z = ρe±iψ , with |ψ| < p+q−1

−1

||λ A(λ + A) α

m=0

≤ M|

λ τ

π 2,

from the strong positivity of A it follows that

[λ τ] (−A)m λ [λ m τ ]−j (λ − τ) ARq,p (τ A)Q−1 q,p (τ A)ϕj τ ||E m! τ j=1

∞

ρ1−α [ λ ]−j+1 τ 2

(1 + 2ρ cos ψ + ρ2 ) Summing the geometric progression, we get 0

j=1

−1

||λ A(λ + A) α

E

λ τ

p+q−1 m=0

(τ λ)α dρ ϕτ Cτ (Eα ) . λτ + ρ

[λ τ] (−A)m λ [λ m τ ]−j (λ − τ) ARq,p (τ A)Q−1 q,p (τ A)ϕj τ ||E m! τ j=1 ≤

M (ψ) ϕτ Cτ (Eα ) α(1 − α)

for any λ > 0. Hence,

p+q−1 m=0

[λ τ] (−A)m λ [λ m τ ]−j (λ − τ) ARq,p (τ A)Q−1 q,p (τ A)ϕj τ Eα m! τ j=1 ≤

M (ψ) ϕτ Cτ (Eα ) . α(1 − α)

(3.19)


241

Finally, using the triangle inequality and estimates (3.15), (3.19), we obtain Au0 Eα ≤ for

λ τ

M (ψ) ϕτ Cτ (Eα ) +M1 (ψ) A (ϕ0 + μ) Eα α(1 − α)

(3.20)

∈ / Z. Estimate (3.3) follows from (3.16) and (3.20). Theorem 2.3 is proved.

Theorem 3.4. Let τ be a sufficiently small number. Then, the boundary value problem (2.1) is coercive stable in Cτα,α (E). Proof. By [10], [Theorem 2.2 in Chapter 3], ⎫N ⎧ p q ⎬ ⎨ αj (−A)j τ j−1 uk − βj (−A)j τ j−1 uk−1 Cτα,α (E) ⎭ ⎩ j=1

j=1

>, - > + > τ −1 (uk − uk−1 ) N > 1

+

Cτα,α (E)

1

M ϕτ Cτα,α (E) ≤ α(1 − α)

(3.21)

1 (I − Rq,p (τ A)) u0 E τ

for the solution of the Padé difference schemes (3.3). Applying formulas (2.12), (2.13) and estimates (2.5), (2.7), we get > > >1 > > (I − Rq,p (τ A)) u0 > (3.22) >τ > E > ⎡> λ > > τ > > 1 λ −j −1 τ > ⎣ ≤ M (δ) > (I − Rq,p (τ A)) Rq,p (τ A)Qq,p (τ A)ϕj τ > > τ > >j=1 > > >1 > > + > (I − Rq,p (τ A)) (ϕ0 + μ)> > τ E for

for

λ τ

λ τ

E

∈ Z, >p+q−1 > > > (−A)m >1 > λ > (I − Rq,p (τ A)) u0 > ≤ M (δ) > (λ − τ )m > >τ > > m! τ E m=0 > > [λ ] τ 1 λ > [ τ ]−j > (I − Rq,p (τ A)) Rq,p × (τ A)Q−1 (τ A)ϕ τ j > q,p τ > j=1 > > E >1 > > +> > τ (I − Rq,p (τ A)) (ϕ0 + μ)> E ∈ / Z.

(3.23)

242

A. Ashyralyev Using estimates (2.4) and (2.5), we obtain > λ > > > > τ 1 > λ −j −1 τ > (I − Rq,p (τ A)) Rq,p (τ A)Qq,p (τ A)ϕj τ > > > > j=1 τ >

(3.24)

E

> > >1 λ > −1 τ −j > (I − Rq,p (τ A)) Rq,p > (τ A)Q (τ A) q,p >τ > λ τ −1

≤

j=1

⎛λ τ −1 ⎝ ≤M j=1

for

λ τ

E→E

> > > > > > > > >ϕj − ϕ λτ > τ + >ϕ λτ > E

E

⎞ M1 τ + 1⎠ ϕτ Cτα,α (E) ≤ ϕτ Cτα,α (E) λ 1−α α α(1 − α) (( τ − j + 1)τ ) (jτ )

∈ Z and

> > > >p+q−1 [λ τ] λ > > (−A)m 1 λ [ τ ]−j m −1 > > (λ − τ ) (I − R (τ A)) R (τ A)Q (τ A)ϕ τ q,p q,p j > q,p > m! τ τ > > m=0 j=1

E

(3.25) > > [λ τ ]−1 >p+q−1 > λ [λ > > (−A)m m1 −1 τ ]−j (λ − τ) (I − Rq,p (τ A)) Rq,p (τ A)Qq,p (τ A)> ≤ > > > m! τ τ m=0 j=1 E→E

> > > > > > > > × >ϕj − ϕ[ λ ] > τ + >ϕ[ λ ] > τ τ E E ⎛ ⎜ ≤M⎝

[ λ τ ]−1 j=1

for

λ τ

⎞ τ M1 ⎟ ϕτ Cτα,α (E) + 1⎠ ϕτ Cτα,α (E) ≤ α(1 − α) (( λτ − j)τ )1−α (jτ )α

∈ / Z. Using the estimate > > >1 > > (I − Rq,p (τ A)) A−1 > ≤M >τ > E→E

and estimates (3.22), (3.23), (3.24) and (3.23), we get

1 (I − Rq,p (τ A)) A−1 u E τ 0

≤ M1 (δ) A(μ + ϕ0 ) E +

M1 ϕτ Cτα,α (E) . α(1 − α)


243

Finally, using the triangle inequality and the last estimate and (3.21), we obtain the coercive stability estimates ⎧ ⎫N >⎨ p q ⎬ > > > j j−1 j j−1 > > αj (−A) τ uk − βj (−A) τ uk−1 >⎩ ⎭ > α,α j=1

j=1

1

>, - > > + > τ −1 (uk − uk−1 ) N 1 C α,α (E) ≤

Cτ

(E)

M (δ) ϕτ Cτα,α (E) α(1 − α) +M (δ) A(μ + ϕ0 ) E . τ

Theorem 2.4 is proved.

4. Applications First, the boundary-value problem on the range {0 ≤ t ≤ 1, x ∈ Rn } for the 2m-order multidimensional parabolic equation is considered: ⎧ |τ | ⎨ ∂v(t,x) + 0 aτ (x) ∂ τ1 v(t,x)τn + σv(t, x) = f (t, x), 0 < t < 1, ∂t ∂x1 ...∂xn (4.1) |τ |=2m ⎩ v(0, x) = v(λ, x) + μ(x), 0 < λ ≤ 1, x ∈ Rn , | τ |= τ1 + · · · + τn , where ar (x), μ(x) and f (t, x) are given as sufficiently smooth functions. Here, σ is a sufficiently large positive constant. It is assumed that the symbol r r B x (ξ) = ar (x) (iξ1 ) 1 . . . (iξn ) n , ξ = (ξ1 , . . . , ξn ) ∈ Rn |r|=2m

of the differential operator of the form ar (x) Bx = |r|=2m

∂xr11

∂ |r| . . . ∂xrnn

(4.2)

acting on functions defined on the space Rn , satisfies the inequalities 0 < M1 |ξ|2m ≤ (−1)m B x (ξ) ≤ M2 |ξ|2m < ∞ for ξ = 0. The abstract theorems given above are applied in the investigation of difference schemes for approximate solution of (4.1). The discretization of problem (4.1) is carried out in two steps. In the first step, the grid space Rnh (0 < h ≤ h0 ) is defined as the set of all points of the Euclidean space Rn whose coordinates are given by xk = sk h, sk = 0, ±1, ±2, . . . , k = 1, . . . , n. The difference operator Axh = Bhx + σIh is assigned to the differential operator Ax = B x + σI, defined by (4.2). The operator s2n−1 s2n 1 2 bxs Δs1− Δs1+ . . . Δn− Δn+ , (4.3) Bhx = h−2m 2m≤|s|≤S

244

A. Ashyralyev

acts on functions defined on the entire space Rnh . Here, s ∈ R nonnegative integer coordinates, Δk± f h (x) = ± f h (x ± ek h) − f h (x) ,

2n

is a vector with

where ek is the unit vector of the axis xk . An infinitely differentiable function of the continuous argument y ∈ Rn that is continuous and bounded together with all its derivatives is said to be smooth. We say that the difference operator Axh is a λth-order (λ > 0) approximation of the differential operator Ax , if the inequality sup |Axh ϕ (x) − Ax ϕ (x)| ≤ M (ϕ) hλ

x∈Rn h

holds for any smooth function ϕ (y) . The coefficients bxs are chosen in such a way that the operator Axh approximates in a specified way the operator Ax . It will be assumed that the operator Axh approximates the differential operator Ax with any prescribed order [38]. The function Ax (ξh, h) is obtained by replacing the operator Δk± in the right-hand side of equality (4.3) with the expression ± (exp {±iξk h} − 1), respectively, and is called the symbol of the difference operator Bhx . It will be assumed that for |ξk h| ≤ π and fixed x the symbol Ax (ξh, h) of the operator Bhx = Axh − σIh satisfies the inequalities π (4.4) (−1)m Ax (ξh, h) ≥ M |ξ|2m , | arg Ax (ξh, h)| ≤ φ < φ0 ≤ . 2 Suppose that the coefficient bxs of the operator Bhx = Axh − σIh is bounded and satisfies the inequalities kh |bx+e − bxs | ≤ M h , s

x ∈ Rnh ,

where ε ∈ (0, 1] is a fixed number. With the help of boundary-value problem ⎧ dvh (t,x) ⎨ + Axh v h (t, x) = f h (t, x), dt ⎩

Axh ,

v h (0, x) = v h (λ, x) + μh (x),

(4.5)

we arrive at the nonlocal

0 < t < 1, (4.6) x ∈ Rnh

for an infinite system of ordinary differential equations. In the second step, problem (4.6) is replaced by the difference schemes p q uhk (x) − uhk−1 (x) + αj (−Axh )j τ j−1 uhk (x) − βj (−Axh )j τ j−1 uhk−1 (x) = ϕhk (x), τ j=1 j=1

(4.7) ϕhk (x) = −

p j=1

+

q j=1

βj

j−1 i=0

αj

j−1

(−Axh )j−i+1 f (i)h (tk , x)τ j−1

i=0

(−Axh )j−i+1 f (i)h (tk−1 , x)τ j−1 ,

1 ≤ k ≤ N,

High-accuracy Stable Difference Schemes for Well-posed NBVP uh0 (x) = uhλ (x) + ϕh0 (x) + μh (x), τ

uh0 (x) =

p+q−1 m=0

ϕh0 (x) =

245

λ ∈ Z, x ∈ Rnh τ

(−Axh )m λ λ (λ − τ )m uh[ λ ] (x) + ϕh0 (x) + μh (x), ∈ / Z, x ∈ Rnh , τ m! τ τ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0, 0p+q−1 m=1

1 m! (λ

−

λ τ

λ τ

τ )m λ τ

∈ Z, x ∈ Rnh , 0m−1 i=0

(−Axh )m−i+1 f (i)h (t[ λ ] , x),

∈ / Z, x ∈ Rnh .

τ

Based on the number of corollaries of the abstract theorems given above, to formulate the result, one needs to introduce the space Ch = C(Rhn ) of all bounded grid functions uh (x) defined on Rhn , equipped with the norm ||uh ||Ch = sup |uh (x)|. xεRn h

Theorem 4.1. Suppose that assumptions (4.4) and (4.5) for the operator Axh hold. Then, the solutions of the difference schemes (4.7) satisfy the following almost coercivity inequalities: −1 {τ −1 (uhk − uhk−1 )}N Cτ (Ch ) 1

≤ M (σ, δ, λ)[ln

1 h

s2n−1 s2n h 1 2 h−2m ||Δs1− Δs1+ . . . Δn− Δn+ ϕ0 + μh ||Ch

2m≤|s|≤S

+ ln

1 ϕτ,h Cτ (Ch ) ]. τ +h

The proof of Theorem 4.1 is based on the abstract Theorem 3.2 and on the estimate : 1 1 min ln , 1 + ln Axh Ch →Ch ≤ M (σ) ln τ τ +h as well as on the positivity of the operator Axh in Ch [38], along with the following theorem on the almost coercivity inequality for the solution of the elliptic difference equation in Ch . Theorem 4.2. [38] Suppose that assumptions (4.4) and (4.5) for the operator Axh hold. Then, for the solutions of the elliptic difference equation Axh uh (x) = ω h (x),

x ∈ Rnh

the following almost coercivity inequality 1 s2n−1 s2n h 1 2 h−2m ||Δs1− Δs1+ . . . Δn− Δn+ u ||Ch ≤ M (σ) ln ||ω h ||Ch h 2m≤|s|≤S

is valid.

(4.8)

246

A. Ashyralyev

The next step in the definition of the result would be to introduce the space Chβ = C β (Rhn ) of all grid functions uh (x), defined on Rhn , equipped with the norm | uh (x) − uh (y) | , |x − y|β x,y∈Rn h,

uh C β = uh Ch + sup h

0 ≤ β < 1.

x=y

Theorem 4.3. Suppose that assumptions (4.4) and (4.5) for the operator Axh hold. Then, the solutions of the difference scheme (4.7) satisfy the coercivity inequalities: {τ −1 (uhk − uhk−1 )}N 1 Cτα,α (C 2mβ ) h

≤ M (α, β, σ, δ, λ)[

s2n−1 s2n h 1 2 h−2m ||Δs1− Δs1+ . . . Δn− Δn+ ϕ0 + μh ||C 2mβ h

2m≤|s|≤S

+ ϕτ,h Cτα,α (C 2mβ ) ], h

0 ≤ α < 1, 0 < β
0. Here, σ is a sufficiently large positive constant.


247

The discretization of problem (4.9) is carried out in two steps. In the first step, the grid sets 1 h = {x = xm = (h1 m1 , . . . , hn mn ), m = (m1 , . . . , mn ), Ω 0 ≤ mr ≤ Nr , hr Nr = 1, r = 1, . . . , n} ,

1 h ∩ Ω, Sh = Ω 1h ∩ S Ωh = Ω

are defined. To the differential space operator A by the formula n ∂ 2 u(x) Au = − αr (x) + σu(x) ∂x2r r=1 acting in the space of functions u(x), defined in Ω and satisfying the Dirichlet boundary condition: u(x) = 0 in S, we assign the difference operator Axh by the formula n Axh uh = − ar (x)uh− + σuh (4.10) r=1

xr xr ,jr

acting in the space of grid functions uh (x), satisfying the conditions uh (x) = 0 for all x ∈ Sh . With the help of Axh , we arrive at the nonlocal boundary value problem ⎧ h ⎪ ⎨ du dt(t,x) + Axh uh (t, x) = f h (t, x), 0 < t < 1, x ∈ Ωh , (4.11) ⎪ ⎩ uh (0, x) = uh (λ, x) + μh (x), x ∈ Ω 1h for an infinite system of ordinary differential equations. In the second step, we replace problem (4.11) by the high order of accuracy Padé difference schemes p q uhk (x) − uhk−1 (x) + αj (−Axh )j τ j−1 uhk (x) − βj (−Axh )j τ j−1 uhk−1 (x) = ϕhk (x), τ j=1 j=1 (4.12) ϕhk (x) = −

p j=1

+

q

βj

j=1

αj

j−1

(−Axh )j−i+1 f (i)h (tk , x)τ j−1

i=0

j−1

(−Axh )j−i+1 f (i)h (tk−1 , x)τ j−1 ,

1 ≤ k ≤ N, x ∈ Ωh ,

i=0

uh0 (x) = uhλ (x) + ϕh0 (x) + μh (x), τ

λ 1 h, ∈ Z, x ∈ Ω τ

(−Axh )m λ λ 1 h, (λ − τ )m uh[ λ ] (x) + ϕh0 (x) + μh (x), ∈ / Z, x ∈ Ω = τ m! τ τ m=0 ⎧ 1 h, ⎪ 0, λτ ∈ Z, x ∈ Ω ⎪ ⎪ ⎨ λ m 0m−1 0p+q−1 1 ϕh0 (x) = x m−i+1 (i)h f (t[ λ ] , x), ⎪ m=1 i=0 (−Ah ) m! (λ − τ τ ) ⎪ τ ⎪ ⎩ λ 1 ∈ / Z, x ∈ Ω . h τ

uh0 (x)

p+q−1

248

A. Ashyralyev

To formulate the results, one needs to introduce the Banach spaces Ch = 1 h ) (β = (β1 , . . . , βn ) , 0 < βr < 1, 1 ≤ r ≤ n) of all the grid 1 h ) and C β = C β (Ω C(Ω 01 h h 1 h , equipped with the norms functions ϕ (x) = {ϕ(h1 m1 , . . . , hn mn )} defined on Ω ϕh Ch = max | ϕh (x) |, ϕh C β = ϕh Ch 1h x∈Ω

+

sup 1h x,x+Δy∈Ω

h

| ϕh (xj1 , . . . , xjn ) − ϕh (xj1 + Δyj1 , . . . , xjn + Δyjn ) | ×

n

(Δyjr )−βr xβjrr (1 − xjr − Δyjr )βr .

r=1

Theorem 4.6. Let τ and |h| = h21 + · · · + h2n be sufficiently small numbers. Then, the solutions of difference schemes (4.12) satisfy the following almost coercivity inequality: n > 1 > −1 > ϕh0 + μh xr xr , jr > {τ −1 (uhk − uhk−1 )}N ≤ M (σ, δ, λ)[ln Cτ (Ch ) 1 Ch |h| r=1 + ln

1 ϕτ,h Cτ (Ch ) ]. τ + |h|

The proof of Theorem 4.6 is based on the abstract Theorem 3.2 and on the estimate : 1 1 x min ln , 1 + ln Ah Ch →Ch ≤ M (σ) ln τ τ + |h| as well as on the positivity of the operator Axh in Ch [39], along with the following theorem on the almost coercivity inequality for the solution of the elliptic difference equation in Ch . Theorem 4.7. [6] For the solutions of the elliptic difference equation + x h Ah u (x) = ω h (x), x ∈ Ωh , uh (x) = 0, x ∈ Sh ,

(4.13)

the following almost coercivity inequality n > h > >u xr xr , jr > ≤ M (σ) ln 1 ||ω h ||C h Ch |h| r=1 is valid.

Theorem 4.8. Let τ and |h| = h21 + · · · + h2n be sufficiently small numbers. Then, the solutions of difference schemes (4.12) satisfy the following coercivity inequalities: n > h > > ϕ + μh xr x , jr > β {τ −1 (uhk − uhk−1 )}N α,α β ≤ M (α, β, σ, δ, λ)[ 1 0 r Cτ (Ch ) Ch r=1

+ ϕ

τ,h

Cτα,α (C β ) ], h

0 < α < 1, 0 < βr < 1, 1 ≤ r ≤ n.


249

The proof of Theorem 4.8 is based on the abstract Theorem 3.4 and the positivity of the operator Axh in Ch [39] and on the following theorem on the coercivity inequality for the solution of the elliptic difference equation (4.13) in Chβ . Theorem 4.9. [6] For the solutions of the elliptic difference equation (4.13), the following coercivity inequalities n > h > u xr xr , r=1

> >

jr C β h

≤ M (σ, β)||ω h ||C β h

are valid. Note that in a similar manner the high order of accuracy Padé difference schemes with respect to one variable for the approximate solution of the nonlocal boundary value problem for the parabolic equation ⎧ ∂2 u ⎨ ∂u 0 < t < 1, 0 < x < 1, ∂t − a(x) ∂x2 + σu = f (t, x), (4.14) u(0, x) = u(λ, x) + μ(x), 0 ≤ x ≤ 1, 0 < λ ≤ 1, ⎩ u(t, 0) = u(t, 1), ux (t, 0) = ux (t, 1), 0≤t≤1 can be constructed. Here, σ is the sufficiently large positive constant and a(x), μ(x), and f (t, x) are given sufficiently smooth functions and a(x) ≥ a > 0. Abstract theorems given above and results of papers [35], [36] and [37] permit us to obtain the stability, the almost coercive stability and the coercive stability estimates for the approximate solutions of these difference schemes. Acknowledgement Many thanks to Prof. Sobolevskii P.E. for his helpful suggestions to improve the paper.

References [1] M. Dehghan, On the Numerical Solution of the Diffusion Equation with a Nonlocal Boundary Condition. Mathematical Problems in Engineering 2003 (2003), no. 2, 81–92. [2] J.R. Cannon, S. Perez Esteva and J. Van Der Hoek, A Galerkin Procedure for the Diffusion Equation Subject to the Specification of Mass. SIAM J. Numerical Analysis 24 (1987), no. 3, 499–515. [3] N. Gordeziani, P. Natani and P.E. Ricci, Finite-Difference Methods for Solution of Nonlocal Boundary Value Problems. Computers and Mathematics with Applications 50 (2005), 1333–1344. [4] R. Dautray and J.L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Technique. Volume 1-11, Masson, Paris, 1988. [5] A. Ashyralyev and Y. Ozdemir, Stability of Difference Schemes for HyperbolicParabolic Equations. Computers and Mathematics with Applications 50 (2005), no. 8-9, 1443–1476.

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[6] P.E. Sobolevskii, The Coercive Solvability of Difference Equations. Dokl. Acad. Nauk SSSR 201 (1971), no. 5, 1063–1066. (Russian). [7] A. Ashyralyev, A. Hanalyev and P.E. Sobolevskii, Coercive Solvability of Nonlocal Boundary Value Problem for Parabolic Equations. Abstract and Applied Analysis 6 (2001), no. 1, 53–61. [8] A. Ashyralyev, I. Karatay and P.E. Sobolevskii, Well-Posedness of the Nonlocal Boundary Value Problem for Parabolic Difference Equations. Discrete Dynamics in Nature and Society 2004 (2004), no. 2, 273–286. [9] A. Ashyralyev, Nonlocal Boundary-Value Problems for Abstract Parabolic Equations: Well-Posedness in Bochner Spaces. Journal of Evolution Equations 6 (2006), no. 1, 1–28. [10] A. Ashyralyev and P.E. Sobolevskii, Well-Posedness of Parabolic Difference Equations. Birkh¨ auser Verlag, 1994. [11] A. Ashyralyev, Well-Posedness on the Modified Crank-Nicholson Difference Schemes in Bochner Spaces, Discrete and Continuous Dynamical Systems-Series B 7 (2007), no. 1, 29–51. [12] A. Ashyralyev, Well-Posedness of the Boundary Value Problem for Parabolic Equations in Difference Analogues of Spaces of Smooth Functions. Mathematical Problems in Engineering 2007 (2007), Article ID 90815, 1–16. [13] A. Ashiraliev and P.E. Sobolevskii, Differential Schemes of High Order of Accuracy for Parabolic Equations with Variable Coefficients. Dopovidi Akademii Nauk Ukrainskoi RSR Seriya A-Fiziko-Matematichni ta Technichni Nauki 6 (1988), 3–7. (Russian). [14] A. Ashyralyev and P.E. Sobolevskii, Well-posed Solvability of the Cauchy Problem for Difference-equations of the Parabolic Type. Nonlinear Analysis- Theory, Methods and Applications 24 (2005), no. 2, 257–264. [15] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations. Birkh¨ auser Verlag, 2004. [16] D. Guidetti, B. Karasozen, and S. Piskarev, Approximation of Abstract Differential Equations. Journal of Math. Sci. 122 (2004), no. 2, 3013–3054. [17] A. Ashyralyev, S. Piskarev, S. Wei, On Well-Posedness of the Difference Schemes for Abstract Parabolic Equations in Lp ([0, 1], E) Spaces. Numerical Functional Analysis and Optimation 23 (2002), no. 7-8, 669–693. [18] M. Grouzeix, S. Larson, S. Piskarev, V. Thomee, The Stability of Rational Approximations of Analytic Semigroups. Preprint, Department of Mathematics, Chalmers Institute of Technology, G¨ oteborg, 1991, 28p. [19] I.P. Gavrilyuk and V.L. Makarov, Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equation. Computational Methods in Applied Mathematics 1 (2001), no. 4, 333–355. [20] I.P. Gavrilyuk and V.L. Makarov, Algorithms without Accuracy Saturation for Evolution Equations in Hilbert and Banach Spaces. Mathematics of Computation 74 (2005), 555–583. [21] I.P. Gavrilyuk and V.L. Makarov, Exponentially Convergent Algorithms for the Operator Exponential with Applications to Inhomogeneous Problems in Banach Spaces. SIAM Journal of Num. Anal. 43 (2005), no. 5, 2144–2171.


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[22] D. Gordeziani,H. Meladze and G. Avalishvili, On One Class of Nonlocal in Time Problems for First-Order Evolution Equations. Zh. Obchysl. Prykl. Mat. 88 (2003), no. 1, 66–78. [23] D. Gordeziani and G. Avalishvili, Time-Nonlocal Problems for Schr¨ odinger Type Equations. I: Problems in Abstract Spaces. Differ. Equ. 41 (2005), no. 5, 703–711. [24] R. Agarwal, M. Bohner and V.B. Shakhmurov, Maximal Regular Boundary Value Problems in Banach-Valued Weighted Spaces. Boundary Value Problems 1 (2005), 9–42. [25] V.B. Shakhmurov, Coercive Boundary Value Problems for Regular Degenerate Differential-Operator Equations. Journal of Mathematical Analysis and Applications 292 (2004), no. 2, 605–620. [26] J.I. Ramos, Linearly-Implicit, Approximate Factorization, Exponential Methods for Multi-Dimensional Reaction-Diffusion Equations. Applied Mathematics and Computation 174 (2006), no. 2, 1609–1633. [27] X.Z. Liu, X. Cui and J.G. Sun, FDM for Multi-Dimensional Nonlinear Coupled System of Parabolic and Hyperbolic Equations. Journal of Computational and Applied Mathematics 186 (2006), no. 2, 432–449. [28] A.V. Gulin and V.A. Morozova, On the Stability of a Nonlocal Finite-Difference Boundary Value Problem. Differ. Equ. 39 (2003), no. 2, 962–967. (Russian). [29] A.V. Gulin, N.I. Ionkin and V.A. Morozova, On the Stability of a Nonlocal FiniteDifference Boundary Value Problem. Differ. Equ. 37 (2001), no. 7, 970–978. (Russian). [30] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh¨ auser Verlag, 1995. [31] Y.G. Wang, M. Oberguggenberger, Nonlinear Parabolic Equations with Regularized Derivatives. Journal of Mathematical Analysis and Applications 233 (1999), no. 2, 644–658. [32] Beyn Wolf-Jurgen, B.M. Garay, Estimates of Variable Stepsize Runge-Kutta Methods for Sectorial Evolution Equations with Nonsmooth Data. App. Num. Math. 41 (2002), no. 3, 369–400. [33] R. Rautmann, H 2,r -Convergent Approximation Schemes to the Navier-Stokes Equations. Nonlinear Analysis – Theory, Methods and Applications 30 (1997), no. 4, 1915–1926. [34] A. Ashyralyev, Method of Positive Operators of Investigations of the High Order of Accuracy Difference Schemes for Parabolic and Elliptic Equations. Doctor of Sciences Thesis, Kiev, 1992, 312p. (Russian). [35] A. Ashyralyev, Fractional Spaces Generated by the Positive Differential and Difference Operators in a Banach Space. in ISMME, Springer, 2006, 10–19. [36] A. Ashyralyev, Nonlocal Boundary Value Problems for Partial Differential Equations: Well-Posedness. AIP Conference Proceedings Global Analysis and Applied Mathematics: International Workshop on Global Analysis 729 (2004), 325–331. [37] A. Ashyralyev and N. Altay, A Note on the Well-Posedness of the Nonlocal Boundary Value Problem for Elliptic Difference Equations. Applied Mathematics and Computation 175 (2006), no. 1, 49–60.

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[38] Yu.A. Smirnitskii and P.E. Sobolevskii, The Positivity of Difference Operators, Vychisl. Sist. 87 (1981), 120–133. (Russian). [39] Kh.A. Alibekov, Investigations in C and Lp of Difference Schemes of High Order Accuracy for Approximate Solution of Multidimensional Parabolic Boundary Value Problems. Ph.D. Thesis, Voronezh, VSU, 1978, 134p. Allaberen Ashyralyev Department of Mathematics Fatih University 34500 Buyukcekmece Istanbul, Turkey e-mail: [email protected]


The Factorization of the Flow, Defined by the Euler-Poisson’s Equations Alexandr Belyaev Abstract. The factorization of the flow, defined by three-body problem, gives us an opportunity to compactificate the phase space and to investigate the global properties of the solutions of this problem. Mathematics Subject Classification (2000). Primary 55R55, 34M30, 74H10; Secondary 34M45, 74H05. Keywords. Three-body problem, factorization of the flow, first integral, singular points, analytic functions, entire functions, compact holomorphic manifold, holomorphic foliation.

1. Introduction The properties of differential equations’ solutions with an analytic right-hand side much depend on the singular points of these solutions, lying in the complex plane of time. It is a well-known fact that S. Kovalevskaya’s classic solution ([1]) of the Euler–Poisson equations was found when she investigated the single-valued solutions of the problem. The systematic research of the singular points of the solutions in the aggregate with the compactification of the flow ([2]), defined by the Euler–Poisson equations, allows us not only to find some partial solutions with given properties ([3]) but also to investigate global properties of these solutions ([4]). That is the reason we should use this approach to investigate the three-body problem (see, for example, [5], [6] and bibliography in [7], [8]). The essence of this method is obtaining the compact holomorphic manifold with the one-dimensional foliation F having the singular points, as the result of the factorization of the flow of the phase space. The compactification of this problem enables us to consider the solutions globally, and that is really important for the research of nonlinear differential equations. Besides the singularities of the foliation F are correspondent to the

254

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singularities of the complex solutions of the initial problem. They can be studied efficiently. In this paper we accomplish the factorization of the flow of the three-body problem, obtain the asymptotics and the classification of the singular points. The main theorem of the paper is following: Theorem 1.1. All the solutions pi (t), qi (t) ∈ C3 , i = 1, 2, 3, t ∈ C, of three body problem ⎧ pi ⎪ ⎨q˙i = m i qi − qj p ˙ = −Gm mj . ⎪ i ⎩ i |q − q |3 j=i

where |qi − qj | =

i

j

(qi1 − qj1 )2 + (qi2 − qj2 )2 + (qi3 − qj3 )2

have the singular points. All the singular but not essentially singular points of the solutions (p(t), q(t)) of three-body problem have the asymptotic behavior ⎧ p1 = q˙1 , σ, ⎪ ⎪ ⎪ ⎪ ⎨mq1 = I t + m1 q1 (0) + G1/3 (m2 x3 − m3 x2 ), σ, σ ⎪ ⎪ ⎪ μk vk tλk +2/3 + · · · ; ˜0 t2/3 + κ2 u2 t8/3 + ⎪ ⎩x(t) = x 1≤k≤4

here σ denotes the circle permutation of indices 0 (1, 2, 3), x(t) = (x1 (t), x2 (t), x3 (t)), κ2 , μk ∈ C are free parameters, I = i mi q˙i is an integral of the kinetic momentum. All the vectors x ˜0 , u2 , vk ∈ (C3 )3 and the parameters λk ∈ C, Re λk > 0, are found efficiently if the solutions of the polynomial s1 (m2 + m3 )ρ5 + s1 (3m2 + 2m3 )ρ4 + (s1 (3m2 + m3 ) − (s2 − s3 )m1 )ρ3 − (s2 (3m1 + m3 ) − (s1 − s3 )m2 )ρ2 − s2 (3m1 + 2m3 )ρ − s2 (m1 + m3 ) = 0, where si = ±1, are known. Either essentially singular point t∗ of the solution pi (t), qi (t) of three-body problem have the following description. Let π is the natural projection π : C18 → C18 /C, defined by the following action: α : (p1 , p2 , p3 , q1 , q2 , q3 ) → (αp1 , αp2 , αp3 , α−2 q1 , α−2 q2 , α−2 q3 ). Then the limit set X∗ = lim π((pi (t), qi (t)) is contained in the set X0 t→t∗

E

X,

where X0 = {π(p1 , p2 , p2 , q1 , q2 , q3 ) : ∃i, j |qi − qj | = 0}, Y = {π(η) : H(η) = 0, I(η) = 0, M(η) = 0} and I, M are the classic first integrals of the energy, of the kinetic momentum and of the torque of three-body problem.

The Factorization of the Flow

255

2. Preliminaries Now we consider the problem (see, for example, [5]) on moving n bodies (m1 , r1 ), . . . , (mn , rn ), mi ∈ R+ , ri ∈ R3 which move by the law of gravity. Kinetic and potential energy are correspondingly equal 1 G mj mk T = . mi r˙i2 , U = − 2 i 2 |rj − rk | j,k,j=k

The Lagrangian L = T − U determines the following system of differential equations: ri − rj m¨ ri = −Gmi mj . (2.1) |ri − rj |3 j=i

In Hamilton form H=

i

r˙i

∂L 1 G mj mk , −L= mi r˙i2 − ∂ r˙i 2 i 2 |rj − rk | j,k,j=k

canonical coordinates are qi = ri ,

pi =

∂L = mi r˙i ∂ r˙i

and the Hamilton system has the following form: ⎧ ∂H pi ⎪ ⎪ ⎨q˙i = − ∂p = m i i qi − qj ∂H ⎪ mj . ⎪ ⎩p˙i = ∂qi = −Gmi |qi − qj |3

(2.2)

j=i

The first integrals of the system (2.2) are H, I = mi q˙i , M = qi × pi . i

i

In the classic notation (2.1), (2.2) of n-body problem there is the module which is the local real-analytic function in the right-hand side of the system. As we want to consider the n-body problem for the complex time, the right-hand side of the differential equations is necessary to be complex-analytical. Therefore the function module for the vector q ∈ C3 should be considered as complex-analytical function which is determined by the formula:

2 + q2 + q2 . |qi | = qi1 i2 i3 We use the following notation for the norm of the vector: √ qi = qi1 q¯i1 + qi2 q¯i2 + qi3 q¯i3 .

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3. The factorization of the flow of n-body problem Let pi (t), qi (t), i = 1, 2, 3 be the solution of n-body problem then αpi (α3 t), α−2 qi (α3 t) is solution too. Due to this fact the factorization on the set of trajectories of the system (2.2) is possible. Remark 3.1. The flow of the problem (2.2) allows the factorization with the action of the orthogonal group So(3, C) ([13]) pi → Api ,

qi → Aqi ,

A ∈ So(3, C),

but using this factorization does not give us any visible technic preferences. Proposition 3.2. Let C act as a transformation group on Cn in the following way: α : (z1 , . . . , zn ) → (αk1 z1 , . . . , αkn zn ), (k )

(k )

k = (k1 , . . . , kn ) ∈ Nn . Then the factor-space Pkn−1 = {(z1 1 : · · · : zn n )} is a compact holomorphic manifold ([14]) with respect to this action. Proof. Let π0 , π be the canonical projections π0 : Cn → P n−1 ,

π : Cn → Pkn−1 .

Then let us consider the mapping of the complex projective space P n−1 f : P n−1 → Pkn−1 . This mapping is assigned as follows: f 0 : (z1 , . . . , zn ) → (z1k1 , . . . , znkn ),

f = π ◦ f 0 ◦ π0−1 .

It is clear that the mapping f is correct but it has singular points zi = 0. The mapping f is open, therefore any atlas of the manifold P n−1 induces atlas Pkn−1 , hence, Pkn−1 is a holomorphic manifold. The compactness of the manifold Pkn−1 follows from the compactness of P n−1 . Proposition 3.3. The projection π : C3n + C3n → P∗6n−1 , F GH In F GH In where ∗ = (1, . . . , 1 , 2, . . . , 2 ), is determined by the next formula: (1)

(2)

(2) π : (p1 , . . . , pn , q1 , . . . , qn ) → (p1 : · · · : p(1) n : w1 : · · · : wn ),

where wi = |qqii|2 and induces the structure of the holomorphic C-one-dimension foliation ([15]) F of the compact holomorphic manifolds P∗6n−1 . Proof. According to the definition of π, the vector (αp1 , . . . , αpn , α−2 q1 , . . . , α−2 qn ) is projected onto {(αp1 , . . . , αpn , α2

q1 qn , . . . , α2 ), α ∈ C}. |q1 |2 |qn |2


257

Remark 3.4. The projection π can be defined in the more natural way: (1)

(−2)

π : (p1 , . . . , pn , q1 , . . . , qn ) → (p1 : · · · : p(1) n : q1

: · · · : qn(−2) ),

but in this case the image of the mapping is a noncompact manifold. Remark 3.5. We can use the mapping π −1 which has the following presentation: π −1 : (p1 : · · · : pn : w1 , . . . , wn ) : w1 −2 wn , . . . , α ), α ∈ C → (αp1 , . . . , αpn , α−2 |w1 |2 |wn |2 if it is necessary. Remark 3.6. The foliation F is integrable as there exists the invariant mapping J : P∗6n−1 \ X → P 6 , J : η → (H(ξ) : I12 (ξ) : I22 (ξ) : I32 (ξ) : M21 (ξ) : M22 (ξ) : M23 (ξ)), ξ = π −1 (η), X = {η ∈ P∗6n−1 : H(π −1 (η)) = 0, I(π −1 (η)) = 0, M(π −1 (η)) = 0} Moreover the surface X is fiber invariant for the foliation F too. Proposition 3.7. The singular points of the foliation F are the following: π-projections of the solutions (˜ p0 , q˜0 ) of the characteristic system ⎧ p˜i ⎪ ⎪ qi + 3 = 0, ⎨−2˜ mi q˜i − q˜j ⎪ mj = 0, ⎪ ⎩p˜i − 3 Gmi |˜ qi − q˜j |3

(3.1)

j

π-projections of the singular points of the system (2.2), i.e., the points {π(p1 , . . . , pn , q1 , . . . , qn ) : ∃i, j |qi − qj | = 0} and (1)

(2)

(2) 6n−1 {(p1 : · · · : p(1) : ∃i |wi | = 0}. n : w1 : · · · : wn ) ∈ P∗

Proof. Evidently singular points of the equation (2.2) {π(p1 , . . . , pn , q1 , . . . , qn ) : ∃i, j |qi − qj | = 0} are projected onto singular points of the foliation F . Moreover if the vector (p, ˙ q) ˙ touches the π-pre-image of π(p, q) the point π(p, q) will be a singular point of the foliation too. Such points satisfy the system (3.1), which we call a characteristic system. At last the points of the manifold P∗6n−1 which does not have the pre(1) (1) (2) (2) image, i.e., the points {(p1 : · · · : pn : w1 : · · · : wn ) ∈ P∗6n−1 : ∃i |wi | = 0} may be singular points. Remark 3.8. The solution of the characteristic system (3.1) determines the central configuration leading to some partial solutions of three-body problem, discovered by Euler and Lagrange ([9], [10], [11]).

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4. The change of variables for the 3-body problem Further we make the calculations for 3-body problem. In this case the problem has the dimension 18. Using the integrals, this dimension can be lowered to 8 ([16]). However differential equations which can be received in this way, are very inconvenient for further investigations. That is the reason we change the variables without lowering of the problem’s dimension in order to get the equations as simple as possible. At first we pass to the relative coordinates: ⎧ (q2 − q3 ) · G −1/3 , σ ⎨x1 = (4.1) p2 p3 ⎩y 1 = · G−1/3 , σ. − m2 m3 Here and lower σ denotes the circle permutation of indices (1, 2, 3). The system (2.2) takes the following form ⎧ ⎨x˙1 = y1 , σ x2 x3 x1 (4.2) + ) − (m2 + m3 ) ), σ ⎩y˙1 = (m1 ( |x2 |3 |x3 |3 |x1 |3 If the system (4.2) is solved it will be simple to find pi , qi : ⎧ ⎨p1 = q˙1 , σ 1/3 ⎩q1 = G (IG−1/3 + m2 x3 − m3 x2 ), σ, m 0 where m = σ m1 . Now let us suppose that x1 , mz1 = y1 − m1 u, σ, u˙ = |x1 |3 σ then

⎧ x1 z˙1 = − ,σ ⎪ ⎪ ⎪ |x1 |3 ⎨ x˙1 = mz1 + m1 u, σ x1 ⎪ ⎪ ⎪ . ⎩u˙ = |x1 |3 σ Since u+

then u = −

0 σ

σ

z1 =

1 1 y1 = m σ m σ

p2 p3 − m2 m3

(4.3)

(4.4)

G−1/3 ≡ 0,

z1 and finally we get the following system: ⎧ 0 ⎨x˙1 = mz1 − m1 σ z1 , σ x1 ,σ ⎩z˙1 = − |x1 |3

(4.5)


259

Theorem 4.1. The system of the differential equations (4.5) is equivalent to threebody problem (2.2) and is the canonical Hamiltonian system with the coordinates xi (m , zi ). Hamiltonian has the form: i ⎛ 2 ⎞ 2 1 mz1 1 . H= ⎝ − z1 ⎠ − 2 m1 m1 |x1 | σ σ σ The system of the differential equations (4.5) has the following first integrals: 1 I1 = x1 , I2 = z1 × x1 . m1 σ σ I1 =

Proof. I˙ 2 =

σ

1 x1 − × x1 + z1 × mz1 − m1 z1 m1 |x1 |3 σ σ z1 × z1 = z1 × z1 = 0. =

1 (z˙1 × x1 + z1 × x˙ 1 ) = m1 σ

(q1 − q2 )G−1/3 ≡ 0.

σ

σ

σ

σ

The fact that the system (4.5) is canonical can be tested by the straight calculation. Now we make one more change of variables ([2]) which allows us to investigate the asymptotics of the singular points of the solutions of three-body problem. Let t∗ ∈ C be the singular point of the solution (xi (t), zi (t)) (i.e., t∗ is a singular point of one of the coordinate functions of (xi (t), zi (t))). It is necessary to get rid of the branching in t∗ , if any, by representing xi (t) = x î (Ln(t − t∗ )α ), α zi (t) = zî (Ln(t − t∗ ) ), where x î (τ ), zî (τ ) is single-valued function if Re τ → −∞. The system (4.5) is transformed into ⎧ ⎪ τ /α 1 ˙ ⎪ ˆ1 = α e mzˆ1 − m1 zˆ1 , σ ⎨x x ˆ1 ⎪ ⎪ ⎩zˆ˙1 = − α1 eτ /α , σ; |xˆ1 |3

σ

where the derivative is taken with respect to τ. In order to make the right-hand side of the equation independent of τ, we make a replacement of the variables in the following form: x ˜i (τ ) = eβτ x î (τ ), γτ z˜i (τ ) = e zî (τ ). Then we have ⎧ ⎪ τ β+τ /α 1 ˙ ⎪ ˜1 = β x mˆ z 1 − m1 ˜1 + α e zˆ1 , σ ⎨x ⎪ ⎪ ⎩z˜˙i = γ z˜1 −

1 τ γ+τ /α αe

x ˆ1 , σ. |x1 |3

σ

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A. Belyaev

We see that the right-hand side is independent of τ, if β + α1 = γ, γ + α1 = 1 2 , β = − 3α , Taking α = 13 , we obtain the following system: −2β, i.e., if γ = 3α ⎧ 0 ⎨x ˜˙ 1 = −2˜ x1 + 3 (m˜ z1 − m1 σ z˜1 ), σ x ˜1 (4.6) ⎩z˜˙1 = z˜1 − 3 , σ. 3 |xˆ1 | The dependence between the differential systems (4.5) and (4.6) is expressed by the following relations: 1 1 −1/3 2/3 Ln(t − t∗ ) , xi (t) = (t−t∗ ) x Ln(t − t∗ ) (4.7) z˜i ˜i zi (t) = (t−t∗ ) 3 3 Theorem 4.2. The solution of the system (4.5) does not have a singularity at the point t∗ if only the corresponding solutions of (4.6) have the asymptotic behavior x˜i ∼ x ˜i0 e−2τ , z˜i ∼ z˜i0 eτ if Re τ → −∞. Proof. It is enough to substitute the asymptotics x ˜i ∼ x ˜i0 e−2τ , z˜i ∼ z˜i0 eτ into (4.7). Remark 4.3. If we know the asymptotic behavior of the solutions x˜i , z˜i we will be able to obtain the asymptotics of the singular points of the solutions (4.5) using (4.7).

5. The singular points of the foliation F for 3-body problem The projection π from Proposition 2.2 for the system (4.5) has the following form: xi (1) (1) (1) (2) (2) (2) . π : (z1 , z2 , z3 , x1 , x2 , x3 ) → (z1 : z2 : z3 : w1 : w2 : w3 ), wi = |xi |2 The correspondent system of differential equations for the fiber of F on the manifold P∗17 takes the following form: + 0 0 w˙1 = −2w1 (w1 , mz1 − m1 σ z1 ) + |w1 |2 (mz1 − m1 σ z1 ), σ (5.1) z˙1 = −w1 |w1 |, σ The system (5.1) is suitable for the investigation of global properties of threebody problem because this system is defined on the compact manifolds and its right-hand side is always determined. At the same time the system (4.5) as essentially more simple is more convenient for calculations. Proposition 5.1. The singular points of the foliation F of the compact holomorphic manifold P∗17 are the projections π(˜ x0i , z˜i0 ) of the roots of the characteristic system ⎧ 0 ⎨−2˜ x1 + 3 (m˜ z1 − m1 σ z˜1 ) = 0, σ x ˜1 (5.2) ⎩z˜1 − 3 = 0, σ. |˜ x1 |3


261

Proof. This proposition is similar to Proposition 3.7. (1) (2) (2) As for the points of the form {(z1 : z2 : z3 : w1 : w2 : w3 ) ∈ P∗17 : ∃i|wi | = 0}, pretending to be singular due to the singularity of the functions |wi |, they form the invariant surface in P∗17 . We can make sure that it is true by finding the derivative of the function |wi |2 along the vector field (5.1). This derivative is identically equal to zero on the surface |wi |2 = 0. The roots of the characteristic system (5.2) were already known to Euler and Lagrange. We present their finding for completing the paper, taking into consideration that this finding is simple. Let us substitute z˜1 into the first equation of (5.1). If omitting the sign ∼ for simplicity we get x1 x1 9 m x1 , σ. = − m1 2 m 1 |x1 |3 |x1 |3 σ Then we subtract these equations one from another and have x1 x1 x2 9m x2 , σ. − = − m1 m2 2 |x1 |3 m1 |x2 |3 m2 or x1 m1

9m x2 9m x3 9m 1− = 1− = 1− . 2|x1 |3 m2 2|x2 |3 m3 2|x3 |3

(5.3)

Let all vectors xi be collinear. Denote x2 = ρx1 , x3 = −(1 + ρ)x1 and substitute x2 , x3 into (5.3). We get x1 9m ρx1 9m (−1 − ρ)x1 9m 1− = 1− = 1− , m1 2s1 |x1 |3 m2 2s2 |x1 |3 m3 2s3 |x1 |3 where si = ±1 and the following quintic polynomials: s1 (m2 + m3 )ρ5 + s1 (3m2 + 2m3 )ρ4 + (s1 (3m2 + m3 ) − (s2 − s3 )m1 )ρ3 −(s2 (3m1 + m3 ) − (s1 − s3 )m2 )ρ2 − s2 (3m1 + 2m3 )ρ − s2 (m1 + m3 ) = 0. (5.4) Now let the vectors xi be non-collinear then |x1 |3 = |x2 |3 = |x3 |3 =

9m 2

and besides x1 + x2 + x3 = 0. The singular points of the foliation F are interesting because they enable us to find the asymptotic behavior of three-body problem’s solution. Definition 5.2. α-singular points are the singular points π(˜ x0 , z˜0 ) of the foliation 0 0 F where (˜ x , z˜ ) is a root of the characteristic system with non-collinear vectors xi (5.2) . β-singular points are the singular points π(˜ x0 , z˜0 ) of the foliation F where 0 0 (˜ x , z˜ ) is a root of the characteristic system with collinear vectors xi (5.2).

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Let (˜ x0 , z˜0 ) be a solution of the characteristic system (5.2). At the same time it is the singular point of the differential equation system (4.6). The linearization of the system (4.6) has the following form in the singular point: ⎧ 0 ˜˙ 1 = −2˜ x1 + 3 (m˜ z1 − m1 σ z˜1 ), σ ⎨x (5.5) x ˜ x ˜0 ⎩z˜˙1 = z˜1 − 3 01 3 + 9 01 5 (˜ x01 , x ˜1 ), σ. |˜ x1 | |˜ x1 |

6. The asymptotic behavior of α-singular points of the foliation F 3 Let us suppose that |˜ x01 |3 = |˜ x02 |3 = |˜ x01 |3 = 9m 2 = a then the linear system for the eigenvectors’ of left-hand side (5.5) finding has the following form: ⎧ ⎨(m2 + m3 )z1 − m1 (z2 + z3 ) = λ+2 x1 , σ 3 (6.1) 3 9 0 0 ⎩− x1 + (˜ x1 , x1 )˜ x1 = (λ − 1)z1 , σ. 3 5 a a

Theorem 6.1. The eigenvalues and the corresponding eigenvectors of the linear system (6.1) are following: λ = 1. The eigenvectors u1 have the form: x1 = 0, z1 = m1 r, σ, r ∈ C3 . The dimension of the eigenspace is equal to 3. λ(λ + 1) = 0. The eigenvectors u0 , u−1 satisfy the conditions: 3x1 , σ. x1 ⊥˜ x01 , z1 = 3 a (1 − λ) The dimension of the eigenspaces for λ = 0 and λ = −1 is equal to 3. (λ + 3)(λ − 2) = 0. The eigenvectors u−3 , u2 have the form: x1 = ρ˜ x01 , z1 = ˜ z10 , σ, 2 = (λ+ 2)ρ. The dimension of the eigenspace for λ = −3 and λ = 2 is equal to 1. The remaining eigenvalues λk , k = 1, . . . , 4 are the roots of the equation (compare with §16, [6]) 27 λ(λ + 1)(λ + 3)(λ − 2) + 2 m1 m2 = 0. m σ The dimension of the eigenspaces for these λ is equal to 1. It is possible to select the eigenbasis from all the eigenvectors mentioned above. Proof. λ = 1. From the second equation of the system (6.1) we get the following presentation: 3 0 3 0 x1 = 2 (˜ x1 , x1 )˜ x01 = κ˜ x01 = 2 (˜ x1 , κ˜ x01 )˜ x01 = 3κ˜ x01 , σ a a We see that κ = 0, hence x1 = 0, σ. We find the vectors zi from the first equation of the system (6.1).


263

λ = 1. We substitute the presentation zi taken from the second equation the sys0 0 of0 tem (6.1) into the first equation. And then we use the relations σ x ˜1 = σ x1 = 0. As a result we obtain the following system: ⎧ m(˜ x01 , x1 )˜ x01 + m1 ((˜ x02 , x1 + x2 ) + (˜ x01 , x2 ))˜ x01 + ⎪ ⎪ ⎪ ⎨m ((˜ 1 0 0 0 5 x2 , x1 ))˜ x2 = 27 a λ(λ + 1)x1 1 x1 , x1 + x2 ) + (˜ (6.2) 0 0 0 ⎪ m(˜ x2 , x2 )˜ x2 + m2 ((˜ x1 , x1 + x2 ) + (˜ x02 , x1 ))˜ x02 + ⎪ ⎪ ⎩ 1 m2 ((˜ x02 , x1 + x2 ) + (˜ x01 , x2 ))˜ x01 = 27 a5 λ(λ + 1)x2 λ = −1 or λ = 0. Let us multiply the first and the second equations of the system (6.2) by m2 and m1 correspondingly and then subtract one from another. We get m2 x ˜01 (˜ x01 , x1 ) − m1 x ˜02 (˜ x02 , x2 ) = 0 ⇒ (˜ x01 ⊥ x1 ), σ. If +

(˜ x01 ⊥

(6.3)

x1 ) then the equations (6.2) will take the form

((˜ x02 , x1 ) ((˜ x01 , x2 )

+ (˜ x01 , x2 ))˜ x01 + ((˜ x01 , x2 ) + (˜ x02 , x1 ))˜ x02 = + (˜ x02 , x1 ))˜ x02 + ((˜ x02 , x1 ) + (˜ x01 , x2 ))˜ x01 =

1 27 1 27

a5 λ(λ + 1)x1 a5 λ(λ + 1)x2

and then 1 5 1 5 a λ(λ + 1)x1 = a λ(λ + 1)x2 27 27 The vectors xi have two components which lie in the plane {˜ x01 , σ} and in the orthogonal plane to this one. 0 There is a free parameter for the component lying in the first plane because σ x1 = 0 and there are two free parameters for the second component for the same condition. λ = −3 or λ = 2. Now let us find the eigenvectors which have the following presentation: x1 = ρ˜ x01 , z1 = ˜ z10 , σ, ρ, ∈ C. Using the characteristic system (5.2) we obtain the next relations: + 2 = (λ + 2)ρ, 2ρ = (λ − 1), x03 , x3 ) = 0 = (˜ x01 + x˜02 , x1 + x2 ) = (˜

from which we get (λ + 3)(λ − 2) = 0. λ = −1, 0, 1. In this case (see (6.2)) vectors xi lie in the plane {˜ x01 , σ}. Let us 3 mark the expression a λ(λ + 1)/27 by μ. Then the linear system (6.2) in the basis 0 0 0 x02 x ˜2 0 x ˜1 x ˜1 + 2˜ , e2 = , e , e = = e1 = 3 4 x ˜01 0 x ˜02 −2˜ x01 − x ˜02 will be presented the following matrix: ⎛ 1 1 2 (2m2 + m3 ) 2 (m2 − m3 ) 1 1 ⎜ (m − m ) 1 3 2 2 (2m1 + m3 ) ⎜ 1 1 ⎝ (m − m ) 1 3 4 4 (m2 − m3 ) 1 1 (m − 2m − m ) (2m 1 2 3 1 − m2 + m3 ) 4 4 where E is a unitary matrix.

0 0 m 0

⎞ 0 0 ⎟ ⎟ − μE, 0 ⎠ 0

(6.4)

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We recall that (˜ x01 , x ˜01 ) = (˜ x02 , x ˜02 ) = a2 , (˜ x01 , x ˜02 ) = a2 cos(2π/3) = −a2 /2 and moreover 0 0 e3 − e4 e3 + e4 x ˜2 0 x ˜1 0 e1 = e2 = , −e +e , +e = −e = 1 2 1 2 x ˜01 x ˜02 0 0 2 2 As a result the characteristic polynomial of (6.4) has the following form: m1 m2 )(μ − m)μ (4μ2 − 4μ + 3 σ

= (λ(λ + 1)(λ + 3)(λ − 2) +

27 m1 m2 )(λ − 2)(λ + 3)λ(λ + 1). m2 σ

Being equal to 15 the total dimension of the eigenspaces is equal to the dimension of the subspace C18 , which is determined by the condition x1 + x2 + x3 = 0.

7. The asymptotic behavior of β-singular points of the foliation F Now we find the solution of the following system: ⎧ ⎨(m2 + m3 )z1 − m1 (z2 + z3 ) = λ+2 3 x1 , σ 9 3 ⎩− 0 3 x1 + 0 5 (˜ x01 , x1 )˜ x01 = (λ − 1)z1 , σ. |˜ x1 | |˜ x1 |

(7.1)

Without the restriction of generality we can suppose that the vectors x ˜0i , z˜i0 have a form (ai , 0, 0), (bi , 0, 0) correspondingly (see (4.2)). In this case the operator (7.1) has three following eigenspaces: xi , zi ∈ V1 = {(∗, 0, 0)}, xi , zi ∈ V2 = {(0, ∗, 0)}, xi , zi ∈ V3 = {(0, 0, ∗)}, Considering the problem (7.1) in the every eigensubspace we get the next theorem. Theorem 7.1. The eigenvalues and the corresponding eigenvectors of the linear system (7.1) are following: λ = 1. The eigenvectors u1 have a form: x1 = 0, z1 = m1 r, σ, r ∈ C3 . The dimension of the eigenspace is equal to 3. The space V1 . (λ + 3)(λ − 2) = 0. The eigenvectors u−3 , u2 have the following form: x1 = ρ˜ x01 , z1 = ˜ z10 , σ, 2 = (λ + 2)ρ. The dimension of the eigenspace for λ = −3, λ = 0 and λ = 2 is equal to 1. λ = −3, 1, 2. The eigenvalues λ1 , λ2 are found as the roots of the equation m1 + m2 λ2 + λ + 2 − 18 = 0. a33 σ The dimension of the eigenspace for every root is equal to 1. The space V2 . λ(λ + 1) = 0. The eigenvectors u0 , u−1 have the following form: x1 = ρ(0, a1 , 0), z1 = (0, b1 , 0), σ, 2 = (λ − 1)ρ. The dimension of the eigenspace for λ = −1, λ = 0 and λ = 1 is equal to 1.


265

λ = 0, 1, 2. The eigenvalues λ3 , λ4 are found as the roots of the equation m1 + m2 = 0. λ2 + λ − 4 + 9 a33 σ The dimension of the eigenspace for every root is equal to 1. In the space V3 the eigenvalues are the same as in the space V2 . The eigenvectors are found similarly. It is possible to select the eigenbasis from all the eigenvectors mentioned above. Proof. λ = 1. The second equation of the system (7.1) for the space V1 and the spaces V2 , V3 has the form 6 x1 = (λ − 1)z1 , σ, |˜ x01 |3

and

−3 x1 = (λ − 1)z1 , σ |˜ x01 |3

correspondingly. In either case xi = 0 and we get the presentation of the eigenvectors for eigenvalue λ = 1. V1 . Repeating the proof of the Theorem 6.1 let us find the eigenvectors in such a form: x1 = ρ˜ x01 , z1 = ˜ z10 , σ. In this case we get the equation (λ + 3)(λ − 2) = 0. To find the remaining roots λ we substitute zi from the second equation (6.1) into the first one. At the same time we denote the product (λ − 1)(λ + 2) by μ, |˜ x0i | by ai and replace −x1 − x2 by x3 . We obtain the following system: + 1 3 3 a1 a3 μ)x1 + a31 m1 (a2 − a3 )(a22 + a2 a3 + a23 )x2 a32 (a33 m2 + a33 m3 + m1 a31 − 18 1 3 3 m2 a32 (a1 − a3 )(a21 + a1 a3 + a23 )x1 + a31 (m1 a33 + a33 m3 + m2 a32 − 18 a2 a3 μ)x2 The determinant of this system is the quadratic polynomial which is exactly divided in (λ + 3)(λ − 2) = (λ − 1)(λ + 2) − 4 = μ − 4. Therefore we write it in the following form: a31 a32 a33 (μ − 4)2 + (8a31 a32 a33 − 18 (m1 a31 a32 + m1 a31 a33 ))(μ − 4) + R1 (mi , ai ), σ

where R1 (mi , ai ) = 0. As a result we have m1 + m2 m1 + m2 = 0 or λ2 + λ + 2 − 18 = 0. μ + 4 − 18 3 a3 a33 σ σ V2 . Now let us pay attention to the fact that system (7.1) differs a bit from the same one for the space V1 . That is why the eigenvectors for V2 in the form xi = ρ(0, ai , 0), zi = (0, bi , 0) and similar ones for V3 can be found. So we get the following system: + 2 = (λ + 2)ρ −ρ = (λ − 1) from which is followed that λ2 + λ = 0.

266

+

A. Belyaev If repeating the same calculations as for the space V1 we get the system

−a32 (a33 m2 + a33 m3 + m1 a31 + 19 a31 a33 μ)x1 − a31 m1 (a2 − a3 )(a22 + a2 a3 + a23 )x2 −m2 a32 (a1 − a3 )(a21 + a1 a3 + a23 )x1 − a31 (m1 a33 + a33 m3 + m2 a32 + 19 a32 a33 μ)x2 ,

and the polynomial a31 a32 a33 (μ

3 3 3 3 3 3 3 + 2) + 9 (m1 a1 a2 + m1 a1 a3 ) − 4a1 a2 a3 (μ + 2) + R2 (mi , ai ). 2

σ

As a result we have μ−2+9

m1 + m2 a33

σ

= 0,

where μ = (λ − 1)(λ + 2). The calculation for the space V3 coincides exactly with the case V2 . Being equal to 15 the total dimension of the eigenspaces is equal to the dimension of the subspace C18 , which is determined by the condition x1 + x2 + x3 = 0. We can hardly ignore the surprising coincides of the properties of the eigenvectors of the α- and β-points. Theorem 7.2. The operators (6.1), (7.1), which linearize the differential equations (5.5) in the α- and β-points, have the following whole eigenvalues and corresponding eigenvectors for any masses mi . λ = 1. The eigenvectors u1 have the following form: x1 = 0, z1 = m1 r, σ, r ∈ C3 . λ(λ + 1) = 0 The eigenvectors u0 , u−1 satisfy the following conditions: x01 , z1 = x1 ⊥˜

3˜ x01 , σ. |˜ x0 |3 (1 − λ)

(λ + 3)(λ − 2) = 0. The eigenvectors u1 have the following form: x01 , z1 = ˜ z10 , σ, x1 = ρ˜

2 = (λ + 2)ρ.

Proof. The verity of the theorem follows from the Theorems 5.1 and 6.1.

8. The analytic properties of the solutions of three-body problem Let us suppose that the functions x1 (t), x2 (t) from the system (4.5) are known. x1 (t) dt, σ. Then x3 (t) = −x1 (t) − x2 (t), z1 (t) = |x1 (t)|3 If we know the functions w1 (t), w2 (t) we shall be able to find x1 (t), x2 (t) as w1 (t) , σ. x1 (t) = |w1 (t)|2

The Factorization of the Flow At last if we know the functions z1 (t), z2 (t), x1 (t) =

267 z˙1 (t) , σ, according |z˙1 (t)|3/2

to (4.5). Taking into consideration the facts mentioned above we can consider any collection x(t) = ((x1 (t), x2 (t)), w(t) = (w1 (t), w2 (t)), z(t) = (z1 (t), z2 (t)), σ, to be the solution of three-body problem. Theorem 8.1. All the singular but not essentially singular points of the solutions (x(t), w(t), z(t)) of three-body problem (5.1) have the asymptotic behavior (compare with 2.4. [7]): ⎧ 4 0 ⎪ ⎪ ⎪ x(t) = x ˜0 t2/3 + κ2 u2 t8/3 + μk vk tλk +2/3 + · · · , ⎪ ⎪ ⎨ k=1 04 0 (8.1) w(t) = |˜xx˜0 |2 t−2/3 + κ1 u1 t1/3 + κ2 u2 t4/3 + k=1 μk vk tλ−2/3 + · · · , ⎪ ⎪ 4 ⎪ 0 ⎪ ⎪ μk vk tλk −1/3 + · · · , ⎩z(t) = z˜0 t−1/3 + κ1 u1 t2/3 + κ2 u2 t5/3 + k=1

0

0

where x ˜ , z˜ is the solution of the characteristic system (5.2), κ1 , κ2 , μk are free parameters, (ui , ui ) are the eigenvectors of the operators (6.1), (7.1), (vk , vk ) are the eigenvectors of the same operators for the eigenvalues λk > 0, k = 1, . . . , 4 (see Theorems 6.1, 7.1). All eigenvectors (ui , ui ), (vk , vk ) are found efficiently for the asymptotics of α-points. The eigenvectors (ui , ui ), (vk , vk ) of β-points may be found if the solutions of the polynomial (5.4) are known. The vectors ui , vk are expressed by (ui , ui ), (vk , vk ). Proof. Let the point t∗ be a singular point of the solution (w(t), z(t)) to threebody problem. By compactness of the manifold P 17 there exist the limit set of the fiber π(w(t), z(t)) for t → ∞. If this set is not a point, t∗ will be the essentially singular point because w(t), z(t) → ∞. In the opposite case we have the fiber π(w(t), z(t)) entering the singular point of the foliation F and we can obtain the asymptotics of the initial solution in the singular point t∗ . With all this going on, the linear operators of the systems (6.1), (7.1) are degenerated by the invariant action of the So(3) group. It causes the fact that the eigenvalue is equal to zero in the linear span of the orbit of x˜0 by So(3) action. The dimension of the orbit for α-point is equal to 3 and for β-point is equal to 2. Consequently the asymptotics of the α and β-singular points is induced by non-degenerate asymptotics with the help of So(3) action. In order to get the necessary asymptotics we consider the trajectories entering the singular points with λ > 0. The trajectories with λ < 0 correspond the series of the degrees of t−1/3 and give the asymptotics for t → ∞. As it may be seen from the Theorem above the asymptotics of the singular points (8.1) is not general.

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Proposition 8.2. The little general perturbation of the solution with asymptotic behavior (8.1) has the essentially singular point. Proof. In fact, it is enough to prove that the little perturbation of the solution (8.1) has a singular point. Indeed if the initial solution had branching, then the perturbation of solution would also have it and then w(t), z(t) → ∞. If the initial solution was single-valued, the principle of maximum would be broken and then it would be also broken for the perturbation solution too. Thus the perturbation solution is unbounded, consequently the proposition is proved. Now we see that the solution of three-body problem of a general form has an essentially singular point. This result is verified by the next proposition. Proposition 8.3. Let m1 = 0 in three-body problem. Then the solution w1 (t) has the essentially singular point. Proof. In the case m1 = 0 three-body problem has the invariant subspace {x1 , z1 } where two-body problem is realized ([7]): ⎧ ⎨x˙1 = mz1 , x1 (8.2) , ⎩z˙1 = − 3 |x1 | We see that the singular point exists if |x1 | → 0. The problem (8.2) is two-dimensional. Let us denote the coordinates in the following way: x1 = (x11 , x12 ) and then x12 x11 = 0. ¨11 x12 − x11 x ¨12 = m x12 − x11 (x˙ 11 x12 − x11 x˙ 12 )˙ = x |x1 |3 |x1 |3 In the polar coordinates we have x˙ 11 x12 − x11 x˙ 12 = (r cos(ϕ))˙ r sin(ϕ) − r cos(ϕ)(r sin(ϕ))˙ = −r2 ϕ˙ = C. 1 2 In accordance with Theorem 4.1 H = m 2 z1 − |x1 | = we get x˙ 11 = r˙ cos(ϕ) − r ϕ˙ sin(ϕ),

1 2m

x˙ 21 − |x11 |

and then

x˙ 12 = r˙ sin(ϕ) + r ϕ˙ cos(ϕ), ⇓

1 2 1 1 1 1 1 C2 2 2 2 2 2 (x˙ 11 + x˙ 12 ) − = (r˙ + r ϕ˙ ) − = r˙ + 2 − H= 2m |x1 | 2m r 2m r r ⇓ = ⎧ 2 ⎪ ⎪ ⎨r˙ = − C + 2m H + 1 r2 r (8.3) ⎪ C ⎪ ⎩ϕ˙ = − . r2


269

The obtained system is integrable but in the same time its solutions r(t) = |x1 (t)| have the essentially singular point for r → 0. We see this, using Picard √ iterations to the function r(t) = 2Cit + · · · . We get the members tk ln(t)l , which describe the essentially singular point. Finally coming back to the variables w1 , w2 we get the necessary result. Essentially singular points cannot be described for each case with the help of some approximation. But for our problem the limit fiber of the compact manifold may characterize the asymptotics of the essentially singular point. Theorem 8.4. Let (w(t), z(t)) is the solution of three-body problem having the essentially singular point t∗ . Then the limit set X∗ = lim π(w(t), z(t)) determined by t→t∗ E this solution is contained in the set X0 X, where X0 = {π(w1 , w2 , w2 , z1 , z2 , z3 ) : ∃i |wi | = 0}, X = {η ∈ P∗17 : H(π −1 (η)) = 0, I1 (π −1 (η)) = 0, I2 (π −1 (η)) = 0} and H, I1 , I2 are the first integrals of the system (4.5). Proof. The necessary and sufficient condition for (w, z) → X0 is

|wi | → 0 by wi

|wi | |xi | = . wi xi The point t∗ is essentially singular, so in accordance with (4.5) z˙ i → ∞, |xi |3 xi xi / → ∞. = hence |xi |3 xi 3 xi 3 xi |xi |3 → 0. Then → ∞ and xi → 0 from The theorem is not true if xi 3 xi 3 which is followed that coordinates xi are as small as possible in the neighbourhood of the singular point. It is impossible for essentially singular point, consequently |wi | → 0. wi ˇ ˇ Let (w(t), z(t)) = ξ(t) = α(t)ξ(t), where ξ(t) = 1. The function H is the ˘ first integral of the system (4.5), hence H(ξ(t)) = α−2 H(ξ(t)) = α−2 const → 0 ˘ ˘ → 0, I2 (ξ(t)) → 0 and the Theorem is for α → ∞ or t → t∗ . Similarly I1 (ξ(t)) proved. relation

The Theorem 8.4 is verified by the example from the Proposition 8.3. √ C In this case r(t) = 2Cit + · · · , ϕ˙ = 2 . Then we have ϕ = ∓ 2i ln(t) + · · · r and J J √ t1/2 + t−1/2 Ci Ci + ··· = ± + ± t+ ··· x11 = r cos(ϕ) = ±2Cit 2 2 2 J J 1/2 −1/2 √ t −t Ci Ci x12 = r sin(ϕ) = ±2Cit + ··· = i ± − i ± t+ ··· 2i 2 2 and consequently |x1 | 2it =± + · · · → 0, t → 0. x1 1 + tt¯

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A. Belyaev

So we have considered all possible singular points of three-body problem’s solutions. Naturally there is a question if the solutions without singular points exist. The next theorem gives an answer. Theorem 8.5. (see [2]) There are no entire solutions of three-body problem. Proof. Let the solution (w(t), z(t)) be an arbitrary solution of the problem (5.1) and Y = π(w(t), z(t)) be a fiber of the foliation F . Assume that the solution (w(t), z(t)) has no singular points t∗ ∈ C. Then the leaf Y has no singular points, otherwise the leaf Y would have a singular point π(w˜0 , z˜0 ) and the solution (w(t), z(t)) would have singular α or β-points or would be bounded in infinity, which is impossible for entire solution. Let y ∈ Y be an arbitrary point and π(w(t0 ), z(t0 )) = y, w(t0 ), z(t0 ) = max{ wi (t0 ) , zi (t0 ) } = 1. i

One can find such a path γ ⊂ C that w(t1 ), z(t1 ) > 2. And what is more, there exists the neighbourhood U in P∗17 such as for all y´ ∈ U if π(w(t0 ), z(t0 )) = y´, w(t0 ), z(t0 ) = 1 then along γ we have w(t0 ), z(t0 ) > 2. Thus we have an open covering of the closure of Y and choose a finite sub-covering Ui . Let |t0i − t1i | < T for all i. Now let us construct a path Γ which contains the points t0 , t1 , . . . , tn , . . . where tk can be found from tk−1 in the same way as t1 was found by t0 . Then the points t0 , t1 , . . . , tn , . . . must be in the circle with the radius T + T2 + 2T2 + · · · = 2T and there exists the limit point t∗ ∈ C which must be a singular point. Acknowledgment Many thanks to Yu.M. Berezansky and F.E.P. Hirzebruch for the attention, they have paid to the paper.

References [1] S.V. Kovalevskaya, Scientific works. AN SSSR, Moscow, 1948. [2] A.V. Belyaev, The factorization of the flow defined by the Euler–Poisson equations. Methods of Functional Analysis and Topology 7 (2001), no. 4, 18–30. [3] A.V. Belyaev, On single-valued solutions of the Euler–Poisson’s equations. Matematychni Studii 15 (2001), no. 1, 93–104. [4] A.V. Belyaev, The entire solutions of the Euler–Poisson’s equations. Ukr. Mat. Journ. 56 (2004), no. 5, 677–686. [5] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris, 1892, 1893, 1899. [6] C.L. Siegel, Vorlesungen u ¨ber Himmelsmechanik. Springer–Verlag, Berlin–G¨ ottingen–Heidelberg, 1956. [7] V.I. Arnol’d, V.V. Kozlov, A.I. Ne˘ıshtadt, Mathematical aspects of the classic and celestial mechanics. Results of the science and technic. Contemporary problems of mathematics. Fundamental direction 3 (1985), 5–304.


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[8] V.M. Alexeev, The final movings in three-body problems and symbolic dynamics. Usp. Mat. Nauk. 38 (1981), no. 4, 161–176. [9] L. Euler, De motu recilineo trium corpurum se mutuo attrahentium. Novi Comm. Sci. Imp. Petrop. 11 (1767), 144–151. [10] J.L. Lagrange, Oeuvres, Bd. 6 (1873), 272–292. [11] A. Wintner, The analytical foundation of celestial mechanics. AN SSSR, Princeton– Oxford, 1941. [12] K.F. Sundman, Recherches sur le problème des trois corps. Acta Soc. Sci. Fenn. 34 (1907), no. 6. [13] D. Husemoller, Fiber bundles. Mir, Moscow, 1970. [14] R.O. Wells, Differential analysis on the complex manifolds. Mir, Moscow, 1976. [15] I. Tamura, Topology of foliations. Mir, Moscow, 1979. [16] E. Whittaker, A treatise on the analytical mechanics. Cambridge: Univ. PressMir, 1927. Alexandr Belyaev Petrovskogo 123b, ap.40 83117 Donetsk, Ukraine e-mail: [email protected]



On a Moment Problem on a Curve Connected with Ill-posed Boundary Value Problems for a PDE and Some Other Problems V.P. Burskii Abstract. This paper is devoted to a connection between ill-posed boundary value problems in a bounded domain for a PDE that isn’t proper elliptic and a new moment problem on a curve that is a generalization of well-known trigonometric moment problem. Some connections with another field of mathematics are given in partial cases of the curve and the equation.

1. Introduction Let Ω be a bounded domain in the plain R2 with smooth boundary ∂Ω. Consider the following moment problem (the curve ∂Ω is parametrized by s, x(s) ∈ ∂Ω): α(s)(x(s) · a ˜j )N ds = μjN ; j = 1, 2; N = 0, 1, 2, . . . , (1) ∂Ω

where on two given vectors a ˜j ∈ C2 and for two sequences of numbers μjN it is found the function α. It is obviously that for the case when ∂Ω is the unit circle and vectors a ˜j , j = 1, 2 are equal a ˜1 = (1, i); a ˜2 = (1, −i) this moment problem turns to the well-known trigonometric moment problem because then (˜ aj · x(s))N = exp(±iN s). Among a lot of problems connected with above moment problem we will consider the problem of indeterminacy (nonuniqueness): for what curve ∂Ω and vectors a ˜j , j = 1, 2 there exists a function α such that α(s) (x(s) · a ˜j )N ds = 0. (2) ∀N ∈ Z+ , j = 1, 2, ∂Ω

We will read this equality also as the equality with arbitrary polynomial of one variable: ∀Q ∈ C[t], j = 1, 2, α(s) Q(x(s) · a ˜j )ds = 0. (3) ∂Ω

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We will see that the problem (2) is close connected with boundary value problems for a partial differential equation in the domain Ω that we will write down in the form (a1 · ∇)(a2 · ∇) u = 0,

(4)

where aj = (aj1 , aj2 ), j = 1, 2 and a ˜j = (−¯ aj2 , a ¯j1 ), j = 1, 2 are unit complex vectors. The solution is assumed to be in a Sobolev space u ∈ H k (Ω), k ≥ 0. The equation (4) will be written also as a

∂2u ∂2u ∂2u + b + c = 0, ∂x21 ∂x1 ∂x2 ∂x22

(5)

Consider homogeneous the Dirichlet problem u|∂Ω = 0

(6)

for the equation (4) and the Neumann problem uν∗ |∂Ω = 0

(7)

for the same equation (4). Introduce a conormal vector ν∗ and a derivative with respect to the conormal by means of an analog of the Green formula for the Laplace operator (Lu · v − u · Lv) dx = (uν∗ v − u v∗ ) ds, (8) Ω

∂Ω

see for instance [1]. One can count up that ∂ 1 ∂ ∂ − [l(ν(s))]s · , = l(ν) ∂ν∗ ∂ν 2k ∂s

(9)

where l(ξ) = (a1 · ξ)(a2 · ξ) is the symbol of the operator L, ν is a unit vector of normal, s is natural parameter on ∂Ω, k = ±|νs | is the curvature, more exactly νs = kτ where τ = (−ν2 , ν1 ) is the tangent vector. Let designate Δ = det (a1 a2 ) where a1 , a2 are columns. Take an arbitrary ¯ aj · x(s)), polynomial Q of one variable. Substitute the solutions u and v = Q(˜ (overline is complex conjugation for coefficients of Q) of the equation (4) in the equality (8), we obtain zero in the left-side part. Below we will show that the rest with i = 1, 2 can be transformed to the equalities that are valid for any solution u of the equation (4): Δ ∀Q ∈ C[t], uν∗ + uτ Q(x(s) · a ˜1 ) ds = 0, (10) 2 ∂Ω Δ uν∗ − uτ Q(x(s) · a ˜2 ) ds = 0. 2 ∂Ω

∀Q ∈ C[t],

(11)

On a Moment Problem on a Curve. . .

275

In short words these formulae arise as follows. Write the Green formula as 2 1 (Lu · v¯ − u · L¯ v) dx = (a · ∇)u (a · ν)¯ v ds − (a2 · ν)u (a1 · ∇)¯ v ds. (12) Ω

∂Ω

∂Ω

Firstly, we have thrown over derivatives a · ∇ and then a2 · ∇. Now let us do it in reversed sequence, then we obtain (Lu · v¯ − u · L¯ v) dx = (a1 · ∇)u (a2 · ν)¯ v ds − (a1 · ν)u (a2 · ∇)¯ v ds. (13) 1

Ω

∂Ω

∂Ω

Adding formulae (12), (13) and comparing the result with (8) we can accept (a2 · ∇)u (a1 · ν) + (a2 · ν) (a1 · ∇)u = 2uν∗ . Besides note that on ∂Ω direct calculations give (a2 · ∇)u (a1 · ν) − (a2 · ν) (a1 · ∇)u = uτ Δ. ¯ a1 · x(s)) (with any polynomial Q) Substitute u that is a solution of (4) and v = Q(˜ 1 that is a solution of the equation (a ·∇)¯ v = 0 into (12), we obtain the equality (10) ¯ a2 · x(s)) with Q(x(s) · a ˜1 ) = v¯. It is analogical, substitute the same u and v = Q(˜ that is a solution of the equation (a2 · ∇)v = 0 into (13), we obtain the equality (11). We will see below that the conditions (12), (13) will be arisen also sufficient in some sort for existence of a solution of the problem (4), (6), (7). We see now that if the Dirichlet problem (4), (6) has a nontrivial solution u then uν∗ = 0 because otherwise both equalities (6) and (7) are fulfilled and the solution u ≡ 0 by virtue of equality (8) and surjectivity of maximal operator L+ (see condition (16) below). From what it follows that the equality (2) holds with α = uν∗ ≡ 0. The same can be done for the Neumann problem. It is valid inverse arguments (see below). They bring the following fact. Theorem 1. Let m ≥ k > 3 and let us have three sets of statements: 1m ) The homogeneous moment problem (2) has a nontrivial solution α ∈ H m−3/2 (∂Ω). 2k ) The Dirichlet problem (6) for the equation (4) has a nontrivial solution u ∈ H k (Ω). 3k ) The Neumann problem uν∗ |∂Ω = 0 for the equation (4) has a nonconstant solution u ∈ H k (Ω). Then 1m ) ⇒ 2m−q ); 1m ) ⇒ 3m−q ); 2m ) ⇒ 1m ); 3m ) ⇒ 1m ) q = 0 for elliptic equation (vectors a1 , a2 aren’t real), q = 1 + 0 for hyperbolic (unit vectors a1 , a2 are real and different) and mixed equation (one of vectors a1 , a2 is real, another isn’t real), q = 2+0 for equation with K real symbol of kind l(ξ) = (a, ξ)2 . k+0 (By definition, for bounded domain H (Ω) = >0 H k+ (Ω), more accurately, it is the inductive limit of spaces with a corresponding topology.)

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We have answers to above problem (2) in cases when the boundary is an ellipse or a bi-quadratic algebraic curve F (x, y) :=

2

aik xi y k = 0.

(14)

i,k=0

It is proved to be that the answers in both cases (an ellipse or a bi-quadratic curve) can be written in the form: Θ ∈ Q where Q is the rational field and Θ is a complex number which can be counted up by coefficients of curve equation. It shows that the problem becomes ill-posed for real Θ and well-posed for nonreal one. Besides we give equivalences of above indeterminacy problem with real vectors aj to famous the Poncelet problem and the Pell-Abel equation for given polynomial of the third or fourth order. Both problems are famous problems that are connected with a lot of problems in analysis. The connections stated by theorem 1 arises as a corollary of a link form of solution traces u|∂Ω , uν∗ |∂Ω for the equation (4) therefore we start with the link form of traces of a solution.

2. Green formula and connection of solution traces Let Ω be an arbitrary bounded domain in the space Rn with the boundary ∂Ω = Ω\Ω, H = L2 (Ω) be and αn n 1 L= aα (x)Dα , Dα = (−i∂)|α| /∂xα αk 1 . . . ∂xn , α ∈ Z+ , |α| = |α|≤m

k

be some differential operation with smooth complex coefficients aα (x) ∈ C ∞ (Ω) = {u ∈ C ∞ (Ω)| ∃U ∈ C ∞ (Rn ), U |Ω = u}, L+ · = Dα (a∗α (x)·), a∗α = aα |α|≤m + be a formally adjoint differential operation. Let L0 , L+ 0 with domains D(L0 ), D(L0 ) + ∗ be minimal operators and L = (L0 ) , L+ = (L0 )∗ be maximal operators of L and L+ respectively. We will consider the equation

Lu = f ∈ H. (15) ˜ We will also need an operator L, which is a contraction of L on the closing of the space C ∞ (Ω) (or of the Sobolev space H m (Ω)) in the norm of the graph ˜ +. u 2L = u 2L2(Ω) + Lu 2L2 (Ω) and the same operator L We will consider the following conditions: operators L0 , L+ 0 have continuous left inverses;

(16)

˜ + = (L0 )∗ . ˜ = (L+ )∗ ; L (17) L 0 Remember that a basis of the theory of general boundary value problems for the equation (15) with a general differential operator L has been put by M.Yo.


277

Vishik in the paper [11], a predecessor of which was the famous work [7] by M.G. Krein. For u, v ∈ H m (Ω) there is the following Green formula: (Lu, v)Ω − (u, L+ v)Ω =

m−1

< Lm−j−1 u, ∂νj v >∂Ω

j=0

0p

p,s s s=0 Lτ ∂ν

where Lp = is an operator of order p, Lp,s is a tangent linear differτ ential operator of order p − s with smooth coefficients. The first question which we discuss: what boundary properties are supposed for each solution of the equation (15), what traces or agglomerates of traces exist, let be even in sense of the theory of distributions? For a case of the Laplace operator L = −Δ we have expressions L(0) u = u|∂Ω , L(1) u = −uν |∂Ω . The belonging u|∂Ω ∈ H −1/2 (∂Ω) for u ∈ D(−Δ) has been shown actually still by M.Yo. Vishik [11]. From [9] it is known that if the part of boundary ∂ Ω has no characteristic points, i.e., points of contact of real characteristic surfaces (as, for example, in an elliptic case), then usual traces (l) u|∂ Ω , uν |∂ Ω , . . . , uν l |∂ Ω exist for each solution from L2 (Ω) of the differential equation (15). Examples show that, generally speaking, usual traces of solutions from L2 (Ω) do not exist in distributions even for elementary equations. So, for the equation Lu = ∂ 2 u/∂x1 ∂x2 = 0 in the unit disk the solution 3u(x) = (1 − x21 )−5/8 belongs to L2 (K) but u|∂K , 1∂K = ∞ in sense that lim |x|=r u(x)dsx = ∞, r→1−0

so that the trace u|∂K is not a distribution. It is possible to show, however, that each solution u ∈ L2 (K) of such equation has a trace of the product L(0) u := −u(x)l(x)|∂K ∈ L2 (∂K) where l(x) = x1 x2 is the symbol of the operator. In just the same way not for all solutions there is a trace uν |∂K but for each solution u ∈ L2 (K) there is a trace L(1) u = l(x)uν (x) + lτ uτ + 1/2lττ u|∂K ∈ H −3/2 (∂K) where τ is angular coordinate. Similar reasonings can be carried out and in a general case. They are based on the following statement. Statement 1. For any pair functions w and ϕ from H m (Rn ) the following Green formula takes place: G(w, ϕ) := −L(θΩ w) − θΩ Lw, ϕRn =

m−1

L(m−q−1) w, ∂νq ϕ∂Ω =: L∂Ω w, ϕ∂Ω ,

(18)

q=0 (q)

where ∂νq ϕ = ϕν q , L(p) =

P 0 s=0

L(ps) ∂νs is the operator of the order p, L(ps) is

some linear differential operator with respect to tangent directions τ with smooth coefficients of degree p − s. + Let’s note that if 3 L is the maximal operator to formally conjugated operation, then G(w, v) = Ω (Lw · ϕ − w · L+ ϕ) dx. Let us more note that in case of the

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elliptic equation of the second order the distributions fq = (−1)q ∂νq (μ · δ∂Ω ), μ ∈ D (∂Ω), acting according with the formula < fq , ϕ >Rn =< μ, ∂νq ϕ >∂Ω and taking place in the Green formula (18) are accepted to name for q = 0 a simple and for q = 1 a double layer on ∂Ω with density μ. The corresponding potential will arise by means of convolution fq ∗ E with a fundamental solution E if it exists there, of course. Let Jqm = Jm−q−1 : H m−q−1/2 (∂Ω) → H m (Rn ) be a continuous operator of continuation with the property ∂νp (Jqm ψ)|∂Ω = δqp · ψ, p, q = 0, 1, . . . , m − 1

(19)

For the case of bounded domain with smooth boundary one can easy build such operator of continuation by solving of polyharmonic equation Δm u = 0 with boundary data (19). We will substitute in (18) the function Jqm ψ instead of ϕ and instead of w-sequence wk ∈ H m (Rn ) converging to the solution u of the equations (15) in sense of norm of the graph w L2 (Ω) + Lw L2(Ω) . The left part of equality (18) 3 will tend to expression Ω (f · Jqm ψ − u · L+Jqm ψ) dx which is linear and continuous on ψ ∈ H m−q−1/2 (∂Ω). Obtained functional we will designate L(m−q−1) u. Distribution L(p) u we will name pth trace of the solution u on ∂Ω associated with the operator L, or simple pth L-trace of function u on ∂Ω and the distribution L∂Ω w from (18) will name L-boundary distribution. We obtain the following ˜ of the operator L ˜ there Statement 2. For each element u from the domain D(L) exist L-traces L(k) u ∈ H −k−1/2 (∂Ω), k = 0, 1, . . . , m − 1 such that for each se˜ to the element u the sequence L(k) wj ∈ quence wj ∈ H m (Ω) converging in D(L) m−k −k−1/2 H (∂Ω) converges in space H (∂Ω) to an element L(k) u continuously depending on u and independent of a choice of sequence wj . For a case of the differential operator L the existence of the associated traces ˜ is shown in [2], se also [4]. So, we see that in distributions for functions from D(L) L-traces of function from a domain D(L) of the maximal operator exist there and L(m−q−1) u ∈ H −m+q+1/2 (∂Ω), q = 0, 1, . . . , m − 1 if the space H m (Ω) is dense in D(L), i.e., if the conditions (17) are fulfilled. The main property of L-traces is that all of them are equal to zero under conditions of density of smooth functions in D(L) and D(L+ ) if and only if they are L-traces of function from the domain of the minimal operator D(L0 ) that is visible from the formula (18) expanded on the domains of maximal L+ and minimal L0 operators. More exactly, Statement 3. In order to an element u ∈ D(L) belong to the space D(L0 ) it is necessary and under conditions (17) it is sufficient that it has trivial L-traces L(k) u = 0, k = 0, 1, . . . , l − 1. The second impotent property of L-traces is the following Statement 4. Let smooth functions be dense in spaces D(L) and D(L+ ) and the operator L0 is normally solvable, i.e., Im L0 is closed in H. In order to a set u0 , u1 , . . . , um−1 of L-traces be a set L-traces of the solution u of the equations Lu = 0 it is necessary and sufficient that for each sequence vk ∈ H m (Rn ) converging in norm v L2 (Ω) + L+ v L2 (Ω) to some solution of the equation L+ v = 0 the


279

following condition takes place: lim

k→∞

m−1

um−q−1 , ∂νq vk ∂Ω = 0

(20)

q=0

This property can easy be proved directly from the Green formula (18). For the operator L with constant coefficients, the symbol l(ξ) of which can be factored in a product of different irreducible polynomials, in a convex domain exponential solutions are dense in the kernel of maximal operator ([12]) therefore the condition (20) (necessary and sufficient condition of belonging u ∈ ker L) can be written in the form (note that normal solvability of the minimal operator L0 is a corollary of the Hörmander’s estimate ∃C > 0, ∀φ ∈ C0∞ (Ω), Lφ H ≥ C φ H ) m−1 ¯ = {ξ ∈ Cn | ¯l(ξ) = 0}, L(m−q−1) u ∂νq e−i ξ· x dx = 0. (21) ∀ξ ∈ Λ q=0

∂Ω

Note more that for the equation of the second order with homogeneous Dirichlet data L(0) u = 0 we obtain a necessary condition of nontrivial solvability of the Dirichlet problem in view ∃α ∈ H −3/2 (∂Ω), ∀ ξ ∈ Λ, α(x) e−i ξ· x dx = 0 (22) ∂Ω

which will be sufficient for enough smooth α. For the case of homogeneous symbol the last condition can understand as a problem of integral geometry ([4]).

3. Problem of indeterminacy of the moment problem (1) Let us consider now the condition (21) for the equation (4). In this case the symbol is l(ξ) = (a1 ·ξ)(a2 ·ξ), the variety Λ = Λ1 ∪Λ2 , Λi = {t˜ ai | t ∈ C}. In the beginning, consider the Cauchy problem u|∂Ω = ψ, uν |∂Ω = χ

(23)

for the equation (4) in a bounded domain that is convex with respect to real characteristics (i.e., in particular without restrictions on convexity if the equation is elliptic). Statement 5. In order that a function u ∈ H m (Ω), m > 3/2 be a solution of the problem (23) it is necessary that functions P = −l(ν(x))ψ(x) ∈ H m−1/2 (∂Ω), C = l(ν(x))χ(x) + b (ν12 − ν22 ) − 2(a − c)ν1 ν2 ψτ + k (a − c)(ν12 − ν22 ) − 2b ν1 ν2 ψ ∈ H m−3/2 (∂Ω) satisfy the condition ∀ξ ∈ Λ, [P (x(s))(−i ξ · ν(s)) + C(x(s))] exp(−i ξ · x(s))ds = 0, ∂Ω

(24) (25)

(26)

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V.P. Burskii

where s is the natural parameter growing in a direction of the vector τ = (−ν2 , ν1 ), d d ds = dτ ∂Ω , k = −|νs | = −|τs | is curvature of the curve ∂Ω. The proof will be clear from above when one checks equalities P = L(0) u, C = L(1) u. Statement 6. In order that a function u ∈ H m (Ω), m ≥ 2 be a solution of the problem (23) it is sufficient that functions (24), (25) satisfy the condition (26) and belong to the spaces P ∈ H m−1/2+q (∂Ω), C ∈ H m−3/2+q (∂Ω)

(27)

where q = 0 for elliptic equation (vectors a1 , a2 aren’t real), q = 1+0 for hyperbolic (unit vectors a1 , a2 are real and different) and mixed equation (one of vectors a1 , a2 is real, another isn’t real), q = 2 + 0 for equation with real symbol of kind l(ξ) = (a, ξ)2 . Correspondence: [H m−1/2+q (∂Ω) × H m−3/2+q (∂Ω) with property (26) ] (ψ, χ) → (P, C) → u ∈ H m (Ω) is continuous. The fact that there exists any solution u from the kernel kerL ⊂ D(L) and with given smooth traces ψ, χ that are under the condition (26), follows from above. But the smooth solvability (P, C) → (ψ, χ) and raising of smoothness of u needs an additional reasonings that are more difficult (see [3], [4]). Let us write down the condition (26) in a little another form. In the first place, we substitute ξ = tai , i = 1, 2 in the formula (26) and take the Taylorseries expansion of the exponential. All powers of t must be zero and we obtain the condition (26) in the view ∀Q ∈ C[z], j = 1, 2, P (x)(˜ aj · ν) Q (x · a ˜j ) + C(x) Q(x · a ˜j ) ds = 0. (28) ∂Ω

It is obviously that the inverse reasoning is valid also. Further, one can understand ¯ aj · x). this last integral as the integral in the right part of equality (18) at v = Q(˜ For our equation this right part of equality (18) can be written as the right part of equality (8). Then we take the 3 Green formula for our operator in the form (8) and count up the second term ∂Ω u v ν ∗ ds. ∂ 1 u Q(x · a ˜ ) ds = (a1 · ν)(a2 · ν)(ν · a ˜1 )u(x(s)) Q (x · a ˜1 )ds ∂ν∗ ∂Ω ∂Ω : d 1 [l(ν(s))]s u(x(s)) Q(x(s) · a + ˜1 )ds 2k ∂Ω ds : d 1 2 1 =− (a · ν)(ν · a [l(ν(s))]s u(x) Q(x · a ˜1 )ds. ˜ )− 2k ∂Ω ds Here it is used that ∂ Q(x · a ˜1 ) = Q (x · a ˜1 )(τ · a ˜1 ) = (a1 · ν) Q (x · a ˜1 ). ∂s


281

Calculate now the expression in square brackets: 1 1 1 (a · ν)(a2 · ν) s = (ν · a ˜1 )(a2 · ν) − (ν · a (ν · a ˜1 )(a2 · ν) − ˜2 )(a1 · ν) 2k 2 - Δ 1, (−ν1 a12 + ν2 a11 )(ν1 a21 + ν2 a22 ) − (−ν1 a22 + ν2 a21 )(ν1 a11 + ν2 a12 ) = . = 2 2 Therefore at j = 1 we obtain the equality Δ ∂ uν∗ Q(x · a uν∗ + uτ Q(x · a ˜1 )ds. ˜1 ) − u Q(x · a ˜1 ) ds = ∂ν 2 ∗ ∂Ω ∂Ω Similarly, at j = 2 we will obtain Δ ∂ 2 2 uν∗ Q(x · a uν∗ − uτ Q(x · a ˜2 )ds. ˜ )−u Q(x · a ˜ ) ds = ∂ν∗ 2 ∂Ω ∂Ω Calculations can be coverted. Thus, the condition (28) is equivalent to the pair of conditions: (10) and (11). Thus, the following theorem is proved. Theorem 2. The pair of conditions (10), (11) is equivalent to the condition (26). Consider a problem uτ |∂Ω = γ,

uν∗ |∂Ω = κ,

(29)

From statements 5, 6 we obtain validity of the following statements 7, 8. Statement 7. In order to a function u ∈ H m (Ω), m > 3/2 be a solution of the problem (29) for the equation (4) it is necessary that functions 1 γ = us (x), κ = l(ν)uν − [l(ν(s))]s · us ∈∈ H m−3/2 (∂Ω), (30) 2k satisfy the conditions (10) and (11). Statement 8. In order to a function u ∈ H m (Ω), m > 2 be a solution of the problem (29) for the equation (4) it is sufficient that its traces γ, κ from (30) satisfy the conditions (10), (11) and belong to the space H m−1/2+q (∂Ω), where q = 0 for elliptic equation (vectors a1 , a2 are not real), q = 1 + 0 for hyperbolic (unit vectors a1 , a2 are real and different) and mixed equation (one of vectors a1 , a2 is real, another isn’t real), q = 2 + 0 for equation with real symbol of kind l(ξ) = (a, ξ)2 . Such solution u is unique to within an additive constant. Correspondence: [H m−3/2+q (∂Ω) × H m−3/2+q (∂Ω) with properties (10), (11) ] (γ, κ) → u ∈ H m (Ω)/{const} is continuous. Now we can give a proof of Theorem 1 from the introduction. Proof. 1) ⇒ 2). Using pair γ = 0, κ = 2α/Δ we build the solution u ∈ H m−q (∂Ω) by means of the statement 8. 2) ⇒ 1). We have κ = 0 by applying the statement 8. Then we put α = κ. The implications 1) ⇒ 3) and 3) ⇒ 1) are similar.

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4. Solving of indeterminacy problem for the moment problem in some algebraic curves and connected topics Let λ1 , λ2 be roots of the equation l(1, λ) = 0. Let us enter (complex) angles of inclinations of characteristics. Any solution ϕ1 of the equation tgϕ1 = −λ1 = ±i will be named an angle of inclination of the characteristic direction corresponding to the root λ1 . Restriction −λ = ±i is connected to that the equation tgϕ = ±i has no any solution in C, in this case we will speak that the characteristic direction has no angle of inclination. Similarly, we determine an angle ϕ2 through the root λ2 as tgϕ2 = −λ2 = ±i and then an angle ϕ0 := ϕ1 − ϕ2 . It is easy to see that sin ϕ0 = det(a1 a2 ) = Δ where a1 , a2 are columns and tg2 ϕ0 = (b2 −4ac)/(a+c)2 = [(λ1 − λ2 )/(1 + λ1 λ2 )]2 . If vectors a1 , a2 are normalized by unit (|aj | = 1) we will aj2 = − cos ϕj , a ˜j2 = a ¯j1 = sin ϕj . suppose that a ˜j1 = −¯ Consider a case of the disk Ω = K = {x ∈ R2 |x2 < 1}. In this case we have ν(x) = x, x · a ˜j = − cos(τ + ϕj ), x = (cos τ, sin τ ), τ = s is angular coordinate. Designate through Tñ , Qn the Chebyshev polynomials: Tñ (cos α) = cos nα, Qn−1 (cos α) = sin nα/ sin α and also we enter functions Tn (τ ) = Tñ (cos(τ + ϕ1 ))/ An = cos n(τ + ϕ1 )/ An = ˜1 )/ An (n > 0), T0 = √12π , Un = (x · a ˜1 )τ · Qn−1 (−x · a ˜1 )/An = Tñ (−x · a sin n(τ + ϕ1 )/An where 2π

2 cos n(τ + ϕ1 ) cos n(τ + ϕ1 ) dτ An = πch (2n Im ϕ1 ), An = 0

2π

sin n(τ + ϕ1 ) sin n(τ + ϕ1 ) dτ.

= 0

It is easy to see that system of functions {Tn (τ ), Un (τ )}∞ n=0 is orthonormalized in L2 (∂K) and total and orthogonal in0each space H l (∂K) because the same is ∞ system sin nτ }, moreover, f = n=0 (fn Tn + gn Un ) ∈ H l (∂K) if and only 0 {cos 2nτ, l 2 2 if n (1 + n ) (|fn | + |gn | ) < ∞. Let us substitute in the condition (3) expansion of the function ∞ # $ 1 α(τ ) = αT0 + αTn Tn (τ ) + αU U (τ ) n n 2 n=0 and in the capacity of the polynomial Q we will take the Chebyshev polynomial Q(t) = Tñ (−t). Then condition (3) will be written down as αTk = 0, (j = 1);

cos kϕ0 αTk + sin kϕ0 αU k = 0, (j = 2); k = 0, 1, 2, . . . .

From what we obtain αTk = 0, sin kϕ0 αU / Q. k = 0. Therefore α(τ ) ≡ 0 if ϕ0 /π ∈ If ϕ0 /π = p/q ∈ Q then α = sin kq(τ + ϕ1 ), k = 1, 2, . . . is a set of nontrivial solutions of the problem (3). The following theorem is proved (by applying of another method [3], [4]).


283

Theorem 3. For λj = ±i, j = 1, 2, λ1 = λ2 the problem (3) has only trivial solution in each space H m (K), m > 2 if and only if the number ϕ0 /π is irrational. Under the condition ϕ0 /π ∈ Q (31) m in each space H (K) the homogeneous problem (3) has infinite number of linearly independent solutions. Below we will consider more complicate domain Ω that the circle or the ellipse, namely, domains which boundary is a biquadratic curve C = ∂Ω with equation: 2 F (x, y) := aik xi y k = a22 x2 y 2 + a21 x2 y + · · · = 0. (32) i,k=0

We will need the following John mapping. Let Ω be arbitrary bounded domain, which is convex with respect to characteristic directions, i.e., it has the boundary C intersected in at most two points by each straight line that is parallel to x- or y-axes. We start from arbitrary point M1 on C and consider a vertical line passing through M1 . Obviously, there are two points of intersection with the curve C: M1 and some M2 , which may be coincided with M1 . We denote I1 an involution which transform M1 into M2 . Then, starting from M2 , we consider a horizontal line passing through M2 . Let M3 be the second point of intersection with the curve C. Let I2 be corresponding involution: I2 M2 = M3 . We then repeat this process, applying step-by-step involutions I1 and I2 . Denote T = I2 I1 , T −1 = I1 I2 . This transformation T : C → C gives us a discrete dynamical system on C, i.e., an action of group Z and each point M ∈ C generates an orbit {T nM |n ∈ Z}. This orbit can be finite or denumerable set. The point M with finite orbit is called a periodic point and smallest n, for which T n M = M , is called a period of the point M . In the paper by John the uniqueness breakdown in the problem have studied in connection with topological properties of the mapping T for the case of even mapping T . The mapping T is called to be even or preserving an orientation if each positive oriented arc (P, Q) with points P, Q ∈ C transforms into positive oriented arc (T P, T Q). Fritz John have proved several useful assertions for even T , among of which we extract the following one. Sufficient condition of uniqueness. The homogeneous Dirichlet problem for the string equation in the bounded domain has only a trivial solution in the space C 2 (Ω) if the set of periodic points on C is finite or denumerable. Below we give several problem settings from different areas of mathematics, each of them proves to be equivalent to each of above given in a domain with a biquadratic boundary. First three of them were noted above. 1) Existence of a nontrivial solution of the homogeneous Dirichlet problem. 2) Existence of a nonconstant solution of the homogeneous Neumann problem. 3) Existence of a nontrivial solution of the homogeneous generalized trigonometrical moment problem.

284

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4) Integration by elementary functions: Let us consider the problem (N.H. Abel, 1826): for what polynomial R(t) of degree 2m and of one variable t exist there polynomials ρ(t), P (t), Q(t) such that √ ρ P + RQ √ dt = A ln √ + C. (33) R P − RQ Later Liouville, Golubev and others have proved that if the primitive from left-side part of this equality is an elementary function (i.e., composition of polynomials, exponents, roots, trigonometric functions) then it must be a function from the right-side part. Abel proved two criteria for this problem: Statement 9. The equality (33) holds if and only if the expansion in a continuous fraction √ 1 R = r0 (t) + 1 r1 (t) + r2 (t)+··· is periodic, that is ∃N, ∀k > k0 , rk = rk+N . Statement 10. The equality (33) holds if and only if the algebraic Pell-Abel equation P 2 (t) + R(t)Q2 (t) = 1

(34)

is solvable. Here for given polynomial R of even order one should find polynomials P, Q such that the equality (34) is valid, then ρ = 2P /Q. Below we will deal with the case m = 2, ord R = 4. Thus we have also the setting of 5) Problem of solvability of Pell-Abel equation. As it was shown in a works by Sodin-Yuditskii [10] the last setting is equivalent to the following: 6) Problem of maximal set with least deviation by Chebyshev-Akhiezer. Consider the Chebyshev problem of finding a polynomial of least deviation on a l−1 K closed set in the real axe. Let I = [−1, 1]\ (aj , bj ) − a system of l closed intervals j=1

and one should find a polynomial of a given order n with leading coefficient 1 which gives a least deviation on the set I, i.e., to find a minimum of the functional tn − Pn−1 (t) C(I) → min . The general polynomial Pn−1 (t) is running a finite-dimensional subspace and we deal with a problem of functional minimization on the nonreflexive Banach space. In 30 years of XX century A. Markov and A. Borel, based on Chebyshev ideas, proved that such polynomial P exist there on each set I. But if the polynomial P is minimal on I then, possible, it will be minimal on a more large set I˜ which is an expansion of I. Such maximal among of I˜ a set E =

On a Moment Problem on a Curve. . . [−1, 1] \

285

m−1 K

(αj , βj ), which is called n-correct. If one take the polynomial R in the

j=1

form R = (t2 − 1)

m−1 L

(t − αj )(t − βj ) then it is valid the following

j=1

Statement 11. (see [10]) The solvability of the Pell-Abel equation P 2 (t) + R(t)Q2 (t) = L2

(35)

where the constant L is unknown also, is equivalent to that the set E is n-correct. In addition, the polynomial P gives us a solution of extremal problem and the number L is the minimal deviation. 7) Spectrum of infinite Jacoby matrix (see references in work [10]). It is interesting that the set E is a continuous spectrum of some infinite selfadjoint real Jacoby (three-diagonal) matrix in the space l2 if and only if the set E is ncorrect set, that is if and only if the equation (35) is solvable. For our aims the case of two intervals (the Akhiezer problem) is appropriated. A connection of the problem (35) with boundary value problems is realized by means of the Poncelet problem. 8) Setting of the Poncelet problem. Recall the Poncelet problem for the case of two ellipses, for simplicity and as it was introduced by Jean-Victor Poncelet himself. We take two arbitrary ellipses A and B, A inside B in the plane R2 of variables ξ, η. Let us have an arbitrary point Q1 on the ellipse A and pass a tangent straight line to A at the point Q1 . This tangent crosses the ellipse B at two points P1 and P2 , P1 before P2 with respect to a standard orientation. Then we take the point P2 on B and pass the second tangent to the ellipse A. We denote as Q2 the point on A where this tangent contacts with A. This tangent meets the ellipse B in two points P2 and P3 . Take the point P3 and repeat this procedure. Then we obtain a mapping UB : B → B which acts by the rule UB : Pk → Pk+1 that will be called the Poncelet mappings below. The point P1 will be named a periodic point and of a period N if PN +1 = UBN P1 = P1 and N is minimal with this property. The big Poncelet theorem says: if there is a periodic point P then each point is periodic with the same period. This construction is projective so that in general we may build it for a pair of conics (conic sections). For any conic A it is possible to find polynomials E0 (x), E1 (x), E2 (x) with 1 (x) 2 (x) deg(Ei (x)) ≤ 2 such that ξ = E η= E E0 (x) , E0 (x) . Quite analogously, the conic B 1 (y) 2 (y) can be parametrized as ξ = G η= G G0 (y) , G0 (y) , where Gi (y) are some other polynomials of most degrees 2. Our observations (in coauthorship with A.S. Zhedanov, see our common work [5]) shows that in the plane of variables x, y the Poncelet mapping UB turn into the John mapping on a biquadratic curve (C) which is given by conics A and B.

286

V.P. Burskii This gives us the following

Theorem 4. A given the Poncelet problem is periodic if and only if the John mapping for corresponding biquadratic curve is periodic. Conversely, any generic biquadratic curve generated a projective class of a conics pair and we obtain. Theorem 5. For generic biquadratic curve the Dirichlet problem has non-unique solution if and only if corresponding John mapping has a periodic trajectory and if and only if corresponding Poncelet problem has a periodic trajectory. Note that for the case of bounded domain with a biquadratic boundary we have proved that John’s sufficient uniqueness condition is also necessary, moreover, it will be so even for cases when the curve C is unbounded but then we should change the setting of the problem. Namely, along with the usual setting of the uniqueness property: The examined bounded domain such that the homogeneous Dirichlet problem (1) has only trivial solution in the space C 2 (Ω) for cases when the curve C is unbounded we examine the following modification of uniqueness property for the homogeneous Dirichlet problem The examined curve C is such that each analytic in real sense solution in R2 of the string equation with the property u|C = 0 is only zero solution. Note, the assumption “analytic” is introduced in order that we can consider such curves C for them there exist characteristic lines which are not intersect C and are between of curve branches, because without this assumption for such curve and, e.g., with an assumption of infinite smoothness one may build a simple example of a smooth nontrivial solution of the problem in sense (1). The Poncelet porism in form of two circles Let a circle A lies inside another circle B. From any point on B, draw a tangent to A and extend it to B. From the point, draw another tangent, etc. For n tangents, the result is called an n-sided Poncelet transverse. This Poncelet transverse can be closed for one point of origin, i.e., there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) n-gon. We could begin with a polygon that is understood as the union of a set of straight lines sequentially joint a given cyclic sequence of points (vertices) on the plane. If there exist two circles, inscribed and circumscribed for this polygon, then this polygon is called a bicentric polygon. Note that sides of the polygon can intersect and the intersection point is not obligatory to be a vertex. Furthermore, the inscribed circle does not obligatory touch a segment between vertices, the contact point can lie on extension of the side and therefore the circles can intersect. Bicentric polygons are popular objects of investigations in geometry. This is most known form of the Poncelet porism. If we denote by r the radius of the inscribed circle, by R the radius of the circumscribed circle and by d a distance between the circumcenter and incenter for a bicentric polygon then these three numbers can not be arbitrary and together with n they satisfy some relations. So, for the case of triangle the relation is sometimes known


287

as the Euler triangle formula R2 − 2Rr − d2 = 0. One of popular notations for such relations (necessary and sufficient for existence of a bicentric polygon) is given in terms of additional quantities a=

1 , R+d

b=

1 , R−d

c=

1 . r

So, for a triangle above the Euler formula has the view: a + b = c, for a bicentric quadrilateral, the radii and distance are connected by the equation a2 + b2 = c2 . The relationship for a bicentric pentagon is 4(a3 + b3 + c3 ) = (a + b + c)3 . In a general case one introduces numbers λ=1+

2c2 (a2 − b2 ) , a2 (b2 − c2 )

ω = cosh−1 λ , 1

K = K(k) = 0

k 2 = 1 − e−2ω ,

dt

2 (1 − k t2 )(1 − t2 )

(36)

and then the relationship can be written by means of elliptic functions in the form √ √ c b 2 − a2 + b c2 − a2 K ,k = sc n a(b + c) (Richelot (1830) – the first edition of the criterion, Kerawala (1947) – the above criterion). The connection with the Pell-Abel equation comes from the well-known Cayley criterion for the Poncelet problem. Recall that the Cayley criterion can be formulated as follows. Let f (λ) = det(A − λB) be a characteristic determinant for the one-parameter pencil of conics A and B presented in the projective form. In more details, assume that the conic A has an affine equation φA (x, y) = 0. We then pass to the projective co-ordinates ξ: x = ξ1 /ξ0 , y = ξ2 /ξ0 and present the equation of the conic A in the form 2

Aik ξi ξk = 0

i,k=0

with some 3 × 3-matrix A. Similarly, the projective equation for the conic B has the form 2 Bik ξi ξk = 0 i,k=0

with some 3 × 3-matrix B. Then we define the polynomial f (λ) = det(A − λB) of the third degree. Note that f (λ) is a characteristic polynomial for the generalized eigenvalue problem for two matrices A, B. Calculate the Taylor expansion

f (λ) = c0 + c1 λ + · · · + cn λn + · · ·

288

V.P. Burskii

and compute the Hankel-type c3 c (1) Hp = 4 ... cp+1 and Hp(2)

c2 c = 3 ... cp+1

determinants from these Taylor coefficients: c4 . . . cp+1 c5 . . . cp+2 , p = 2, 3, 4, . . . ... ... . . . cp+2 . . . c2p−1 c3 c4 ... cp+2

... ... ... ...

cp+1 cp+2 ... c2p

,

p = 1, 2, 3, . . .

Then the Cayley criterion is: the trajectory of the Poncelet problem is pe(1) (2) riodic with the period N if and only if Hp = 0 for N = 2p, and Hp = 0 for N = 2p + 1. Moreover, we have done the following observation: the Cayley condition coincides with a solvability criterion of the Pell-Abel equation by V.A. Malyshev [8]. A2 (λ) + f˜(λ)B 2 (λ) = 1 with deg f˜ = 4, f (0) = 0, (for details concerning solvability of the Pell-Abel equation and its relations with other problems of mathematics see, e.g., papers by Malyshev) if one takes f (x) = x4 f˜(x−1 ). We have the following proposition: Theorem 6. ([5]) The Poncelet problem is periodic with an even period iff corresponding the Pell-Abel equation is solvable. Methods are based on the theory of elliptic functions. Such functions are generated by the biquadratic curve from the Dirichlet problem, to which each of above problem can be reduced. The John mapping of the Dirichlet problem acts on a biquadratic curve generated of the Poncelet problem for two conics which are built by data of the Pell-Abel equation. This biquadratic curve, possible after a projective transformation of the plane, may be parametized by an elliptic function φ(z) of the second order: x = φ(z), y = φ(z + η). Then the John mapping on the complex biquadratic curve of the complex space C2 (that is a Riemann surface of a genus 1, i.e., a torus) may be given as a shift z → z +2η. Because the periods ω1 , ω2 of the elliptic function φ (as η also) are counted up by data of the corresponding biquadratic curve then we obtain our periodicity criterion of the John mapping in the view: 2ηN = m1 ω1 + m2 ω2 (9) with some integer N, m1 , m2 . Further we have done an analysis on reality of the functions that gives us a criterion for each of the problems in the form: θ m = ∈Q (37) 2K n where the number θ is counted up by data of the corresponding problem as, for instance, the number K may be counted up by the formula (36). We see that this


289

criterion is similar to the criterion (31) of uniqueness breakdown for the Dirichlet problem in an ellipse. Note that a part of these results and references one can find in the works: [4], [5], [6]. Along of above writing problems there are other problems that allows their investigation by means of this approach.

References [1] Yu.M. Berezanskii, Decomposition on eigenfunctions of self-ajoint operators. – Kiev: Naukova dumka, 1965. (In Russian) [2] V.P. Burskii, Boundary properties of L2 -solutions of linear differential equations and duality equation-domain. Doclady Academii Nauk SSSR 309 (1989), no. 5, 1036–1039. (In Russian) [3] V.P. Burskii, On boundary value problems for differential equations with constant coefficients in a plane domain and a moment problem. Ukr. Math. Journal 48 (1993), no. 11, 1659–1668. [4] V.P. Burskii, Investigation methods of boundary value problems for general differential equations. Kiev, Naukova dumka, 2002. (In Russian) [5] V.P. Burskii, A.S. Zhedanov On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems. Ukr. Math. Journal 58 (2006), no. 4, 487–504. [6] V.P. Burskii, A.S. Zhedanov, Dirichlet problem for string equation, Poncelet problem and Pell-Abel equation. Symmetry, Integrability and Geometry: Methods and Applications. 4p. arXiv:math.AP/0604278 – Apr 2006. [7] M.G. Krein,Theory of selfadjoint expansions of semibounded Hermitian operators and its applications. I. – Mathematical Sbornik 20:3 (1947), 431–495. (In Russian) [8] V.A. Malyshev, Abel equation, Algebra and analysis 13 (2001), no. 6, 1–55. (In Russian) [9] Ya.A. Roitberg, On boundary values of generalized solutions of elliptic systems by Duglis-Nirenberg . 18 (1977), no. 4, 845–860. (In Russian) [10] L.M. Sodin, P.M. Yuditskii, Functions least deviating from zero on closed sets of real axis. Algebra and analysis 4, no. 2, 1–61. (In Russian) [11] M.Yo. Vishik, On general boundary value problems for elliptic differential equations. Trudy Moskowskogo Mathematicheskogo Obshchestva 1 (1952), 187–246. (In Russian) [12] J.L. Treves, Lectures on linear partial differential equations with constant coefficients. Rio de Janeiro: Instituto de Mathematica, 1961. V.P. Burskii Institute of Applied Mathematics and Mechanics NASU Donetsk 83114, Ukraine



The Construction and Analysis of a Posteriori Error Estimators for Piezoelectricity Stationary Problems Fedir Chaban and Heorgiy Shynkarenko Abstract. This paper considers the building of a posteriori error estimator for finite element method approximation of piezoelectricity boundary problem. The construction of the estimator is based on the error problem and properties of bubble-functions. The efficiency and reliability of estimator is illustrated by numeric results of solved model problems. Mathematics Subject Classification (2000). Primary 65N30; Secondary 65N15. Keywords. Finite element method (FEM), piezoelectricity problem, a posteriori error estimator (AEE).

1. Introduction The important feature of modern numeric schemes, which are used for solving boundary problems, is a possibility to receive a posteriori error estimation for obtained solutions. In most cases this possibility is realized via construction of a posteriori error estimator. Constructed AEEs are indicators of reliability and efficiency of the developed scheme. In case of a solution search with help of FEM, constructed AEE can be used to improve obtained solutions (h-adaptive schemes and post processor refinement schemes.) The examples of efficient AEE application are described in [2, 5, 8, 10]. The paper contains an analysis of error functional and a definition of the variational problem on Galerkin’s discretization error for stationary piezoelectricity problems. Bubble approximation of Galerkin’s discretization error was constructed for detection of AEE. Solutions of experimental problems, where a shank made of piezoelectric PZT-4 was used as a model, prove efficiency of a proposed This work was completed with the support of department of information systems of Ivan Franko National University of Lviv.

292

F. Chaban and H. Shynkarenko

methods. Preliminary numeric results are presented in [9]. The paper can be also treated as a research extension of [1, 2, 4].

2. Piezoelectricity problems Let Ω is the limited bounded domain of points x = (x1 , . . . , xd ) from Euclidean space Rd with continuous by Lipshyts boundary Γ and unit normal vector n = (n1 , . . . , nd ), where ni = cos(n, xi ). 2.1. Physical and mechanical properties of a piezoelectric Let the material of a piezoelectric be characterized by: (i) mass density ρ = ρ(x) > 0; (ii) elastic coefficients {aijkm (x)}di,j,k,m=1 with common properties of symmetry and ellipticity aijkm = ajikm = akmij , (2.1) aijkm ij km ≥ a0 ij ij a0 = const > 0 ∀ij = ji ∈ R in Ω. (iii) piezoelectric coefficients {ekij (x)}dk,i,j=1 with symmetry properties of a sort ekij = ekji in Ω;

(2.2)

(iv) dielectric susceptibility coefficients {gkm (x)}dk,m=1 with properties of symmetry and ellipticity + gkm = gmk , (2.3) gkm ξk ξm ≥ g0 ξk ξk g0 = const > 0 ∀ξk ∈ R, in Ω. In general in a three-dimensional problem there are 18 piezoelectric constants ekij , 21 elastic coefficients aijkm and 6 dielectric susceptibility constants gkm . As far as a polar tensor disappears in centrosymmetrical crystals, a piezoelectric effect does not appear in them. 2.2. Boundary problem of piezoelectricity Equilibrium state of piezoelectric is characterized by vector of elastic deformations u = {ui (x)}di=1 , scalar electric potential p = p(x), strain tensor ε = {εij (x)}di,j=1 , stress tensor σ = {σij (x)}di,j=1 , vector of electric induction D = {Di (x)}di=1 and vector of electric field E = {Ei (x)}di=1 , which are described by the next boundary

AEE for Piezoelectricity Problems value problem: ⎧ find vector of elastic deformation u = {ui (x)}di=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ and electric potential p = p(x) which ⎪ ⎪ ⎪ ⎪ ⎪ satisfies the system of fundamental equations of piezoefect ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ − {σij (u)} = ρfi , ⎪ ⎪ ⎪ ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ σij (u) := aijkm εkm (u) − ekij Ek (p), ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ εij (u) := 12 ( uj + ui ), ⎪ ⎪ ∂xi ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∇. D(u, p) = ρ∗ , ⎪ ⎪ ⎪ ⎨ D (u, p) := e ε (u) + g E (p), k

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

kij ij

km

∂ Ek (p) := − p in ∂xi and boundary conditions u=0

m

293

(2.4)

Ω,

on Γu ⊂ Γ, mes Γu > 0,

σij nj = σ î

on Γσ := Γ \ Γu ,

p = 0 on Γp ⊂ Γ, mes Γp > 0, D(p). n = 0 on ΓD ⊂ Γ, Γp ∩ ΓD = {∅}, E(p) − (n. E(p))n = 0 D(p). n dγ = ρe ,

on Γe ⊂ Γ, mes Γe > 0, Γp ∩ Γe = {∅}.

Γe

2.3. Variational problem of piezoelectricity The main object of our analysis is Proposition 2.1. About variational problem of piezoelectricity. Boundary value problem of piezoelectricity (2.4) has the following variational formulation ⎧ ⎪ ⎨ find a pair {u, p} ∈ V × Q such that ∀v ∈ V, c(u, v) − e(p, v) = ls , v (2.5) ⎪ ⎩ e(q, u) + g(p, q) = le , q ∀q ∈ Q, where spaces of admissible functions are + , V := v ∈ [H 1 (Ω)]d : v = 0 on Γu , , Q := q ∈ [H 1 (Ω)] : q = 0 on Γq , q = const on Γe ,

(2.6)

294


bilinear forms are described by expressions ⎧ 3 3 ∂uk ∂vi ⎪ ⎪ c(u, v) := Ω aijkm εkm (u)εij (v) dx = Ω aijkm dx, ⎪ ⎪ ∂xm ∂xj ⎪ ⎪ ⎪ ⎨ 3 3 ∂q ∂vi e(q, v) := Ω ekij Ek (q)εij (v) dx = − Ω ekij dx, ∂x ⎪ k ∂xj ⎪ ⎪ ⎪ ⎪ 3 ∂p ∂q ⎪ ⎪ dx ⎩ g(p, q) := Ω gij ∂xi ∂xj and linear functionals + 3 3 ˆ . v dγ ∀v ∈ V, ls , v := Ω ρf. v dx + Γσ σ 3 le , q := Ω ρ∗ .qdx + ρe q|Γe ∀q ∈ Q.

(2.7)

(2.8)

In order to simplify notations on Cartesian space product of admissible functions Φ := V × Q let’s define the bilinear form s(ψ, φ) := c(u, v) − e(p, v) + g(p, q) + e(q, u) ∀ψ = {u, p}, φ = {v, q} ∈ Φ

(2.9)

and linear functional χ, φ := ls , v + le , q ∀φ = {v, q} ∈ Φ.

(2.10)

Then variational problem of piezoelectricity(2.5) will be written in a state of common variational equation: + find a pair ψ = {u, p} ∈ Φ such that (2.11) s(ψ, φ) = χ, φ ∀φ = {v, q} ∈ Φ. 2.4. Minimax problem Now the attention is paid to specific property of variational structure of piezoelectricity equations. For this purpose, let us define quadratic functional (of Lagrange) on space Φ := V × Q L(v, q) := 12 {c(v, v) − g(q, q)} − e(q, v) − ls , v + le , q ∀φ = {v, q} ∈ Φ and examine the following variational minimax problem: + find a pair ψ = {u, p} ∈ Φ such that L(u, q) ≤ L(u, p) ≤ L(v, p) ∀φ = {v, q} ∈ Φ. The above-formulated problem is characterized by

(2.12)

(2.13)

AEE for Piezoelectricity Problems

295

Theorem 2.2. About saddle point of Lagrangian functional. The problem of variational equations of piezoelectricity (2.5) is equivalent to minimax problem (2.13). In addition, each of their solutions ψ = {u, p} ∈ Φ is such, that 1 L(u, p) = {g(p, p) − c(u, u)} + e(p, u). (2.14) 2 Proof. Let a pair ψ = {u, p} be a solution of minimax problem (2.13) in space Φ; then for any ∈ R we make the following expansion L(u + v, p + q) = L(u, p) +{[c(u, v) − e(p, v) − ls , v] − [g(p, q) + e(q, u) − le , q]} + 12 2 {c(v, v) − g(q, q) − 2e(q, v)}

(2.15)

∀φ = {v, q} ∈ Φ. Taking into account ellipticity of bilinear forms c(· , · ) and g(· , · ), we accept q = 0 (correspondingly v = 0) in expansion (2.15). 2.5. Correctness of variational formulation of piezoelectric effect problem We should remember that piezoelectricity problem (2.5) can be written in a terms of variational equation + find a pair ψ = {u, p} ∈ Φ such that (2.16) s(ψ, φ) = χ, φ ∀φ = {v, q} ∈ Φ. Theorem 2.3. About the correctness of piezoelectricity problem formulation. Let the following conditions be true mes Γu > 0,

mes Γp > 0,

conditions of regularity of the following functions ⎧ 2 d ˆ ∈ [L2 (Γσ )]d ⎪ ⎨ f ∈ H = [L (Ω)] , σ ρe ∈ L2 (Ω), |σe | < +∞, ⎪ ⎩ aijkm , ekij , gij , ρ ∈ L∞ (Ω)

(2.17)

(2.18)

and also the conditions of symmetry and ellipticity (2.1), (2.2) and (2.3). Then: (!) bilinear forms c( · , · ) : V × V → R and g( · , · ) : Q × Q → R create scalar product on admissible function spaces V and Q correspondingly, and, as a corollary of that, new norms (energy) +

|v|V := a(v, v) ∀v ∈ V, (2.19)

|q|Q := g(q, q) ∀q ∈ Q; (!!) variational piezoelectricity problem (2.5) has single solution ψ = {u, p} ∈ Φ, and in addition (2.20) |ψ|Φ ≤ |χ|∗ ,

296 where

F. Chaban and H. Shynkarenko ⎧

2 2 ⎪ ⎪ ⎨ |φ|Φ := c(v, v) + g(q, q) = |v|V + |q|Q , |χ, φ| ⎪ ⎪ ⎩ |χ|∗ := sup |φ| Φ 0=φ∈Φ

∀φ = {v, q} ∈ Φ.

(2.21)

2.6. Abstract Galerkin’s scheme For all values of discretization parameter h > 0 let us define Galerkin’s approximation ψh = {uh , ph } in subspace Φh := Vh × Qh ∈ Φ as a solution of the following problem ⎧ approximation spaces Vh ⊂ V, dim Vh < +∞ ⎪ ⎪ ⎪ ⎪ and Qh ⊂ Q, dim Qh < +∞ are defined; ⎪ ⎪ ⎨ find a pair {uh , ph } ∈ Vh × Qh such that (2.22) ⎪ ⎪ ⎪ a(uh , v) − e(ph , v) = ls , v ∀v ∈ Vh , ⎪ ⎪ ⎪ ⎩ e(q, uh ) + g(ph , q) = le , q ∀q ∈ Qh . Let us assume, that the series of linear independent functions + v1 (x), . . . , vN (x) of admissible deformation space V, (2.23) q1 (x), . . . , qM (x) of admissible potential space space Q is selected as a system of base function of approximation spaces Vh and Qh correspondingly. In addition, it means, that Galerkin’s approximation ψh = {uh , ph } is retrieved as a linear combination + 0N uh (x) := j=1 uj qj (x) ∈ Vh , (2.24) 0M ph (x) := m=1 pm qm (x) ∈ Qh with unknown coefficients U = (u1 , . . . , uN ) i P = (p1 , . . . , pM ).

3. Galerkin’s discretization error problem 3.1. Analysis of error functional In addition to energy norm | · |Φ from (2.21) let us provide admissible deformation space Φ with 3 a standard norm φ Φ := { [φ.φ + φi,j φi,j ]dx}1/2 = Ω 3 { [v.v + vi,j vi,j + q.q + qi,j qi,j ]dx}1/2 ∀φ ∈ Φ Ω

conjugated space Φ with norm χ ∗ := sup

0=φ∈Φ

| χ, φ | . φ Φ

Proposition 3.1. About error functional. Let ψ ∈ Φ be a solution of a problem (2.11). For all fixed w ∈ Φ let us define linear functional (error source) according to the following rule φ ∈ Φ → ρ(w), φ := χ, φ − s(w, φ) ∈ R

(3.1)


297

Then the following statements are correct α ψ − w Φ ≤ ρ(w) ∗ ≤ s ψ − w ∀ ∈ Φ,

(3.2)

where α = const > 0 which s(φ, φ) ≥ α φ 2Φ ∀φ ∈ Φ. | ρ(w), φ | |φ|Φ 0=φ∈Φ

|ψ − w|Φ = |ρ(w)|∗ := sup

(3.3) ∀w ∈ Φ.

(3.4)

Proof. From the definition (3.2) and the problem (2.11) we get ρ(w), v = χ, φ − s(w, φ) = s(ψ − w, φ) ∀φ, w ∈ V.

(3.5)

From the previous statement, we can directly calculate |ρ(w), v| ≤ s ψ − w Φ φ Φ and ρ(w) ∗ ≤ s ψ − w Φ

(3.6)

Next, take into account that φ = ψ − w, from inequality (3.5) we get, that α ψ − w 2Φ ≤ s(ψ − w, ψ − w) = ρ(w), ψ − w ≤ ρ(w) ∗ ψ − w Φ ∀w ∈ Φ, (3.7) that with the (3.6) gives us the bidirectional estimate of functional ρ(w) norm in Φ in a terms of standard norm of space Φ. At last, we replace norm · Φ with energy norm | · |Φ in our estimations (3.6) and (3.7) and get desired equality (3.4). For the purpose of mesh adaptation in FEM, the following corollary is important. Corollary 3.2. About error functional decomposition. Let τh = {K} be any regular triangulation of region Ω and hk := diam K, h := max hk , which we used to build approximation space Φh ⊂ Φ of FEM scheme. K∈τh

Then, error functional (3.1) allows the following decomposition 0 3 ρ(w), ϕ = {[ρf + divσ (w)] .v + [ρ∗ + divD (w)] .q} dx −

K∈τh K

0 3

[vi σij (w) nj + qk Dk (w) nk ] dγ

K∈τh ∂K

+

0

3

K∈τh ∂K∩Γσ

σvdγ +

0

3

K∈τh ∂K∩Γe

ρe qdγ ∀ w, ϕ ∈ V.

(3.8)

298


Proof. If we start from (3.5) and use integration by parts on each finite element, we get ρ(w), ϕ = s(ψ − w, ϕ) u p 0 3 cijkm εij (u − w ) εkm (v) + gkm Ek (p − w ) Em (q) = dx −ekij Ek (p − wp ) εij (v) + ekij Ek (q) εij (u − wu ) K∈τh K ⎫ ⎧ 3 ⎪ [σji,j (w − ψ)vi + Dk (w − ψ) qk ] dx+ ⎪ ⎬ 0 ⎨K 3 ∀w, ϕ ∈ V = [vi σij (ψ − w)nj + qk Dk (ψ − w)nk ] dγ ⎪ K∈τh ⎪ ⎭ ⎩

(3.9)

∂K

Now take into account and apply equilibrium equation and electric field equation of boundary piezoelectricity problem and continuity of function vi σij (u)nj and qk Dk (w)nk in a case of transition thought the combined border of the adjacent finite elements. As a result of appropriative simplification we get declared decomposition (3.8) of error functional. Let’s characterize connection between functional (3.1) and Galerkin’s discretization error. Corollary 3.3. About Galerkin’s discretization errors functional structure. Let conditions of proposition (3.1) and corollary (3.2) be fulfilled and ψh ∈ Φh – Galerkin’s approximation, which was obtained as a solution of problem (2.11). Then the following statements are correct ρ(ψh ), φ = 0 ∀φ ∈ Φh

(3.10)

ρ(ψh ), φ = s(e, φ) ∀φ ∈ E := Φ\Φh , E = E u × E p ,

(3.11)

|e|Φ = |ψ − ψh | = |ρ(ψh )|∗ ∀h > 0.

(3.12)

Proof. This corollary can be directly proved using calculation based on (3.5) with w = ψh . 3.2. Variational problem for Galerkin’s discretization error The results, provided above, characterize Galerkin’s discretization error e = ψ−ψh from the position of functional ρ(ψh ), defined in (3.1). In particular, now we are able to formulate variational problem about finding Galerkin’s discretization error using (3.11): ⎧ Galerkin’s approximation ψh ∈ Φh for solution ⎪ ⎪ ⎨ ψ ∈ Φ of the problem (2.11) Φh ⊂ Φ is given; (3.13) find the error e := ψ − ψh = (eu , ep ) ∈ E := Φ\Φh such that ⎪ ⎪ ⎩ s(e, ϕ) = ρ(ψh ), ϕ ∀ϕ ∈ E. It is as hard to find the solution of the problem (3.13) as to find the solution of the start piezoelectricity problem (2.11). Therefore, we should perform Galerkin’s discretization on it in some finite-dimensional subspace of errors space E. In order to simplify usage of that procedure in a future, we link up it realization to the


299

triangulation τh = {K}, which was used to find the solution of the piezoelectricity discrete problem (2.22), and formulate the following problem: ⎧ Eh ⊂ E, Eh = Ehu × Ehp , ⎪ ⎪ ⎨ dim Ehu = M u < ∞, dim Ehp = M p < ∞ is given; (3.14) u p ⎪ ⎪ find the eh = (eh , eh ) ∈ Eh such that ⎩ s(eh , ϕ) = ρ(ψh ), ϕ ∀ϕ ∈ Eh . Now, returning to an estimation (3.2), we are capable of defining an important result. Theorem 3.4. About correctness of discretization error problem. Let variational problem of piezoelectricity (2.11) have the single solution ψ ∈ Φ and its Galerkin’s approximation is ψh ∈ Φh , which was found using (2.22) with any h > 0. Then, the variational problem (3.14) for finding approximation eh of discretization error e = ψ − ψh ∈ E = Φ\Φh is well posed and |eh |Φ ≤ |ρ(ψh )|∗ ,

(3.15)

|eh |2Φ = |ρ(ψh )|2∗ − |e − eh |2Φ ∀h > 0.

(3.16)

So, energy norm of Galerkin’s discretization error approximation does not exceed the norm of the error source functional, in addition, as (3.16) testifies, we can estimate difference between that norms, if we are able to calculate norm |e − eh|V . For this purpose, we can use interpolation estimates, which is typical for a priori FEM errors. Furthermore, we propose the procedure to calculate energy norms 2 |eh |K := [cijkm εij (euh ) εkm (euh ) + gkm Ek (eph ) Em (eph ) ] dx∀K ∈ τh (3.17) K

which use bubble-functions to build spaces Eh . The orthogonality, which is inherent to that functions, can reduce computation efforts to find (3.17) on each finite element K, indeed, it needs to solve the d + 1-order system of algebraic equations on each of them. 3.3. Bubble-approximation of FEM discretization error On the set triangulation τh = {K} let us examine piecewise-defined function b ∈ H01 (Ω) of the following sort: b|K := bK ∈ H01 (K), (3.18) b(xK ) = 1 ∀K ∈ τh ∀h > 0, where xK is a point, for example, mass center of the finite element K. Now, we select the system of functions {bK }K∈τh as a basis of space Ehu and series of function systems {{bK }K∈τh }di=1 as a basis of space Ehp in the problem (3.14). Let us define error approximation in spaces Ehu , Ehp , as euh (x) =

300


d 0 0 i=1 K∈τh

p ΛK u {bK (x)}i and eh (x) =

0

ΛK p bK (x). When x = xK we get

K∈τh

p K euh (xK ) = ΛK u and eh (xK ) = λp

(3.19)

This fact makes it possible to formulate the sequence of problems to find a posteriori error estimators ⎧ it is set the ψh = (uh , ph ) ∈ Φh ⎪ ⎪ ⎪ ⎪ ⎪ d K ⎪ to find the ΛK ⎪ u ∈ R , λp ∈ R such that ⎪ ⎪ d ⎪ ⎨ 0 [c (euh , {bK }i ) − e (eph , {bK }i )] (3.20) i=1 ⎪ ⎪ d 0 ⎪ ⎪ ⎪ = [ls , {bK }i − c (uh , {bK }i ) + e (ph , {bK }i )] ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩ g (ep , b ) + e (b , eu ) = l , b − g (p , b ) − e (b , u ) . K h e K h K K h h K The system (3.20) can be represented C −E T E G

in the following matrix notation K Λu L K = λp r

where C = c (bK , bK ) S, E T = e (bK , bK ) I T , G = g (bK , bK ) , E = e (bK , bK ) I, L = l, bK I T − c (uh , bK ) + e (ph , bK ) I T , r = r, bK − g (ph , bK ) − e (bK , uh ) I S-unity matrix, I-unity vector. Thus, the usage of bubble-functions enables us to decompose the problem of K finding the vectors system {ΛK u }K∈τh and scalar values {λp }K∈τh into the independent sequence of solving of the linear algebraic equations d + 1-order systems on each finite element of triangulation τh . Theorem 3.5. About discretization of the posteriori error estimators. Let ψh ∈ Φh is Galerkin’s approximation of the piezoelectricity problem (2.11) solution ψ ∈ Φ, which was found on triangulation τh = {K}. Let {bK }K∈τh be the sequence of functions with properties (3.18). Then, the approximation eh of the error e = ψ − ψh which is uniquely defined by linear combination (3.19), which was received as a result of solving linear algebraic equations system (3.20), so in that case the values ηK (ψh ) := |eh |V %3 &1 2 ≡ [cijkm εij (euh ) εkm (euh ) + gkm Ek (eph ) Em (eph )] dx ∀K ∈ τh

(3.21)

K

and η(ψh ) :=

:1 2 ηK (ψh )

2

∀h > 0

(3.22)

K∈τh

are the local a posteriori error estimator of approximate solution on each finite element and on whole triangulation correspondingly.


301

4. Numerical experiments The developed numerical schemes was applied to the sequence of the model onedimensional piezoelectricity problems. PZT-4 material was used as a piezoelectric. It has the following physic properties: density ρ = 7500 kg/m3 , elastic coefficient c = 13, 9·108 H/m2 , piezoelectric coefficient e = 15, 1 C/m, dielectric susceptibility coefficient g = 730 F/m. In order to calculate norms of the found numerical results we use the following rules (2.19). Also these norms define the value of the potential energy of piezoelec-

tric Π = 2−1 u V + p Q . Let us denote, that the order of the convergence of the magnitude w in space H can be calculated as 2

2

pH (w) = log2 ( w0 H − w1 H ) − log2 ( w1 H − w2 H ) . The values of the relative error of magnitude H can be calculated as εH = −1 eH S H S . In examples, we used bubble-functions of the second order for the linear approximation and the fourth-order bubble-functions for the quadratic approximation. The advantage of using bubble-functions is the ability to find AEE on each finite element independently from the other elements of triangulation. This has positive effect on available computational resources usage. In the next figures you can see appearance of the above-described bubble-functions.

xJ+1

xJ

Figure 1. The second-order bubble-function

xJ

xJ+1

Figure 2. The fourth-order bubble-function

Let us determinate formulas used to calculate the scalar values of the posteriori error estimators in case of the one-dimensional piezoelectricity problems. Let m m 0 0 uh = uih vi and ph = pih qi be the basis combination in the approximation i=1

i=1

spaces Vh and Qh of the deformation and potential correspondingly. Let’s define error approximation in spaces Vk , Qk , as euh = e¯uh bK and eph = e¯ph bK and use them in (3.20). After some transformations we get the following system of two equations which we use to find numerical values of the e¯uh , e¯ph u e¯h c (bK , bK ) − e¯ph e (bK , bK ) = l, bK − c (uh , bK ) + e (ph , bK ) (4.1) e¯ph g (bK , bK ) + e¯uh e (bK , bK ) = r, bK − g (ph , bK ) − e (bK , uh ) Taking into account the properties of the bilinear form system (4.1), the determinant D = c (bK , bK ) g (bK , bK ) + e (bK , bK ) e (bK , bK ) of the system is greater then zero and that means, that the system (4.1) has single solution. As a result of

302


system solving we get a posteriori error estimators of the found FEM approximation of the deformation and potential e¯uh =

l, bK − c (uh , bK ) + e (ph , bK ) + e¯ph · e (bK , bK ) c (bK , bK )

e¯ph = {c (bK , bK ) g (bK , bK ) + e (bK , bK ) e (bK , bK )}−1 ·{(r, bK − g (ph , bK ) − e (bK , uh )) c (bK , bK ) − (l, bK − c (uh , bK ) + e (ph , bK )) e (dK , bK )}. Let’s examine one meter piezoelectric shank, which is fixed and grounded on the left and right sides, that is Ω = [0, 1], u(0) = u(1) = 0, p(0) = p(1) = 0. Inside of the shank the density of the electric charge is ρ∗ = 100 C/m3 and the inside force is f = 100 H. On the following figures you can see graphics of the numeric results for deformation and potential which we received using linear approximation on 8 finite elements. Also you can see appropriative values of the calculated errors, which were found using second-order bubble-functions.

Figure 3. The value of deformation (1) and its error (2) In Table 1 the norms of the found estimators in a case, when we use linear and quadratic approximation are presented. Also, indexes of the convergency order p are shown here. Table 1. Error norms and indexes of convergency order 8 16 32 p eu V (uh ∈ P1 (K)) 0,72589 0,36294 0,18147 1 p e Q (ph ∈ P1 (K)) 0,13356 0,06678 0,03339 1 eu V (uh ∈ P2 (K)) 3,71951e-6 1,85975e-6 9,29878e-7 1 ep Q (ph ∈ P2 (K)) 6,84396e-7 3,42198e-7 1,71099e-7 1


303

Figure 4. The value of potential (1) and its error (2) In case of linear approximation, received numerical results are conformed with theoretical ones. The indexes of convergency order are less then we have been waiting for in case of the quadratic approximation. The explanation of that is a quadratic character of the found result. That is that, the usage of the quadratic approximation automatically enables sufficiently accurate determination of unknown coefficients of the solution combination. Therefore, the value of the estimator norm actually is an estimate of the calculation inaccuracy on the set triangulation. Table 2 gives us possibility to analyze dynamics of the relative error reducing in case when the triangulation twice as dense. Table 2. Relative error values 8 16 εu (uh ∈ P1 (K)) 12,6% 6,26% εp (ph ∈ P1 (K)) 12,5% 6,25% εu (uh ∈ P2 (K)) 6, 5 · 10−5 % 3, 1 · 10−5 % εp (ph ∈ P2 (K)) 6, 4 · 10−5 % 3, 2 · 10−5 %

32 3,12% 3,12% 1, 6 · 10−5 % 1, 6 · 10−5 %

The values of relative errors indicate, that the usage of the quadratic approximation makes possible to get better results then the linear one on triangulation with equal number of nodes.

5. Conclusions The paper deals with FEM application to the piezoelectricity stationary problems solving. Variational formulation, Galerkin’s discretization and building of numerical schemes to find the solution was developed for these problems. Error problem was formulated. The bubble-functions’ properties and the error problem were used to construct the posteriori error estimators. Received scheme was realized as a soft-

304


ware. The efficiency and reliability of proposed scheme was verified by the series of numerical experiments which use developed software. The received AEE can be used to analyze built numerical scheme, to build h-adaptive and postprocessing refinement FEM scheme. Acknowledgment The paper was developed due to the grant of Ukrainian education science and ministry.

References [1] O. Dan’ko, H. Shinkarenko, The numerical researching of 1D problems of piezoelectricity. Visnyk Lvivskogo universytetu. Seriya mekh.-mat. 46 (1997) [2] H. Kvasnytsya, H. Shinkarenko, The comparison of simple aposteori error estimators of FEM for elastoplasticity problems. Visnyk L’vivs’kogo universytetu. Seriya prikl. mat. ta informat.7 (2003), 162–174. [3] V. Nowacki, Electromagnetic effects in the solids. M., (1986). [4] Yu. Tokar, The numerical research of the forced acoustical oscillations of piezoconvectors. Thesis for Ph. Degree in phys. math. science, L’viv, 1988. 120p. [5] F. Chaban, H. Shinkarenko, The postprocessing accurate determination of half-linear of approximation MSE for the admixture migration problems. Visnyk L’vivs’kogo universytetu. Seriya prikl. mat. ta informat. 13 (2007), 164–176. [6] H. Shinkarenko, The net-projection approximation for the variation problems of piezoelectricity. 1. The problem formulation and analysis of steady forced vibrations. Differentsyal’nye uravneniya 29 (1993), 7, 1252–1260. [7] H. Shinkarenko, The net-projection methods of solving of the initial-boundary problems. Kyiv: NMK VO, 1991. – 88p. [8] H. Shinkarenko, Yu. Kozarevs’ka, The regularization of the numerical solutions of variant migration problems admixture: h-adopted MSE. Part 1.Visnyk L’vivs’kogo universytetu. Seriya prikl. mat. ta informat.5 (2001), 153–164. [9] F. Chaban, H. Shynkarenko, Finite element method approximations for the boundary valued problems of piezoelectricity. Modern Analysis and Application. Book of abstracts. Odessa (2007), p. 32–33. [10] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method. Vol. 1. The Basis. 5th Edition. Oxford: Butterworth-Heinemann, 2000. – 68 p. Fedir Chaban Ivan Franko National University of Lviv Universytetska street 1, 79000 L’viv, Ukraine e-mail: [email protected] Heorgiy Shynkarenko Opole University of Technology Mikolaychuka Stanislava street 5, 45-271 Opole, Poland e-mail: [email protected]


Remarks about Observables for the Quantum Mechanical Harmonic Oscillator Heinz Otto Cordes To the memory of M. Krein

Abstract. A comparison algebra A for the self-adjoint differential operator 2 d2 was introduced by H.Sohrab [So1] [This is a C ∗ -algebra of H = − dx 2 + x singular integral operators with symbol in the sense of our book [Co1]. The symbol space is a (topological) circle. Operators in the algebra are Fredholm if and only if their symbol does not vanish.] Here we ask the question whether A or a convenient subalgebra HS ⊂ A (or the algebra of operators within reach of HS) might be suitable to serve as algebra of precisely predictable observables for the quantum mechanical harmonic oscillator. We identify such algebra HS, with symbol space also a circle, and show that (i) creation and annihilation certainly are within reach of HS and (ii) HS is invariant under conjugation with the propagator e−iHt while (iii) this conjugation induces rotation as associate dual map of the symbol space. The “operators within reach” form an algebra HS ∞ as well, with many properties of the algebra P introduced for Dirac’s equation in [C2]. Introducing it as an algebra of precisely predictable observables, we find that here location and momentum are precisely predictable. Comparing this with results for Dirac’s equation discussed in Ch. 5 of [Co2], and with the attempt for the harmonic oscillator in Ch. 8 there, we find a remarkably different behaviour, for each case. Mathematics Subject Classification (2000). 35Q40, 35S99, 81Q99. Keywords. Harmonic oscillator, precisely predictable observable.

1. Generals about the harmonic oscillator We look at the ordinary differential operator H=−

d2 + x2 = D2 + x2 , dx2

−∞ < x < ∞

(1.1)

306

H.O. Cordes

to be considered as an unbounded self-adjoint operator of the Hilbert space H = L2 (R). The spectral theory of this operator is well known: there is discrete spectrum only; eigenvalues are λj = 2j + 1, j = 0, 1, . . ., all of them simple. Corresponding (normalized) eigenfunctions ϕj are given by the Hermite functions – in detail, −j/2 1 2 1 2 (1.2) ψj (x) = γj e− 2 x Hj (x), γj = π − 4 √ , j! with the Hermite polynomials Hj (x). The operator H and its spectral theory play its special role as the Schr¨ odinger operator of the quantum mechanical harmonic oscillator. However, for physical purposes it might be better to redefine H by setting 1 d2 1 d d 1 2 H= − 2 +x − = x− x+ . (1.1 ) 2 dx 2 2 dx dx This new operator H – i.e., the operator 12 (H − 1) with H of (1.1) – clearly has the same spectral theory, except eigenvalues now are λj = 0, 1, . . . while eigenfunctions still are given by (1.2). Also, H of (1.1 ) is obtained from the classical total energy d , respectively [and E = 12 (p2 + q 2 ) by replacing q and p with x and Dx = 1i dx 1 subtracting a “rest energy” 2 ], following the “standard quantization rule”. d In addition yet, setting ∂x = dx , it turns out that the two operators 1 A = √ (x + ∂x ), 2

1 A∗ = √ (x − ∂x ) 2

(1.3)

satisfy the commutator condition [A, A∗ ] = AA∗ − A∗ A = 1

(1.4)

while (1.1 ) amounts to H = A∗ A. From (1.4) and (1.5) we get [H, A] = −A, H(Aψj ) = (j − 1)(Aψj ),

(1.5) ∗

∗

[H, A ] = A which has the effect that H(A∗ ψj ) = (j + 1)(A∗ ψj ).

(1.6)

Or, in words, application of A to an “eigenstate” ψj converts ψj into an eigenstate of the next lower integer j − 1 (or to 0, if j = 0). On the other hand, application of A∗ converts ψj into a multiple of ψj+1 – a state belonging to the eigenvalue j + 1. This is the reason for calling A and A∗ the annihilation operator and creation operator, respectively. Namely, in the physical interpretation, the unit vector ψj represents a physical state where j “energy quanta” exist. Accordingly, application of A destroys one energy quantum [of the j quanta present] while application of A∗ creates one more quantum [so that a state is reached where j + 1 quanta are present]. The time-independent Schr¨ odinger equation is just the eigenvalue equation Hψ = Eψ,

(1.7)

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So, for a normalized eigenfunction ψj the energy is given as E = j. For an arbitrary unit vector ψ ∈ H one must expand ∞ ψ(x) = ej ψj (x) with ej = ψj , ψ. (1.8) 0

In that state, a measurement of energy will result in the value E = j with probability |ej |2 = |ψj , ψ|2 so that the expectation value of the measurement will be ∞ ˘ = ψ, Hψ = E j|ψj , ψ|2 . (1.9) 0

The time-dependent Schr¨ odinger equation is of the form ∂ψ + iHψ = 0. (1.10) ∂t The initial value problem of this partial differential equation [of order 2] – i.e., finding a solution ψ(t, x) assuming a given initial value ψ(0, t) = ψ 0 ∈ H – is solved by ψ = e−iHt ψ0 with the “evolution operator” or “propagator” U (t) = e−iHt representing a (strongly continuous) group of unitary operators on the Hilbert space H. This propagator U (t) is explicitly known as integral operator U (t)ψ = u(x, y; t)ψ(y)dy, (1.11) R

with kernel1

2 2 i 1 u(x, y; t) = √ (1.12) e 2 sin t {(x +y ) cos t−2xy} . 2πi sin t Existence of this integral for general ψ ∈ H – and even the meaningfulness of the kernel (1.12) for t = jπ with integers j might require some discussions we will not pursue right now. The law of conservation of energy then requires that all physical states [i.e., all unit vectors ψ ∈ H] propagate in time along the Schr¨ odinger equation (1.10), assuming that observables are kept constant. In other words, the state ψ 0 at time t = 0 will become the state ψ(t) = e−iHt ψ 0 at time t. Or, equivalently, if states are kept constant, then all observables – i.e., all self-adjoint operators T – will propagate by the rule T → eiHt T e−iHt .

2. Remarks about observables We already noted some facts about the (total) energy observable represented by the self-adjoint operator H. Other standard observables, in this case, would be location – represented by the (unbounded self-adjoint) multiplication operator M of multiplication by x – i.e., the operator ψ(x) → xψ(x), and momentum represented 0 −iλ t j ψ ψ using a formula found (1.12) follows from the standard Ansatz U (t) = e j j in [MOS] (last fla. on p. 252). A derivation without that fla. may be based on a direct calculation of the Feynman path integral for this special case, using “time slicing approximation”. 1 Formula

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H.O. Cordes

d by the first-order differential operator D = 1i dx – again unbounded self-adjoint. Both these operators have simple continuous spectrum (even absolutely continuous) extending over the entire real line. The location operator is “diagonal” in the present “representation” [often called the “configuration representation”]. Since we have F¯ DF = M = multiplication by x a conjugation by the Fourier transform [of all above operators] would bring us into a representation where the momentum is diagonal – called the momentum representation. An observation of M or D in a physical state ψ would yield the expectation value ψ, M ψ resp. ψ, Dψ. The observed value would lie in an interval Δ = [λ , λ ] with probability ψ, EΔ ψ, where EΔ denotes the spectral projection (of M or D, resp.) belonging to the interval Δ. This is according to a theory developed by J. v. Neumann [Ne1], for the general Schr¨ odinger equation with general potential, and in general dimension ≥ 1. The leading example, at that time, might have been the hydrogen atom where then we have H = − 12 Δ + γ/|x| with the 3-dimensional Laplace operator Δ and the “fine structure constant” γ ≈ 1/137. The Schr¨ odinger equation is intended for nonrelativistic applications. As such it must be regarded an approximation. For the hydrogen atom it was replaced by the relativistically covariant Dirac equation. For Dirac’s theory the above scheme of prediction of observables of v. Neumann is known to lead into some physical paradoxes we have tried to cure in [Co2] by introducing some algebras of global pseudodifferential operators containing the “precisely predictable observables”. Location and momentum above are not precisely predictable, but they are “approximately predictable” insofar as they differ in operator norm from some precisely predictable observable only by a small error quantity – for location the error seems of the order of the Compton wave length. Crudely speaking our algebra of precisely predictable observables for Dirac’s theory consist of pseudodifferential operators T which remain pseudodifferential operators under the conjugation T → eiHt T e−iHt , – i.e., the Heisenberg representation – where now H denotes the Dirac operator. Returning to our Schr¨ odinger equation of the harmonic oscillator, note this must be regarded as the wave equation of an entirely different particle – namely a photon, said to be “of spin 0”, assuming that it can be called a particle at all [rather one might just look at it as a quantization of energy]. In that respect then we must regard the Schr¨ odinger equation of the harmonic oscillator as an exact wave equation. The question of relativistic covariance comes up only in connection with quantum field theory, looking at (3-dimensional) electromagnetic waves, where light quanta of different frequencies come into play, but where also an infinite tensor product (infinite with the cardinality of the continuum) gets involved. Here we plan to stay restricted to the 1-dimensional oscillator (of frequency 1) but there will ask the question whether there is a similar need [or only possibility] of introducing an algebra of precisely predictable observables.

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A first attempt, in this respect, was made in [Co2], Ch. 8, where we investigated an algebra of 1-dimensional ψdo-s similar to the one used for Dirac. In a sense that√algebra was generated by only 2 operators – namely the multiplication s(x) = x/ 1 + x2 and the singular convolution S = F¯ s(x)F . However, an investigation shows that such algebra is not invariant under the Heisenberg transform T → eiHt T e−iHt . In a sense that algebra “rotates” in an abstract sense, with every operator turning into its negative at t = π and to itself at t = 2π, etc. Such operators might be precisely predictable only at periodic times. On the other hand, we were studying different algebras of global singular integral operators in [Co1] (appeared in 1987). The point then was to investigate the Fredholm property of operators in a (so-called) comparison algebra generated by some (global) second-order elliptic differential operator. In that connection the differential operator H of (1.1) was investigated (even for the n-dimensional case) by H. Sohrab [So1]. We thus propose now to investigate the (bounded and unbounded self-adjoint operators) “within reach” of Sohrab’s algebra, in the 1dimensional case, trying to find out the suitability of such global ψdo-s as an algebra of special importance for theory of observables of our present harmonic oscillator.

3. Sohrab’s algebras The operator H of (1.1 ) has all eigenvalues λj = 0, 1, 2, . . . nonnegative2 , hence H + 1 is positive definite ≥ 1, hence it is invertible, and it has a unique positive square root 1 1 ∞ − 12 Λ = (H + 1) = (H + 1 + λ)−1 λ− 2 dλ. (3.1) π 0 Furthermore H is a second-order elliptic differential operator on R. A “comparison algebra” for such an operator (in the sense of [Co1]), is generated as a C ∗ subalgebra of L(H) by 2 kinds of operators: (i) some multiplications a(M):ψ(x) → a(x)ψ(x) by a (smooth) bounded continuous function a(x) over R. (ii) some operators of the form DΛ with Λ of (3.1) and a (≤) first-order linear differential operator D (with smooth coefficients). Sohrab – in [So1], here restricted to one dimension – in effect uses one multiplicax tion operator s(M ) with s(x) = √1+x , and two operators of the second form (ii), 2 namely, 1 d . (3.2) xΛ and DΛ , with D = i dx generating a comparison algebra he calls A. He also discusses another (smaller) such algebra, using only the operators (3.2), without any multiplications. Both these algebras have compact commutators and contain the entire ideal K(H) of operator H has a unique self-adjoint realization in L2 (R) – i.e., the minimal operator is essentially self-adjoint.

2 This

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H.O. Cordes

compact operators H → H, so that the quotients A/K(H) and A# /K(H) are function algebras, by the Gelfand-Naimark theorem. In effect Sohrab proves Theorem 3.1. (H. Sohrab) The symbol space M# of A# is a (topological) circle. It is best realized by looking at the “directional compactification” B2 of the (x, ξ)space R2 , obtained by adding one point ∞(x0 , ξ 0 ) at the end of each ray (ρx0 , ρξ 0 ) as ρ → ∞, where (x0 , ξ 0 ) is any unit vector. The circle M# then is given as the boundary 2 2 M# = ∂B2 = {∞(x0 , ξ 0 ) : x0 + ξ 0 = 1}. (3.3) # With this construction of M the two generators (3.2) have the symbols x ξ 1 1 σxΛ = √

|M# , σDΛ = √

|M# , (3.4) 2 2 2 1+x +ξ 2 1 + x2 + ξ 2 to be interpreted as taking the continuous extension to B2 of the function over R2 given, and restricting that to ∂B2 = M# . The symbol space of A also is a (topological) circle. It is best described by looking at Fig. 1, below: we take the two straight lines {−∞ ≤ x ≤ ∞} × {ξ = ∞} ' Ms ⊂ M &

Mp ⊂ M ξ 6 x -

Mp ⊂ M

$ Ms ⊂ M %

Figure 1. The symbol space M of the algebra A is an oval at |x|+|ξ| = ∞, consisting of the principal symbol space Mp (at |ξ| = ∞) and the secondary symbol space Ms (at |x| = ∞). The space M is defined as the maximal ideal space of the (commutative) quotient algebra A/K(H) An operator A ∈ A is Fredholm if and only if its symbol does not vanish on M. and {−∞ ≤ x ≤ ∞} × {ξ = −∞} and extend at the ends (∞, ±∞) by attaching a right semi-circle {∞(x2 + ξ 2 ) : x2 + y 2 = 1, x > 0}, and, similarly at the other two ends (−∞, ±∞), by attaching the other half-circle {∞(x2 + ξ 2 ) : x2 + y 2 = 1,x < 0}. The symbol of the generator s(x) then is given as the function s(x) on the two lines [−∞, ∞] × ±∞ extended constant on the two semi-circles. The symbol of the generators (3.2) is given by (3.4) on the two semi-circles, extended constant on the two straight lines. We remind of the use for the above: Corollary 3.2. An operator T ∈ A (or ∈ A# ) is Fredholm if and only if its symbol never vanishes. There also is an “index theorem” supplying a formula for the Fredholm index in terms of homotopy invariants of the symbol. For this we refer to [So1].

Harmonic Oscillator

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As a side remark: we, of course, also may adjoin the operator S = s(D) = ˘ still with compact comF¯ s(x)F to the algebra A obtaining then a C ∗ -algebra A, mutators, and all above things still apply – all generators are ψdo-s in our algebra Opψc0 of [Co3] (cf. our remarks, below, regarding xΛ and DΛ of (3.2). In fact, A˘ will contain the C ∗ -algebra of [Co1], Sec. 8.1 (called A there, but to avoid confusion let it be called B here). It might be interesting then to look at the symbol ˘ as indicated in Fig. 2, below. spaces of the 4 algebras A# , B, A, A, ' $ '$ ξ ξ 6 6 x x A# A &%

&

%

'

$

ξ 6 x -

ξ 6 x -

B

A˘ &

%

Figure 2. Symbol spaces of the four C ∗ -algebras A# , A, B, A˘ as boundaries of different compactifications of R2 . All these spaces are boundaries of certain compactifications of R2 . For A# we add one point in each direction ρx0 , ρ → ∞, with a unit vector |x0 | = 1. For B two points (x, ±∞) are added for each x ∈ R, and also two points (±∞, ξ) for ξ ∈ R, but all “directions” of A# generate only the 4 corners of the square generated by the lines (x, ±∞) , (±∞, ξ), according to the 4 quadrants ±x, ±ξ > 0. On the other hand, for A and A˘ both types of “point generation” are used. For A˘ the directions of the 4 quadrants generate their individual point each, rounding the corners of the square. For A, on the other hand, the generator S is missing, hence the vertical lines (±∞, ξ) collapse into a point each, giving us the oval of Fig. 1.

4. On global pseudodifferential operators Let us note that the generators of A are global pseudodifferential operators over R (with strictly classical symbol) in our algebra Opψc0 of [Co2] or [Co3]. Indeed, we trivially have H = h(x, D) with 2h(x, ξ) = x2 + ξ 2 − 1 ∈ Opψc(2,2) . But H or H + 1 are not md-elliptic, as immediately seen. However, one finds that H + 1 is formally md-hypo-elliptic, in the sense of [Co3], p. 85. As such H + 1 possesses

312

H.O. Cordes

a K-parametrix with somewhat weaker properties, by II, Thm. 2.5 of [Co3], and the L2 -inverse (H + 1)−1 must differ from it by an operator of order −∞, so that (H + 1)−1 itself is a ψdo with the same properties3 . Checking about the order of this operator, one finds that (H + 1)−1 ∈ Opψc(2,0) ∩ Opψ(0,2) .

(4.1)

In a similar way, using the parametrix method (cf. also Sec. 8, below) one finds 1 that the positive square root (H + 1)− 2 is a global ψdo, and that 1

Λ = (H + 1)− 2 ∈ Opψc(1,0) ∩ Opψc(0,1) .

(4.2)

Relation (4.2) is just enough to verify that the two generators xΛ and DΛ belong to Opψc0 . The same, evidently, is true for the multiplier s(x). We thus find that the finitely generated algebra A0 ⊂ A is an algebra of (strictly classical) pseudodifferential operators of order 0 = (0, 0), in the sense of [Co2] or [Co3]. Such operators also have a symbol as pseudodifferential operators (here called ψdo-symbol), a smooth function of x, ξ defined over R2 . Comparing the ψdo-symbols and the algebra symbols of the generators we find that [within the algebra A0 ] the algebra symbol of A is obtained as limit ξ → ±∞ of the ψdo-symbol, for finite fixed x on the principal symbol space Mp , and as limit ρ → ∞ of ρ(x0 , ξ 0 ) on the point (x0 , ξ 0 ) of one of the two semi-circles making the secondary symbol space Ms . Taking closure within the Fréchet topology of Opψc0 – obviously stronger than the operator norm topology of A will get us an (adjoint invariant) larger subalgebra Aψ ⊃ A0 of 0-order strictly classical ψdo-s contained in A. We would like to make the point here that there exist two slightly different ways of introducing an algebra of “pseudodifferential operators” to this case: Either (i) we work within the class ΨH of all operators of the form Λ−s A where s ∈ R, and A ∈ Aψ . For this we need to confirm that Λ−s ∈ Opψc2se for s > 0, and Λ−s ∈ Opψcse for s < 0. Moreover, it is to be shown that ΨH is an algebra, and that we can execute some calculus of ψdo-s within ΨH. Or else, (ii) we consider the operators “within reach” of our C ∗ -algebra A (or one of the other algebras of our list of fig.2). For this it is natural also to introduce the L2 -Sobolev spaces (4.3) Hs = {u ∈ S : Λ−s u ∈ H} −s −s with their Hilbert space structure and inner product u, vs = Λ u, Λ v. We notice the following Lemma 4.1. For A ∈ A0 we have Λ−s AΛs − A ∈ K(Hm ) for all s, all m.

(4.4)

Evidently we only must show that the statement holds for the generators. To prove this lemma the ψdo-calculus is not sufficient (although one finds that 3 For

more details, also relating to the “parametrix method” mentioned below, cf. Sec. 8.

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powers Λs belong to Opψc). However the resolvent integral techniques described in [Co1], p. 163f will supply the necessary tools. Clearly Lemma 4.1 will imply that A0 also may be regarded as a ∗-invariant subalgebra of L(Hs ). Its commutators are compact there as well. We may take the C ∗ -algebra closure – the norm closure – to obtain algebras A ⊂ L(Hs ). All As have compact commutators and are algebras with symbol. The symbol space is the same, for all s, and an operator A ∈ As ∩ At has the same symbol in both algebras. We also may speak of the Fréchet algebra A∞ obtained by completing A0 under the topology induced by all Hs - norms – this is a Fréchet topology. Clearly, an operator A ∈ A∞ belongs to all algebras As . it has the same (algebra-)symbol in all these algebras. Definition 4.2. Let s ≥ 0. An operator A ∈ L(Hs , H) of the form A = Λ−s A0 with A ∈ A∞ is said to be within reach of the algebra A (and of order s). Proposition 4.3. The operators within reach of A form a graded algebra, called ΨA, (with subspaces ΨAm of order m) due to Lemma 4.1 above. We have explained above concepts, using the C ∗ -algebra A of Sec. 3. However, the principles may be applied to any of the other 3 algebras of Sec. 3 – i.e., A# , B, A˘ – as well. We shall deal with the algebras of the second kind (ii) here – i.e., with ΨA, etc., rather than with ΨH – because their environment was studied closely in [Co1], while pursuit of (i) would need more work. We are revisiting the above facts here, asking the question whether the algebras A, A# , ΨA, ΨA# are “Heisenberg invariant” – that is, are invariant under conjugation by the propagator e−iHt of the Schr¨ odinger equation (1.10). In Section 4, below, we shall find that the answer is positive for the “#-algebras”, but not for the algebras A, ΨA obtained by adjoining the multiplication s(x), or the ˘ remaining two B, A.

5. The holomorphic representation and the invariance of A# In [Co2] Ch. 8.5 we already discussed the fact that the multiplication operator s(x) does not remain within our algebra there when conjugated with the propagator e−iHt of the Schr¨ odinger equation (1.10), and it appears evident that this remains true when substituting our present algebra A, looking at the symbol space of Fig. 1. Thus we will not expect our algebra A to stay invariant under such conjugation. However, the algebra A# and the algebra ΨA# of its operators within reach will have this property. To see this it may be convenient to introduce the (so-called) holomorphic representation of this harmonic oscillator problem [to be found in Faddeev-Slavnow [FS], for example].

314

H.O. Cordes Consider the sequence of (entire) holomorphic functions 1 ϕj (z) = 2−j/2 √ z j , j!

j = 0, 1, . . . ,

(5.1)

and the space X of entire analytic functions spanned by this sequence. Also define the 3 operators d ˘ = z d = A˘∗ A. ˘ , A˘∗ = multiplication by z, H A˘ = dz dz As a trivial observation, we find that J

j ˘ Aϕj = ϕj−1 , A˘∗ ϕj = 2(j + 1)ϕj+1 , 2 while ˘ j = jϕj , j = 0, 1, . . . , Hϕ

(5.2)

(5.3)

(5.4)

while also ˘ A˘∗ ] = 1. [A,

(5.5) ˘ A˘∗ , H ˘ At a glance it seems that the system of vectors and operators ψ˘j , A, has exactly the same properties than the quantities without the breve, defined in Sec. 1. This is reinforced when we now introduce a Hilbert norm such that the ϕj are an orthonormal base while the two operators A˘ and A˘∗ are mutual adjoints. Such inner product is given by setting 2 2 1 1 f, g˘ = dxdy f¯(z)g(z)e− 2 (x +y ) dxdy. (5.6) 2π Introducing polar coordinates it is immediately clear that the vectors ϕj are mutually orthogonal. Their normalization follows due to ∞ 2 1 dxdy|z|2j e− 2 |z| = 2π e−s sj ds = 2πΓ(j + 1) = 2πj !, (5.7) 0

using polar coordinates and the integral substitution r2 = s, rdr = ds. With ˘ of all entire functions f (z) satisfying the product (5.7) goes the Hilbert space H ˘ f < ∞, easily seen to be complete. Looking at the matrix representations of ˘ A˘∗ with respect to the orthonormal base (5.1) it also follows that, indeed, A˘ A, ˘ is self-adjoint. and A˘∗ are adjoints of each other, so that H It is then confirmed at once that the expression (with ψj of (1.2)) ˘ = U ϕj ψj (5.8) j

˘ : H → H ˘ such that U ˘ ∗H ˘U ˘ = H, U ˘ ∗ A˘U ˘ = A, defines a unitary operator U ∗ ˘∗ ˘ ∗ ˘ U A U =A . ˘ form just another representation of our In other words, the operators X problem of Sec. 1, called the holomorphic representation, while we were using the “configuration representation”, so far.

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Evidently then the time-dependent Schr¨ odinger equation (1.10) now has become a first-order PDE, namely, ∂ψ ∂ψ + iz = 0. (5.9) ∂t ∂z [We are dropping ˘. from now on. In fact, the initial value problem of eq. (5.9) can be easily solved explicitly]: Proposition 5.1. The unique solution ψ(t, z) of eq. (5.9) assuming the initial values ψ(0, z) = ψ 0 (z) is given by the fla. ψ(t, z) = ψ 0 (ze−it ).

(5.10)

To prove this proposition we may just substitute ψ(t, z) of fla. (5.10) into the transformed Schr¨ odinger equation (5.9) [uniqueness is a matter of standard theorems]. The theorem below then becomes trivial. Theorem 5.2. The algebras A# = HS and ΨA# = HS ∞ remain invariant under conjugation by the propagator of eq. (1.10) In particular, setting Tt = eiHt T e−iHt , for general operators T , we have Λt = Λ and (xΛ + iDΛ)t = e−it (xΛ + iDΛ) , (xΛ − iDΛ)t = eit (xΛ − iDΛ) .

(5.11)

Moreover, the creation and annihilation operator as well as location x and momentum D, all are within reach of A# = HS. They are of order 1, of course, and we get A∗t = eit A∗ , At = e−it A , (x)t = x cos t + D sin t , (D)t = −x sin t + D cos t . (5.12) In particular then, the associate dual map M# → M# of the automorphism HS → HS is just given as the rotation (x + iξ) → e−it (x + iξ).

6. Precisely predictable observables In [Co2] we discussed some algebras of pseudodifferential operators invariant under conjugation with the Dirac propagator. The fact was discussed that the unbounded self-adjoint operators of such an algebra have a well-defined Heisenberg transform A → At → eiHt Ae−iHt . Their ψdo-symbol propagates along the classical particle orbits, as |x|+|ξ| is large. Even the electron spin propagates as a magnetic moment in the existing electromagnetic field. We were introducing the concept “precisely predictable” for such operators, in view of the fact that the expectation value of such an observable should behave exactly as prescribed by J. v. Neumann’s rules. Most dynamical observables were not precisely predictable, however. In particular, location and momentum were only approximately predictable, in the sense that they were “close” to a precisely predictable observable – with a “built-in uncertainty”.

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In view of our results of Sec. 5, above, we now are tempted to generalize this concept to the harmonic oscillator (as the only Schr¨ odinger-type Hamiltonian, so far). In Ch. 8 of [Co2] we examined the above algebra B, but found it not “Heisenberg invariant”. However, in view of Thm. 5.2, we now may focus on the algebra ΨA# , and define its self-adjoint operators as precisely predictable observables. We then see that total energy H as well as location x and momentum d Dx = 1i dx belong to ΨA# . So – in the above language – they are precisely predictable. Or, in other words, displacement as well as momentum of the oscillation are precisely predictable. The field of classical orbits for the harmonic oscillator consists of the concentric circles around the origin, in the x, ξ-plane. We did not prove existence of a ψdo-symbol for operators in ΨA# . So we can study symbol propagation under the particle flow on special examples only – i.e., operators in ΨA# also known as ψdos. However, Thm. 5.2 also implies that the algebra symbol space M# at ∞(x, ξ) certainly experiences the classical rotation under the Heisenberg conjugation, just as indicated by the particle flow in ΨA# .

7. Linear perturbations of the harmonic oscillator Our algebras of Sec. 4 are also useful if we perturb the operator H of (1.1 ) by an expression linear in x and D, i.e., deal with a Hamiltonian 1 H = (D2 + ω 2 x2 − ω) + c1 D + c2 x, (7.1) 2 where c1 , c2 are real constants – possibly depending on t, for a time-dependent Hamiltonian. That is, using our holomorphic representation, we consider d d + γ¯ + γz, (7.1 ) dz dz with an arbitrary fixed (complex) constant γ, again allowed to (possibly) depend on t but not on z. In the latter case, the Hamiltonian H = H(t) will be timedependent. Then we may not speak of the propagator e−iHt but still will have a propagator U (t) = U (0, t) – or, more generally U (τ, t) – solving the initial-value problem for t = 0 (or t = τ , resp.). The corresponding Schrödinger equation H = ωz

ψ|t + i(ωzψ|z + γ¯ ψ|z + γzψ) = 0.

(7.2)

still may be solved explicitly, as a first-order scalar PDE. For the propagator U (τ, t) we get the formula (U (t0 , t)ψ)(z) = e

−i{z

3t t0

dτ γ(τ )eiω(τ −t) +i

ψ(e−iω(t−t0 ) z − ieiωt0

t

3

t t0

dτ

3

t τ

dκeiω(τ −κ) γ(τ )¯ γ (κ)}

dτ e−iωτ γ¯ (τ )).

× (7.3)

t0

It may be interesting to observe that the Heisenberg S-matrix comparing the perturbed Hamiltonian H(t) with H of (1.1 ) (or (5.2)), called H0 for a moment,

Harmonic Oscillator

317

also allows an explicit form. Guided by [FS], we get Sψ(z) = e

−i{z

3

∞ −∞

γ(τ )eiωτ +i

3

∞ −∞

dτ

3

∞ τ

dκeiω(τ −κ) γ(τ )¯ γ (κ)}

ψ(z − i

∞

dτ e−iωτ γ¯ (τ )).

−∞

(7.4) Indeed, this will define a meaningful operator if we only assume that γ(t) ∈ L1 (R), because then the two constants t t ∞ iω(τ −κ) b= dτ dκe γ(τ )¯ γ (κ), c = γ(τ )eiωτ (7.5) −t

−∞

τ

are well defined. Note that we may write ∗

S = eib e−icA e−i¯cA ,

(7.6)

with above constants b, c. For us it may be more important to observe that also our propagator U (t) = U (0, t) may be written as (U (t)ψ 0 )(t, z) = eib(t) ec(t)z ψ 0 (e−iωt z − c¯(t)),

(7.7)

with certain b(t), c(t) independent of z. This will allow us to calculate the Heisenberg transforms of creation and annihilation again: we get e−c(t)z A∗ ec(t)z = A∗ ,

e−c(t)z Aec(t)z = A + c(t),

(7.8a)

and, using the substitution operator S : u(z) → u(e−iωt z − c¯(t)), for a moment, S −1 A∗ S = eiωt A∗ + eiωt c¯(t),

S −1 AS = e−iωt A.

(7.8b)

U ∗ (t)AU (t) = e−iωt A + e−iωt c(t).

(7.9)

Combining this we get U ∗ (t)A∗ U (t) = eiωt A∗ + eiωt c¯(t),

Finally, it may be observed (without proof, at this time) that the “perturbed 1 1 generators” x(1 + H(t))− 2 , D(1 + H(t))− 2 generate the same algebras A# = HS # and HS ∞ = ΨA – simply because the difference H − H0 is of first order, hence compensated by the operator Λ. For constant γ(t) – i.e., constant H(t) – the operator (H(t) + 1)s again commutes with the propagator U (t) = e−iHt , so that it remains unchanged under conjugation. As a consequence we find that Thm. 4.2 essentially remains intact and the linear perturbation [for variable γ(t) a slightly different argument may lead to the same result]. Theorem 7.1. The algebras HS and HS ∞ both remain invariant under conjugation by the propagator U (t). While formulas (4.11) and (4.12) do not remain precisely intact, they still remain intact modulo (relatively compact) lower-order additional terms, and we still have the last sentence of Thm. 4.2 correct: The automorphism of the symbol space M# generated by the conjugation T → U ∗ (t)T U (t) is just given as the rotation z → eiωt z.

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Since the generators of the algebra HS = A# are (strictly classical) global pseudodifferential operators (in our algebra Opψc0 ) there will be a subalgebra A# ψ of A# consisting of pseudodifferential operators only. It thus might be interesting to also find out whether there is a global Egorov-type theorem attached to the conjugation by U (t). That is, one might ask how the ψdo-symbol of T ∈ A# ψ propagates under Heisenberg conjugation, assuming that Tt remains in the algebra A# ψ. We have not yet attempted to find an answer to these questions.

8. A discussion of the parametrix method In this section we are going to sketch some details of discussions leading to the proof of the following facts: 1) The operator Λ of (3.2) belongs to Opψc−e1 and to Opψc−e2 [and, in fact to Opψc−τ e1 −(1−τ )e2 for all 0 ≤ τ ≤ 1]. 2) The generators xΛ and DΛ belong to Opψc0 . 3) The powers Λs belong to Opψc−s(τ e1 +(1−τ )e2 ) for all s > 0 and 0 ≤ τ ≤ 1. They belong to Opψc|s|e for all s < 0. Here “Opψcm ” denotes the class of all strictly classical ψdo-s of order m = (m1 , m2 ), and we set e = (1, 1) , e1 = (1, 0) , e2 = (0, 1). The class of symbols a(x, ξ) of order m is denoted by ψcm . The corresponding class of operators A = a(x, D) is called Opψcm . The symbol class ψcm is just the class of C ∞ √ (k) functions a with a(l) (x, ξ) = O(xm2 −l ξm1 −k ), for all k, l, where x = 1 + x2 , (α)

and where the (x, ξ)-derivative of order β [in x] and α [in ξ] is be written as a(β) . The ψdo A = a(x, D) is defined by Au(x) = (2π)−1/2 dξ dyeiξ(x−y) a(x, ξ)u(y), u ∈ S(R). (8.1) We have to make extensive use of the calculus described in [Co3]. In the following let a(x, ξ) be any positive real-valued C ∞ -function over R2 – normally we are thinking of a(x, ξ) = 1 + x2 + ξ 2 . Lemma 8.1. (i) For any real μ we have (α)

(aμ )(β) = aμ pα,β , pα,β ∈ RSPN α β (a) = (αl ) αl = α, β l = β} . RSPN { (a(β l ) /a) :

(8.2)

l

Here RSPN {. . .} denotes the “real span” of the collection {. . .} – that is, the set of all real linear combinations of {. . .}. (k) α+k (ii) For any p ∈ RSPN α β (a) and any k, l we have p(l) ∈ RSPN β+l (a). The proof of this lemma is a matter of induction.

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Note, for a(x, ξ) = (1 + x2 + ξ 2 ), in the language of the lemma, we have RSPN kl (a) ⊂ ψc−ke1 −le2 , and aμ ∈ ψcμe or ∈ ψc−τ e1 −(1−τ )e2 ,

(8.3)

as μ > 0 or μ < 0, respectively. This first implies existence of an inverse B mod O(−∞) of A = 1 + x2 + D2 where B ∈ Opψc−2τ e1 −2(1−τ )e2 , by II, Thm. 2.5 of [Co3]. To recall the construction of B – by the “parametrix method”: One just starts with a 0th approximation B0 = b0 (x, D) where b0 = 1/a = a−1 is a symbol in ψc−e1 , for example, by (8.3), where we set τ = 0, expecting the same procedure for other 0 ≤ τ ≤ 1. Using ψdo-calculus this gives AB0 = 1 + c0 (x, D) with c0 ∈ ψc−e . One then finds a “correction” B1 = b1 (x, D) of order −2e1 − e such that AB1 = −c0 (x, D)+c1 (x, D) where c1 ∈ ψc−2e , so that A(B0 +B1 )−1 ∈ ψc−2e . 1 Iterating 0 one obtains a sequence Bj of order −e − je, and the asymptotic sum B = Bj has AB − 1 of order −∞. Similarly we construct a right inverse, hence an inverse – all mod O(−∞). This “parametrix method” may be generalized to construct a square root √ of a positive operator – related to the above just as extracting the spare root 2 is related to “long division”. But to obtain a “positive square root” some precautions are needed. First, we will not use the “left-multiplying” representation A = a(x, D) but the “Weyl representation” A = aw (Mw , D) (cf. [Co3], p. 61) with a different symbol aw ∈ ψcm . For our special a(x, ξ) = 1+x2 +ξ 2 we have a(x, D) = aw (x, D). In general, for a ∈ ψcm one defines x+y , ξ)u(y), u ∈ S(R). (8.4) (a(Mw , D)u)(x) = (2π)−1/2 dξ dyeiξ(x−y) a( 2 The advantage of this representation is that an operator A = a(Mw , D) is Hermitian if and only if its symbol is real-valued. Furthermore, we must insist that the “parametric square root” we are going construct – i.e., a ψdo B = b(Mw , D) satisfying B 2 − A ∈ O(−∞), – is at least semi-bounded below in the Hilbert space H = L2 (R). We achieve this by starting with B0 = f0 (Mw , D)2 where f0 (x, ξ) = (1 + x2 + ξ 2 )1/4 . That operator is positive, as a square of two self-adjoint operators. Moreover, we find that B0 = b0 (Mw , D) with b0 (x, ξ) = c0 (x, ξ)(1 + p1 (x, ξ)) 1 where c0 = (1 + x2 + ξ 2 ) 2 and p1 ∈ ψc−2e , using the asymptotic composition g(Mw , D)h(Mw , D) = q(Mw , D) where q(x, ξ) =

∞ (−i/2)l l=0

l!

[(∂ξ ∂y − ∂x ∂η )l g(x, ξ)h(y, η)]x=y,ξ=η .

(8.5)

Indeed, setting g = h = f0 in (8.5) the odd order terms cancel and the sum is real, and we find that b0 (x, ξ) = c0 (x, ξ){1 +

∞ j=1

as stated, using (8.3) and Lemma 8.1.

RSPN 2j 2j (a)} = c0 (1 + p1 ),

(8.6)

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H.O. Cordes

Note then that the operator C0 = c0 (Mw , D) is a self-adjoint operator, semibounded, below, in our Hilbert space H, since it differs from B0 by a bounded operator – even a compact operator in Opψc−e . Similarly we then find that C02 = A0 = a0 (Mw , D) with ∞ RSPN 2j a0 (x, ξ) = a(x, ξ)(1 + d0 (x, ξ)), d0 ∈ 2j (a).

(8.7)

1

Next set C1 = c1 (Mw , D) with c1 (x, ξ) = − 12 c0 (x, ξ)d0 (x, ξ). Then get (C0 + C1 )2 = a0 (Mw , D) + (C0 C1 + C1 C0 ) + C12 .

(8.8)

In the second terms symbol the odd powers so that the symbol is 0∞ cancel again, 2l real. That symbol is of the form −a(d0 + 2 RSPN 2l (a)), while the last term is 0∞ of the form 2 RSPN 2l 2l . Hence we find that ∞ RSPN 2l (8.9) (C0 + C1 )2 − A = (ad1 )(Mw , D) with d1 ∈ 2l (a). 2

It is clear now, how this iteration proceeds: We next will use the operator C2 = − 21 (c0 d1 )(Mw , D). Writing (C0 + C1 + C2 )2 = (C0 + C1 )2 + {(C0 + C1 )C2 + C2 (C0 + C1 )} + C22 we have the first term described by (8.9). Each of 1 1 the other terms is of the form (a 2 r)(Mw , D)(a 2 s)(Mw , D) with symbols r, s beβ longing to some RSPN α α (a) and RSPN β (a), respectively. The symbol of the 1 1 (real part of the) product (a 2 r)(Mw , D)(a 2 s)(Mw , D) then will be of the form 0∞ α+β+2l (a)}. This follows from the composition formula (8.5) a{rs + 1 RSPN α+β+2l together with (8.3) and Lemma 8.1. But we have chosen the symbol of C2 in such a way that 2*(C0 C2 ) = C0 C2 + C2 C0 cancels the perturbation term (ad1 )(Mw , D) occurring in (8.9) (modulo a term of lower order). Furthermore, the other two terms C22 and *(C1 C2 ) also are of lower order. Therefore we get ∞ RSPN 2l (8.9 ) (C0 + C1 + C2 )2 − A = (ad2 )(Mw , D) with d2 ∈ 2l (a). 3

Iterating we0then get a sequence {Cj : j = 0, 1, 2, . . .}, and may take the asymptotic ∞ sum C = 0 Cj . This operator C = c(Mw , D) will be self-adjoint and it will 2 satisfy C = A + Φ with A = 1 + x2 + D2 and a ψdo Φ of order −∞. Note that C differs from the positive ψdo B0 only by a ψdo of order −e – that is, a compact operator of H. So, C will be semi-bounded, below. The operator A has discrete spectrum. The perturbation A + Φ must have discrete spectrum as well, since Φ is compact. We must have (Spectrum(C))2 ⊂ Spectrum(A + C). Hence the spectrum of C also is discrete. Since C is semibounded below, there can only be finitely many eigenvalues (of finite multiplicity) in the set {λ ≤ 0}. But for such a ψdo all eigenvectors to isolated eigenvalues belong to S(R). It follows that the orthogonal projection P− onto the non-positive part of the spectrum of C belongs to O(−∞). Defining C˘ = C + 2P− C we then finally get a positive definite ψdo C˘ ∈ Opψce satisfying C˘ 2 = A + Φ.

Harmonic Oscillator

321 1

Finally, to prove that the inverse positive square root A− 2 belongs to Opψc−τ e1 −(1−τ )e2 , note that 1 ∞ −1 − 12 −1 ˘ λ 2 dλ{(A + λ)−1 − (A + Φ + λ)−1 } (8.10) A −C = π 0 ∞ 1 1 = λ− 2 dλ(A + λ)−1 Φ(A + Φ + λ)−1 . π 0 Using the “ψdo-properties” of A and A + Φ it is easily seen that Λm XΛm ∈ L(H) for all m = (m1 , m2 ) where Λm = xm2 Dm1 , and where X denotes the right1 hand side of (8.10). This implies that X = A− 2 − C˘ is a ψdo of order −∞. Knowing the symbol of C˘ up to lower order, we find it to be formally md-hypo-elliptic with a parametrix of order −τ e1 − (1 − τ )e2 for all 0 ≤ τ ≤ 1. This implies that also 1 C˘ −1 is a ψdo of that order, and, hence, also A− 2 . Q.E.D. This will prove point (1) of our initial program, in this section, and point (2) follows as well. It should be clear that the same technique may be employed to also prove (3) – at least for any rational s.

References [1] Yu.M. Berezanski, Expansions in Eigenfunctions of Self-adjoint Operators, AMS transl. of Math. monographs vol. 17, Providence 1968. [2] H.O. Cordes, Spectral theory of Linear Differential Equations and Comparison Algebras, Lecture Notes London Math. Soc. Vol. 76; Cambridge Univ. Press 1987. [3] H.O. Cordes, Precisely Predictable Dirac Observables, Springer, Dordrecht, 2007. [4] H.O. Cordes, The Technique of Pseudodifferential Operators, Lecture Notes London Math. Soc. Vol. 202; Cambridge Univ. Press 1997. [5] L.D. Fadeev and A.A. Slavnov, Gauge fields, Benjamin, Reading, MA, 1980. [6] I. Gohberg, On the theory of multi-dimensional singular integral operators, Soviet Math. 1 (1960), 960–963. [7] L. H¨ ormander, Pseudodifferential operators and hypoelliptic equations, Proc. Symposium Pure Appl. Math. 10 (1966), 138–183. [8] M.G. Krein, Hermitian operators with direction functionals, Sbornik Praz. Inst. Mat. Akad. Nauk Ukr. SSR 10 (1948), 83–105. [9] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York inc. 1966. [10] J. v. Neumann, Die Mathematischen Grundlagen der Quantenmechanik, Springer 1932 New York; reprinted Dover Publ. inc. 1943; English translation 1955 Princeton Univ. Press. [11] H. Sohrab, The C ∗ -algebra of the n-dimensional harmonic oscillator, Manuscripta Math. 34 (1981), 45–70. Heinz Otto Cordes University of California, Berkeley, CA 94707, USA e-mail: [email protected]



Remark on Spectral Rigidity for Magnetic Schr¨ odinger Operators Gregory Eskin and James Ralston Abstract. We give a simple proof of Guillemin’s theorem on the determination of the magnetic field on the torus by the spectrum of the corresponding Schr¨ odinger operator. Mathematics Subject Classification (2000). Primary 35P99, Secondary 35R30. Keywords. Inverse spectral problems, wave trace.

1. Introduction This note is on inverse spectral theory for the Schr¨ odinger operator on a flat two-dimensional torus with electric and magnetic potentials. This problem can be remarkably rigid. For generic flat tori, if the variation of the magnetic field is strictly less than its mean, and the total magnetic flux on the torus is ±2π, then the spectrum of the Schrödinger operator determines both the electric and magnetic fields. This is in marked contrast to both the Schr¨ odinger operator without a magnetic field (see [3]) and the case of a magnetic field of mean zero (see [1]). In both those problems there are large families of isospectral fields, and rigidity results are much more difficult to obtain (see also [2]). The observation that there can be spectral rigidity when the total flux is ±2π is due to Guillemin ([5]). Here we give a short proof of the slightly stronger result stated above. Instead of thinking of the Hamiltonian as acting on functions with values in a line bundle over the torus R2 /L, we think of the Hamiltonian as acting on functions on R2 which are invariant with respect to the “magnetic translations” associated to L. However, these two settings are completely equivalent. Our assumption that the variation of the magnetic field B(x) is strictly less than its mean b0 takes the simple form |B(x) − b0 | < |b0 | for all x. The spectrum of the Laplacian plus lower-order perturbations on flat tori has the feature that there are large families of spectral invariants corresponding to sets of geodesics with a fixed length. In analogy with results on S 2 Guillemin

324

G. Eskin and J. Ralston

proposed the name “band invariants” for these families. The nice feature of the problem discussed here is that only the simplest of the band invariants are needed to prove rigidity. The first complete solution of an inverse spectral problem was Mark Krein’s definitive analysis of the “weighted string”, [9], [10]. Since that time many other inverse spectral problems in one space dimension have been solved (see [11]). In higher dimensions it is widely believed that, modulo natural symmetries and deformations like gauge transformation, most problems will be spectrally rigid. However, so far there have been relatively few settings where this has been proven (for instance those in [6] and [14]) and many interesting examples where it fails (see [12] and [4]). This should remain an active field of research for many years to come, and one can reasonably say that it began with the work of Mark Grigor’evich Krein.

2. Proof of Guillemin’s theorem We begin with the smooth magnetic field B, periodic with respect to the lattice L in two dimensions, expanded in a Fourier series in terms of the dual lattice L∗ B(x) = bβ e2πiβ·x . β∈L∗

For this magnetic field we introduce the magnetic potential A = (A1 , A2 ) with ∂x2 A1 − ∂x1 A2 = B, chosen to be as periodic as possible, i.e., b0 bβ e2πiβ·x (β2 , −β1 )(2πi(β12 + β22 ))−1 . A = A0 + A1 = (x2 , −x1 ) + 2 ∗ β∈L \0

We also have a mean zero periodic electric field which is the gradient of the mean zero periodic potential vβ e2πiβ·x . V (x) = β∈L∗ \0

The quantum Hamiltonian for an electron in these fields (with all physical constants set to 1) is H = (i∂x + A)2 + V. Let D be a fundamental domain for L. To define the domain of H as an operator in L2 (D) we will use “magnetic translation operators” (see [13]). Letting {e1 , e2 } and {e∗1 , e∗2 } be a basis for L and the corresponding dual basis for L∗ , define for linearly independent vectors v1 and v2 Tj u(x) = eivj ·x u(x + ej ),

j = 1, 2.

Then the commutator [T1 , T2 ] is given by [T1 , T2 ]u(x) = (eiv2 ·e1 − eiv1 ·e2 )ei(v1 +v2 )·x u(x + e1 + e2 ),

Spectral Rigidity for Magnetic Schr¨ odinger Operators

325

and the periodicity of A1 and V implies that the commutator [H, Tj ] is given by [H, Tj ]u(x) = eivj ·x ((i∂x + A(x) + A0 (ej ))2 − (i∂x + A(x) − vj )2 )u(x + ej ). Thus, in order for the Tj ’s to commute with H we require vj = −A0 (ej ), and in order for the Tj ’s to commute with each other we require A0 (e1 )·e2 = −A0 (e2 )·e1 = πl for some integer l. Note that this implies A0 (e31 ) = πle∗2 and A0 (e2 ) = −πle∗1 . Moreover, 2π|l| = |b0 |Area (D) and b0 Area (D) = D B(x)dx is the total magnetic flux. Hence the assumption b0 = 0 is equivalent to nonzero flux, and it implies l = 0. Defining the domain of H to be the subspace of H 2 (R2 ) such that Tj u = u, j = 1, 2, we make H a self-adjoint operator in L2 (D). As in many previous works we will look for spectral invariants for H by studying the wave trace. Letting E(x, y, t) be the distribution kernel for the fundamental solution for the initial value problem utt + Hu = 0 in R2x × Rt , u(x, 0) = f (x), ut (x, 0) = 0, the distribution kernel for the corresponding initial value problem in D × Rt is T1m T2n E(x, y, t), (2.1) ED (x, y, t) = (m,n)∈Z2

where the operators Tj act on the x variable. Note that, since the principal part of ∂t2 + H is ∂t2 − Δ, E(x, y, t) = 0 when |x − y|2 > t2 and the sum in (2.1) has only a finite number of nonzero terms for t in a bounded interval. Thus [T1 , T2 ] = 0 implies Tj ED (x, y, t) = ED (x, y, t), j = 1, 2. The fundamental spectral invariant for this problem is the distribution trace of the operator ED (t) corresponding to the kernel ED (x, y, t). Conventionally (with all terms to be interpreted in distribution sense) this is written T r(t) = ED (x, x, t)dx. D

To avoid degeneracies in the contributions to T r(t) from the terms in (2.1), we assume that vectors in L have distinct lengths, i.e., d, d ∈ L and |d| = |d | implies d = ±d . Since E(x, y, t) is singular as a distribution in (x, y) only when |x − y|2 = t2 , it now follows that the singularity of T r(t) at t = |me1 + ne2 | comes from just two terms [T1m T2n E(x, x, t) + T1−m T2−n E(x, x, t)]dx.

(2.2)

D

To determine the spectral invariants coming from the leading terms in the expansion of this singularity it is convenient to use the Hadamard-H¨ ormander expansion [7], [8] for E(x, y, t). Beginning with the forward fundamental solution, E+ , defined by (∂t2 + H)E+ = δ(t)δ(x − y) and E+ = 0 for t < 0 one has E+ (t, x, y) ∼

∞ ν=0

aν (x, y)eν (t, |x − y|)

(2.3)

326


where eν is chosen so that (∂t2 − Δ)eν = νeν−1 for ν > 0 and e0 (t, |x − y|) is the forward fundamental solution for ∂t2 − Δ. In two space dimensions this means eν (t, |x − y|) = 2−2ν−1 π −1/2 X+

ν−1/2

(t2 − |x − y|2 )

for t > 0, eν = 0 for t < 0. For a > −1 the distribution X+a is defined by X+a (s) = (Γ(a + 1))−1 sa for s > 0 and X+a (s) = 0 for s < 0. Hence the coefficients aν are determined by the recursion νaν + (x − y) · ∂x aν − iA(x) · (x − y)aν + Haν−1 = 0, where H acts in the variable x. Solving this with the requirement that a0 (y, y) = 1, we have 1 a0 (x, y) = exp(i (x − y) · A(y + s(x − y))ds) and (2.4) 0

a1 (x, y) = −a0 (x, y)(

1

V (y + s(x − y))ds + b(x, y)),

(2.5)

0

where b(x, y) is determined by A(x). The fundamental solution E(x, y, t) is given by (2.6) E(x, y, t) = ∂t (E+ (t, x, y) − E+ (−t, x, y)). We define 3 31 0 1 −iA0 (d)·x+i 01 d·A(x+sd)ds I(d) = e dx = ei(2A (x)·d+ 0 d·A (x+sd)ds) dx, D

and

D

# J(d) = D

1

$ 31 0 1 (V (x + sd) + b(x + sd, x))ds ei(2A (x)·d+ 0 d·A (x+sd)ds) dx.

0

From (2.2)–(2.6) one sees that I(d)+I(−d) and J(d)+J(−d) are spectral invariants for H. However, the periodicity implies that I(d) = I(−d) and J(d) = J(−d). The rest of this article is devoted to studying I(d) and J(d). We have d = me1 + ne2 = k(m0 e1 + n0 e2 ), k ∈ N and gcd(m0 , n0 )=1. Let δ = −n0 e∗1 + m0 e∗2 . Then we have b0 A0 (d) = (d2 , −d1 ) = πklδ. 2 3 1 2πisβ·d Since 0 e ds = 0 when β · d = 0, the terms in the Fourier series for A1 which contribute to I(d) have β · d = 0, and this implies β = pδ =

pb0 (d2 , −d1 ), p ∈ Z\0. 2πkl

Hence, d · (β2 , −β1 )(2πi(β12 + β22 ))−1 = ikl(pb0 )−1 , and I(d) reduces to ibpδ exp(2πikl(−δ · x + e2πipδ·x ))dx. 2πpb0 D p∈Z\0

Spectral Rigidity for Magnetic Schr¨ odinger Operators Defining Bδ (s) =

bpδ e2πips and A1δ (s) =

p∈Z\0

(note that

p∈Z\0

327

bpδ 2πips e 2πip

d 1 ds Aδ (s)

= Bδ (s)), we have 1 exp(−i2πkl(δ · x + A1δ (δ · x))dx. I(d) = b 0 D

Extending δ to a basis for L∗ , {δ, δ }, and letting {γ, γ } be the dual basis for L, we make the change of variables x = sγ + uγ , and choose D = {sγ + uγ : 0 ≤ s, u ≤ 1}. Then we have

1

I(d) = c(d)

exp(−2πikl(s + 0

1 1 A (s))ds, b0 δ

where c(d) is the Jacobian factor, and only depends on d. Since we have this spectral invariant for all k = 0, it follows that 1 1 f (s + A1δ (s))ds (2.7) b0 0 is a spectral invariant for any function f which can be expanded in terms of {e−2πikly }k∈Z , i.e., for any f ∈ L2loc (R) which has period 1/l. Theorem 2.1. Assume that l = 1 and |b0 | > max |B(x) − b0 |. Then the spectrum of H determines B. Remark 2.2. Since b0 is the average of B(x) on a fundamental domain, the hypothesis here is a constraint on how much B varies instead of constraint on the size of B. Remark 2.3. Since we assume 1 = l = A0 (e1 ) · e2 /π = |b0 |Area(D)/2π for a fundamental domain D, this assumption fixes |b0 | when L is fixed. 31 Proof of Theorem 2.1. Since Bδ (x) = 0 (B(x + sd) − b0 )ds, the hypotheses imply that the derivative of s + A1δ (s)/b0 is strictly positive and the inverse function s(y) to y = s + A1δ (s)/b0 is defined on the range of s + A1δ (s)/b0 for s ∈ [0, 1]. Since A1δ has period 1, the range is I = [A1δ (0)/b0 , A1δ (0)/b0 + 1]. Letting f in (2.7) tend to the δ-function at y, the limit of (2.7) is s (y) if y = y(s) for s ∈ [0, 1]. If y = y(s) for s ∈ [0, 1] then the limit of (2.7) is s (y ∗ ), where y ∗ ∈ I, and y ∗ = y mod 1. In other words taking these limits we recover a function of period 1 in y which agrees with s (y) on I. Thus we recover A1δ (s) modulo an additive constant, and we obtain Bδ (s) by taking the derivative. Since we can carry out this argument for all prime elements δ ∈ L∗ , we recover the full Fourier expansion of B.

328


We now turn to the recovery of V . The preceding analysis shows that, keeping the same d ∈ L as above, the spectral invariant J(d), modulo terms determined by A(x), reduces to 1 ˜ = c(d) J(d) Vδ (s)e−2πik(s+Aδ (s)) ds, (2.8) 0

where Vδ (s) =

vpδ e2πips .

p∈Z\0

This immediately gives the following: Theorem 2.4. Under the hypotheses of Theorem 2.1 the spectrum of H determines V . Proof of Theorem 2.4. Since we are assuming the hypotheses of Theorem 2.1, we have the function s(y) and can make the substitution s = s(y) in (2.8). That gives A1δ (0)+1 ˜ J(d) = c(d) Vδ (s(y)e−2πiky s (y)dy, A1δ (0)

but, since y(s + 1) = y(s), we can extend s(y) smoothly to the whole line by defining s(y + 1) = s(y) + 1. Thus, since Vδ (s) has period 1 in s, we have 1 ˜ = c(d) J(d) Vδ (s(y))s (y)e−2πiky dy. 0

Since we have this spectral invariant for k ∈ Z\0, we recover the Fourier series of Vδ (s(y))s (y), and, hence, since s(y) is determined by A1δ (s), we have Vδ (s). As before, since we can carry out this argument for all prime elements δ ∈ L∗ , we recover the full Fourier expansion of V . Remark 2.5. If l = p/q, p, q ∈ N, for the lattice L, then l = 1 for the lattice L0 generated by c1 e1 + c2 e2 and d1 e1 + d2 when p(c1 d2 − c2 d1 ) = q. So if B(x) and V (x) are periodic with respect to L0 , Theorems 1 and 2 apply in the sense that the spectrum of H on the torus R2 /L0 determines B(x) and V (x). Note that B(x) and V (x) will automatically be periodic with respect to L0 when l = 1/q.

References [1] G. Eskin, Inverse spectral problem for the Schr¨ odinger equation with periodic vector potential. Comm. Math. Phys. 125 (1989), 263–300. [2] G. Eskin, J. Ralston, Inverse spectral problems in rectangular domains. Commun. PDE 32 (2007), 971–1000. [3] G. Eskin, J. Ralston, E. Trubowitz, On isospectral periodic potentials in Rn , I and II. Commun. Pure and Appl. Math. 37 (1984), 647–676, 715–753. [4] C. Gordon, Survey of isospectral manifolds. Handbook of Differential Geometry 1 (2000), North Holland, 747–778.

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[5] V. Guillemin, Inverse spectral results on two-dimensional tori. Journal of the AMS 3 (1990), 375–387. [6] V. Guillemin, D. Kazdhan, Some inverse spectral results for negatively curved 2manifolds. Topology 19 (1980), 301–312. [7] J. Hadamard, Le Problème de Cauchy et les Equations aux Dérivées Partielles Linéaires Hyperboliques. Hermann, Paris, 1932. [8] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, III. SpringerVerlag, Vienna, 1985. [9] M. Krein, Solution of the inverse Sturm-Liouville problem (Russian). Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 21–24. [10] M. Krein, Determination of the density of a nonhomogeneous cord by its frequency spectrum. Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348. [11] V.A. Marchenko, Operatory Shturma Liuvilliˆ a i ikh prilozhen’iˆ a, Naukova Dumka, Kiev, 1977. [12] T. Sunada, Riemannian coverings and isospectral manifolds. Ann. of Math. 121 (1985), 169–186. [13] J. Zak, Dynamics of electrons in solids in external fields. Phys. Rev. 168 (1968), 686–695. [14] S. Zelditch, Spectral determination of analytic, bi-axisymmetric plane domains. Geometric Funct. Anal. 10 (2000), 628–677. Gregory Eskin and James Ralston Department of Mathematics UCLA, Los Angeles CA 90095-1555, USA e-mail: [email protected] [email protected]



On Holomorphic Solutions of Some Implicit Linear Differential Equations in a Banach Space Sergey Gefter and Tatyana Stulova To the 100th anniversary of Mark Krein

Abstract. Let A be a bounded quasinilpotent linear operator on a Banach space and f be a vector-valued function holomorphic in a neighborhood of zero. The problem of the existence and uniqueness of holomorphic and entire solution of the implicit differential equation Aw + f (z) = w is considered. Mathematics Subject Classification (2000). Primary 34A20; Secondary 34A25, 34G10. Keywords. Quasinilpotent operators, holomorphic solutions, linear differential equations.

1. Introduction Differential equations in a Banach space was one of the areas related to Mark Krein’s research interests (see his joint monograph with Ju. Dalec’kii [3]). In the present paper, we consider a question of holomorphic solutions of the implicit linear differential equation Aw (z) + f (z) = w(z),

(1.1)

where A is a bounded linear operator on a complex Banach space and f (z) is a vector-valued function holomorphic in a neighborhood of zero. In this paper, the term ‘solution of Equation (1.1)’ will stand for a holomorphic solution, i.e., a vector-valued function which is holomorphic in a neighborhood of zero and satisfies Equation (1.1) in this neighborhood. Equation (1.1) is a particular case of the general implicit linear differential equation Aw (z) + f (z) = Bw(z), (1.2) where A and B are closed linear operators acting between two Banach spaces, and f is a vector-valued function with values in the corresponding space. In the

332

S. Gefter and T. Stulova

finite-dimensional case, this equation (i.e., an implicit system of scalar differential equations) was studied in the classical works by Weierstrass and Kronecker (see, for example, [15]). In the case of an infinite-dimensional Banach space, the theory related to Equation (1.2) on the semiaxis t ≥ 0 was basically developed in the papers by A.G. Rutkas and his team [18], [19], [20], [17], [22], [23], [24] (see also [4]). If A is an invertible operator and A−1 is bounded, then Equation (1.1) can be written in the form w (z) = A−1 w(z) + g(z), where g(z) = −A−1 f (z). The properties of solutions of the later equation are well known. In particular, the Cauchy problem for this equation always has a unique holomorphic solution (see [3], Chapters II, VI and [13]). The analytical properties of solutions of the explicit equation w (z) = Aw(z) + f (z) for the case when the operator A is unbounded were studied in the papers [9], [10], [11] and [12]. We refer to [14] concerning the general theory of such equations. We consider the case which is opposite to the case where A is invertible as a bounded operator, namely, where A is quasinilpotent (i.e., the spectrum σ(A) of A reduces to the only point λ = 0). Let us formulate the main results of our paper. Theorem 2.5. Let E be a Hilbert space and A : E → E be a Volterra operator. Suppose A is of trace class, or the asymptotic estimate sn (A) = o( n1 ) for the sequence {sn (A)} of the singular values of A is valid (see [7], Chapter II). If f is an E-valued function, which is holomorphic in the disk |z| < R, then Equation (1.1) has a unique holomorphic solution in the same disk. Theorem 2.6. Let E be a Banach space, A : E → E be an arbitrary bounded quasinilpotent linear operator, and f : C → E be an entire function of exponential type (i.e., f (z) ≤ Ceσ|z| for some C > 0, σ > 0). Then Equation (1.1) has a unique entire solution of exponential type which is at most as that of f . We also present an example of a quasinilpotent operator which does not satisfy the assumptions of Theorem 2.5 and such that Equation (1.1) has no holomorphic solutions (see Example 1 after Theorem 2.5). We note that operators satisfying the assumptions of Theorem 2.5 were studied in details at Odessa School on the operator theory (see [7] and [8]). The proofs of our results are based on some approaches contained in the book by Ju. Dalec’kii and M. Krein [3], Chapter VI, § 2. Therein, the question of solutions of the explicit equation w (z) = Aw(z) + f (z) is considered for the case when a bounded operator A has a bounded inverse and f is an entire function of zero exponential type. Note that § 2 of this book is devoted to a detailed presentation of results contained in the paper by Ju. Dalec’kii and I. Korobkova [2]. We note that the use of methods of Chapter VI, § 2 [3] is one of the possible ways to investigate holomorphic solutions of Equation (1.1). There are other methods of studying Equation (1.1) in the complex domain. They will be considered in the papers that follow. We also note that the proofs of the uniqueness for holomorphic solutions of Equation (1.1), described in Theorems 2.4, 2.5, 2.6, and 2.7, are obtained by simple

On Holomorphic Solutions of Linear Differential Equations

333

arguments. This is sharply different from the context of [23], [20] (see also [21]), where the theory of entire and meromorphic functions was applied. The present paper is related to [5] and [6], where the problem of existence of holomorphic ∂2u ∂u solutions of the equations z 2 Aw + f (z) = w and = A 2 was considered for ∂t ∂x the case where A is quasinilpotent.

2. Main results Let E be a complex Banach space, A : E → E be a linear operator, and w(z) = ∞ 0 cn z n be a formal power series with coefficients of E. We denote the formal n=0

power series w(z) =

∞ 0

(Acn )z n by (Aw)(z).

n=0

In fact, the following two statements were obtained in Subsection 1, § 1 of Chapter VI [3]. Lemma 2.1. Let f (z) =

∞ 0

bn z n and w(z) =

n=0

∞ 0

cn z n be formal power series with

n=0

coefficients of E. Then w(z) is a solution of Equation (1.1) if and only if (n + 1)Acn+1 + bn = cn ,

n = 0, 1, 2, . . . .

(2.1)

Moreover, if w(z) is a solution of the homogeneous equation Aw = w, then ck =

(n + k)! n A cn+k , for all k and n. k!

(2.2)

Proof. Substituting w(z) into Equation (1.1) and equating the coefficients of like powers, we obtain (2.1). The equality (2.2) follows directly from (2.1). From Lemma 2.1 we obtain the following propositions concerning solutions of Equation (1.2) for two diametrically opposite algebraic situations. Lemma 2.2. Let A : E → E be an arbitrary linear operator and f (z) =

m 0

bn z n be

n=0

a polynomial with coefficients of E. Then Equation (1.1) has a unique polynomial m m 0 0 1 solution w(z) = cn z n where cn = n! s!As−n bs s=n

n=0

Proof. By Lemma 2.1 the polynomial w(z) =

m 0 n=0

cn z n is a solution of Equation

(1.1) if and only if bm = cm and (n + 1)Acn+1 + bn = cn for all n = 0, . . . , m − 1. m 0 1 From this we obtain that cn = n! s!As−n bs . s=n

334


Lemma 2.3. Let A : E → E be a nilpotent linear operator, Am+1 = 0, and ∞ 0 f (z) = bn z n be a formal power series with coefficients of E. Then Equan=0

tion (1.1) has a unique formal solution w(z) =

∞ 0

cn z n where

n=0

cn = Proof. Let cn =

1 n!

∞ 0

∞ n+m 1 s!As−n bs = s!As−n bs . n! s=n s=n

s!As−n bs =

s=n

n+m 0 s=n

s!As−n bs , n ≥ 0. It is easy to check that

(n + 1)Acn+1 + bn = cn for all n ≥ 0. Hence, the equality (2.1) of Lemma 2.1 is ∞ 0 satisfied and the formal series w(z) = cn z n is a solution of Equation (1.1). Let n=0

us prove the uniqueness of this solution. Let w be a solution of the homogeneous Am+1 cm+k+1 = 0 for all k ≥ 0, equation. It follows from (2.2) that ck = (m+k+1)! k! i.e., w = 0. Theorem 2.4. Let A : E → E be a bounded quasinilpotent linear operator and f be an E-valued function

holomorphic in the disk |z| < R. Assume that A satisfies the condition lim n n! An = 0. Then Equation (1.1) has a unique holomorphic n→∞

solution in the same disk. ∞ ∞ 0 0 bn z n , bn ∈ E and 0 < r < R. Then bn rn < ∞. Proof. Let f (z) = n=0

Therefore, there exists M > 0 such that bn ≤

n=0

for all n ≥ 0. Let us show ∞ 0 cn z n where that a solution of Equation (1.1) can be found in the form w(z) =

cn =

1 n!

∞ 0

M rn

n=0 s−n

s!A

bs (see the proof of Lemma 2.2). For this purpose, let us prove

s=n

that the coefficients cn are defined correctly, the series

∞ 0

cn z n is convergent in

n=0

the disk |z| < R, and the

sum of this series satisfies Equation (1.1). Let |z| < r and n 0 < ε < r − |z|. Since n n! An → 0, there exists C > 0 such that An ≤ C · εn! εs−n C·M for all n. From this we obtain that s!As−n bs ≤ s! (s−n)! for all s ≥ n. rs Therefore, ∞ ∞ s! εs−n s! As−n bs ≤ C · M (s − n)! rs s=n s=n =

∞ n! C · M (k + n)! ε k C ·M ( ) = , n n r k! r r (1 − εr )n+1 k=0

because

∞ (k + n)! k=0

k!

xk =

n! , |x| < 1. (1 − x)n+1


335

Hence

|z|n C ·M rM · C |z| n ( ) . = n r (1 − rε )n+1 r−ε r−ε ∞ 0 cn z n converges in the disk Since |z| < r and ε < r − |z|, the series w(z) = cn z n ≤

n=0

|z| < r. Therefore, the function w(z) is holomorphic in the disk |z| < R. It is not difficult to check that the coefficients cn satisfy the recursion relation (2.1). Hence, w(z) is a solution of Equation (1.1) (see Lemma 2.1). Let us prove the uniqueness of this solution. Let w(z) be a solution of the homogeneous Equation (1.1). Then n cn = (n+k)! k! A cn+k (see Lemma 2.1). Hence J

n+k (n + k)! n+k n+k cn ≤ n! An n+k cn+k . k!n!

Since lim n+k n! An = 0, lim n+k cn+k < ∞, and lim n+k (n+k)! k!n! = 1 for n→∞ n→∞

n→∞ all k ≥ 0, we have that lim n+k ck = 0. Thus, ck = 0 for all k ≥ 0, i.e., n→∞ w = 0. For the case of a Hilbert space, we obtain the following sufficient conditions for existence of a holomorphic solution. Theorem 2.5. Let E be a Hilbert space and A : E → E be a Volterra operator. Suppose A is of trace class, or the asymptotic estimate sn (A) = o( n1 ) for the sequence {sn (A)} of the singular values of A is valid . If f is an E-valued function, which is holomorphic in the disk |z| < R, then Equation (1.1) has a unique holomorphic solution in the same disk. ∞ 0 Proof. Let F (z) = (1 − zA)−1 = An z n be the Fredholm resolvent of A n=0

and A is of trace class. Since A is quasinilpotent, F is an entire function and ∞ L ∞ (1 + rsj (A)), where {sj (A)}j=1 is the sequence of MA (r) = max F (z) ≤ |z|≤r

j=1

the singular values of A (see [7], Chapter V, § 5, inequality

(5.3)). Hence, F (z) is of zero exponential type (see [16], Chapter I, § 11). Thus, n n! An → 0 (see [3], Chapter I, Problem 22 and [1], Appendix B), and we see that A satisfies the assumptions of Theorem 2.4. Let now sn (A) = o( n1 ). It follows from V. Matsaev’s results that log MA (r) = o(r), r → ∞ (see [7], Chapter V, § 5). Hence, in this case F (z) is a function of zero exponential type as well. Let us show that Equation (1.1) may have no holomorphic solutions if A does not satisfy the assumptions of Theorem 2.4. ∞

Example (1). Let E be a Hilbert space with an orthonormalized basis {ek }k=0 and A : E → E be the forward weighted shift operator such that Aek = k1 ek+1 for all

1 k ≥ 0. It is easy to see that A is quasinilpotent and An = n! , i.e., n n! An = 1.

336


If f (z) =

∞ 0

fk (z) ek and w (z) =

k=0

∞ 0

wk (z) ek , then Equation (1.1) takes the

k=0

form of an infinite system of differential equations ⎧ f0 (z) = w0 (z) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w0 (z) + f1 (z) = w1 (z) 1 ⎪ w (z) + f2 (z) = w2 (z) ⎪ ⎪ ⎪ 2 1 ⎪ ⎩ ...

(2.3)

e0 . Then the solution of the system (2.3) is found easLet now f (z) = 1−z ∞ 1 ily: wk (z) = (1−z)k+1 . However, the set of functions {wk (z)}k=0 is not a solution of Equation (1.1) in the neighbourhood of zero for the space E because ∞ 0 2 ∞ |wk (0)| = +∞. It is interesting to note that w (z) = {wk (z)}k=0 is a holok=0

morphic solution of Equation (1.1) outside the disk |z − 1| ≤ 1. An additional point to make is that in this example, Equation (1.1) has no differentiable solution on the semiaxis [0, +∞) at all. In the next theorem, we consider Equation (1.1) with an arbitrary quasinilpotent operator and impose stronger conditions on the function f . Theorem 2.6. Let A : E → E be a bounded quasinilpotent linear operator and ∞ 0 f (z) = bn z n , bn ∈ E, be an entire function of exponential type. Then Equation n=0

(1.1) has a unique entire solution of the exponential type which is at most as that of f . Proof. Let the exponential type of the function f be equal to σ. Then for any ε1 > 0 there exists N > 0 such that n! bn < (σ + ε1 )n for all n > N (see [1], Appendix B and [3], Chapter I, Problem 22). Since A is quasinilpotent, for an 1 arbitrary ε2 satisfying the inequality 0 < ε2 < σ+ε there exists C > 0 such that 1 n n A < Cε2 for all n ∈ N. Let us show that a solution of Equation (1.1) can be ∞ ∞ 0 0 1 found in the form w(z) = cn z n with cn = n! s!As−n bs . For this purpose, n=0

s=n

we prove that the coefficients cn are defined correctly, the series w(z) =

∞ 0 n=0

cn z n

is convergent for all z ∈ C, w(z) is of exponential type no higher than σ, and w(z) satisfies Equation (1.1). Let us estimate cn . For n > N we have: cn ≤

∞ ∞ s > 1 > C s−n (σ + ε1 ) s! >As−n > · bs ≤ s!ε2 n! s=n n! s=n s! ∞

=

(σ + ε1 ) C C k . ε2 (σ + ε1 )n+k = n! n! (1 − ε2 (σ + ε1 )) k=0

n


337

Hence, the coefficients cn are defined correctly whenever n > N . Now, we set: cN = (N + 1)AcN +1 + bN , cN −1 = N AcN + bN −1 , ... c0 = Ac1 + b0 . It is easy to check that the equality cn = 0, 1, . . . , N . Since cn ≤

(σ+ε1 )n C n!(1−ε2 (σ+ε1 )) ,

1 n!

∞ 0

s!As−n bs is preserved for n =

s=n

n > N , we see that the series

∞ 0

cn z n is

n=0

convergent for arbitrary z ∈ C, i.e., w (z) is an entire function. Let us show that the exponential type of w (z) is no higher than σ: =

(σ + ε1 )n C n = σ + ε1 . lim n! cn ≤ lim n n→∞ n→∞ 1 − ε2 (σ + ε1 )

Since ε1 is arbitrary, lim n n! cn ≤ σ. Thus, the inequality for the exponential n→∞ ∞ 1 0 type of w(z) is proved. Since cn = n! s!As−n bs for all n ≥ 0, it is not difficult s=n

to see that the coefficients bn and cn satisfy the equality (n + 1)Acn+1 + bn = cn . Thus, by Lemma 2.1, the function w(z) satisfies Equation (1.1). Let us prove the uniqueness of this solution. Let w(z) be a solution of the homogeneous Equation n (1.1). Then ck = (n+k)! k! A cn+k (see (2.2) in Lemma 2.1) and J

n+k (n + k)! n+k n+k n cn+k . ck ≤ A k! Since lim

n→∞

An = 0

n+k

and

we have lim n+k ck = 0, i.e.,

lim

n→∞

(n + k)! cn+k < +∞,

n+k

n→∞

ck = 0

and w(z) = 0. ∞

Example (2). Let E be a Hilbert space with an orthonormalized basis {ek }k=0 . Consider the a weighted shift operator A such that Ae0 = 0, Aek = k1 ek−1 for all k ≥ 1. It is obvious that the operator A is a bounded quasinilpotent operator with ∞ 1 An = n! . In the basis {ek }k=0 , A is given by the matrix ⎞ ⎛ 0 1 0 0 ... 1/ 0 ⎜ 0 0 ... ⎟ 2 ⎟ ⎜ ⎝ 0 1 0 0 /3 . . . ⎠ ... ... ... ... ...

338


Since Ae0 = 0, Equation (1.1) is degenerate. If w(z) =

∞

wk (z) ek

k=0

and f (z) =

∞

fk (z) ek ,

k=0

then this equation transforms into the following infinite system of scalar differential equations: ⎧ w1 (z) + f0 (z) = w0 (z) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 w (z) + f (z) = w (z) ⎨ 1 1 2 2 ⎪ 1 ⎪ ⎪ w3 (z) + f2 (z) = w2 (z) ⎪ ⎪ ⎪3 ⎩ ... According to Theorem 2.6, for an arbitrary sequence of entire functions ∞ 0 {fk (z)}∞ |fk (z)|2 ≤ C1 e2σ|z| for some C1 > 0, σ > 0 and all k=0 such that n=0

∞

z ∈ C, this system has a unique solution {wk (z)}k=0 consisting of the entire functions for which ∞ 2 |wk (z)| ≤ C2 e2σ|z| , z ∈ C. n=0

Using the approach to the proofs of Theorem 2.4 and Theorem 2.6, we can obtain two more propositions related to the case where f is an entire function. Theorem 2.7. Let A : E → E be a bounded quasinilpotent linear operator with

n lim n! An < +∞ and f be an E-valued entire function. Then Equation n→∞

(1.1) has a unique entire solution. Theorem 2.8. Let A : E → E be an arbitrary bounded linear operator and f be an E-valued entire function. If f is a function of zero exponential type (i.e., for every ε > 0 we conclude that f (z) ≤ Cε eε|z| for some Cε > 0), then Equation (1.1) has a unique entire solution of zero exponential type.

References [1] W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations. Springer-Verlag New York Inc., 2000. [2] Ju. Dalec’kii and I. Korobkova, On certain operator differential equation with regular singular point. Dopovidi Akad. Nauk URSR, Ser. A., no. 2 (1968), 972–977 (in Ukrainian). [3] Ju. Dalec’kii and M. Kreˇin, Stability of differential equations in Banach space. Amer. Math. Soc., Providence, R.I., 1974. [4] A. Favini and A. Yagi, Degenerate differential equations in Banach spaces. New York-Basel-Hong Kong: Marcel Dekker, Inc., 1999.

On Holomorphic Solutions of Linear Differential Equations [5] S. Gefter and V. Mokrenyuk, The power series

∞ 0 n=0

[6]

[7] [8] [9]

[10] [11] [12]

[13] [14] [15] [16] [17]

[18] [19] [20] [21] [22] [23]

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of some differential equations in a Banach space. J. of Math. Phis., Anal., Geom., no. 1 (2005), 53–70 (in Russian). S. Gefter and A. Vershynina, On holomorphic solutions of the heat equation with a Volterra operator coefficient. J. Methods Funct. Anal. Topology 13 (2007), no. 4, 329–332. I. Gohberg and M. Kreˇin, Introduction to the theory of linear nonselfadjoint operators. Amer. Math. Soc., Providence, R.I., 1969. I. Gohberg and M. Kreˇin, Theory and applications of Volterra operators in Hilbert space. Amer. Math. Soc., Providence, R.I., 1970. M. Gorbachuk, On the well-posedness of the Cauchy problem for operator-differential equations in classes of analytic vector functions. Dokl. Akad. Nauk. 374 (2000), no.1, 7–9 (in Russian). M. Gorbachuk, An operator approach to the Cauchy-Kovalevskay theorem. J. Math. Sci. (2000), no. 5, 1527–1532. M. Gorbachuk, On analytic solutions of operator-differential equations. Ukrainian Math. J. 52 (2000), no. 5, 680–693. M. Gorbachuk and V. Gorbachuk, On the well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space. Methods Funct. Anal. Topology 11 (2005), no. 2, 113–125. E. Hille, Ordinary differential equations in the complex domain. A Wiley-Interscience publication, New York, London, 1976. S. Kreˇin, Linear differential equations in Banach space. Translations of Mathematical Monographs, Vol. 29. American Mathematical Society, Providence, R.I., 1971. P. Kurkel and V. Mehrmann, Differential-algebraic equations. Analysis and numerical solution. Europ. Math. Soc., 2006. B.Ya. Levin, Distribution of zeroes of entire functions. Amer. Math. Soc., Providence, R.I., 1964. N.I. Radbel’, On an initial manifold and dissipativity of Cauchy problem for the equation Ax (t) + Bx(t) = 0. J. Differential Equations 15 (1979), no. 6, 1142–1143 (in Russian). A.G. Rutkas, The Cauchy problem for the equation Ax (t) + Bx(t) = f (t). Diff. Uravn. 11 (1975), 1996–2010 (in Russian). A.G. Rutkas, Classification and properties of solutions of the equation Ax +Bx(t) = f (t). Differential Equations 25 (1989), no. 7, 809–813. A.G. Rutkas and L.A. Vlasenko Uniqueness and approximation theorems for a degenerate operator-differential equation. Math. Notes 60 (1996), no. 3-4, 445–449. I.V. Tikhonov, Abstract differential null-equations. Funct. Anal. Appl. 38 (2004), no. 2, 133–137. L.A. Vlasenko, Construction of solutions of some classes of the equations Au (t) + Bu(t) = f (t). Vestnik Kharkov. Gos. Univ., no. 286 (1986), 24–28 (in Russian). L.A. Vlasenko, The completeness of normal solutions of the equation Au (t)+Bu(t) = f (t). Teor. Funktsii, Funktsional. Anal. i Prilozhen., no. 48 (1987), 46–51 (in Russian); translation in J. Soviet Math. 49 (1990), no. 2, 883–886.

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[24] L.A. Vlasenko Evolutionary patterns with implicit and singular differential equations. Systemnye Technology, Dnepropetrovsk, 2006 (in Russian). Sergey Gefter Department of Mechanics and Mathematics Kharkiv National University 4 Svoboda Square 61077 Kharkiv, Ukraine e-mail: [email protected] Tatyana Stulova National AeroSpace University (KhAI) 17 Chkalova Avenue 61085 Kharkiv, Ukraine e-mail: [email protected], [email protected]


Groups of Operators for Evolution Equations of Quantum Many-particle Systems V.I. Gerasimenko Abstract. The aim of this work is to study the properties of groups of operators for evolution equations of quantum many-particle systems, namely, the von Neumann hierarchy for correlation operators, the BBGKY hierarchy for marginal density operators and the dual BBGKY hierarchy for marginal observables. We show that the concept of cumulants (semi-invariants) of groups of operators for the von Neumann equations forms the basis of the expansions for one-parametric families of operators of various evolution equations for infinitely many particles. Mathematics Subject Classification (2000). Primary 35Q40; Secondary 47d06. Keywords. Quantum dynamical semigroup, quantum many-particle system, cumulant (semi-invariant), cluster expansion, BBGKY hierarchy, dual hierarchy, von Neumann hierarchy.

1. Introduction Recently we observe significant progress in the study of the evolution equations of quantum many-particle systems [2], [3]. In particular it is involved in such fundamental problem as the rigorous derivation of quantum kinetic equations [4]–[9]. The construction of solutions of such equations is based on the theory of differential equations in Banach spaces (in particular, the theory of semigroups of operators [18], [13]). In the case of evolution equations with coefficients which are bounded operators, first results on the well-posed solvability and also the foundation of the stability theory in the infinite-dimensional Banach space was established by M. Krein [1]. As is well known, there are various possibilities to describe the evolution of quantum many-particle systems [2]. The sequence of the von Neumann equations This work was partially supported by the WTZ grant No M/124 (UA 04/2007) and by the project of NAS of Ukraine No 0107U002333.

342

V.I. Gerasimenko

for density operators [3], [11], the von Neumann hierarchy for correlation operators [23], the BBGKY hierarchy for marginal density operators [11] and the dual BBGKY hierarchy for marginal observables [2] give the equivalent approaches for the description of the evolution of finitely many particles. Papers [23]–[27] constructed the one-parametric families of operators that define solutions of the Cauchy problem for these evolution equations. It was established that the concept of cumulants (semi-invariants) of groups of operators for the von Neumann equations forms the basis of the one-parametric families of operators of various evolution equations of quantum systems of particles, in particular, the BBGKY hierarchy for infinitely many particles [23]. The aim of the paper is to investigate properties of groups of operators for evolution equations of quantum many-particle systems related with their cumulant structure on suitable Banach spaces. In the beginning we will formulate some necessary facts about the description of quantum many-particle systems. Let a sequence f = I, f1 , . . . , fn , . . . is an infinite sequence of self-adjoint M ⊗n operators fn (I is a unit operator) defined on the Fock space FH = ∞ n=0 H 0 over the Hilbert space H (H = C). Operators fn defined in the n-particle Hilbert space Hn = H⊗n we will denote by fn (1, . . . , n). For a system of identical particles obeying Maxwell-Boltzmann statistics, one has fn (1, . . . , n) = fn (i1 , . . . , in ) if {i1 , . . . , in } ∈ {1, . . . , n}. M∞ n 1 1 Let n=0 α L (Hn ) be the space of sequences f = I, f1 , . . . , Lα (FH ) = fn , . . . of trace class operators fn = fn (1, . . . , n) ∈ L1 (Hn ), satisfying the abovementioned symmetry condition, equipped with the trace norm f L1α (FH ) =

∞ n=0

αn fn L1 (Hn ) =

∞

αn Tr1,...,n |fn (1, . . . , n)|,

n=0

where α > 1 is a real number, Tr1,...,n is the partial trace over 1, . . . , n particles. We will denote by L1α,0 the everywhere dense set in L1α (FH ) of finite sequences of degenerate operators [14] with infinitely differentiable M∞ kernels with compact supports. We will also consider the space L1 (FH ) = n=0 L1 (Hn ). We note that the sequences of operators fn ∈ L1 (Hn ), n ≥ 1, whose kernels are known as density matrices [12] defined on the n-particle Hilbert space Hn = H⊗n = L2 (Rνn ), describe the states of a quantum system of non-fixed number of particles. The space L1 (FH ) contains sequences of operators more general than those determining the states of systems. The evolution of all possible states of quantum systems is described by the initial-value problem to the von Neumann equation [10], [11]. A solution of such Cauchy problem is defined by the following one-parametric family of operators on L1 (FH ) R1 t → G(−t)f := U(−t)f U −1 (−t),

(1.1)

Groups of Operators of Quantum Many-particle Systems where f ∈ L1 (FH ) and U(−t) =

343

M∞

n=0 Un (−t),

Un−1 (−t) := e tHn , (1.2) M∞ U0 (−t) = I is a unit operator. The Hamiltonian H = n=0 Hn in (1.2) is a self∞ adjoint 0 operator 2with domain D(H) = {ψ = ⊕n=0 ψn ∈ FH | ψn ∈ D(Hn ) ⊂ Hn , n Hn ψn < ∞} ⊂ FH [14]. M 2 νn Assume H = L2 (Rν ) then an element ψ ∈ FH = ∞ n=0 L (R ) is a sequence of functions ψ = ψ0 , ψ1 (q1 ), . . . , ψn (q1 , . . . , qn ), . . . such that ψ 2 = 0∞ 3 2 |ψ0 | + n=1 dq1 . . . dqn |ψn (q1 , . . . , qn )|2 < +∞. On the subspace of infinitely differentiable functions with compact supports ψn ∈ L20 (Rνn ) ⊂ L2 (Rνn ), n-particle Hamiltonian Hn acts according to the formula (H0 = 0) Un (−t) := e

Hn ψn = −

− i tHn

i

,

n n 2 Δqi ψn + 2 i=1 i k=1

n

Φ(k) (qi1 , . . . , qik )ψn .

(1.3)

1 At (f ) n (Y ) − fn (Y )>L1 (Hn ) t→0 > > ≤ (|P | − 1)! lim > G|Zk | (−t, Zk ) − I f|Xi | (Xi )>L1 (Hn ) . P: K Y= i Xi

t→0

PK : YP = k Zk

Zk ⊂P

Xi ⊂P

In view of the the fact that group {Gn (−t)}t∈R (1.1) is a strong continuous group, which implies that, for mutually disjoint subsets Xi ⊂ Y , if fn ∈ L10 (Hn ) ⊂ L1 (Hn ) in the sense of the norm convergence L1 (Hn ) there exists the limit G|Zk | (−t, Zk )fn − fn ) = 0. lim ( t→0

Zk ⊂P

Thus if f ∈ L1 (FH ) we finally obtain lim At (f ) n − fn L1 (Hn ) = 0. t→0

We now construct the infinitesimal generator N of group (2.1). Taking into account that for fn ∈ L10 (Hn ) equality (1.4) holds, let us differentiate the |P|thorder cumulant A|P| (t, YP ) for all ψn ∈ D(Hn ) ⊂ Hn in the sense of the point-bypoint convergence. According to equality (1.12) for |P| ≥ 2 we derive 1 lim A|P| (t, YP )fn ψn = (−1)|P |−1 (|P |−1)! (−N|Zk | (Zk ))fn ψn t→0 t K P : YP = k Zk Zk ⊂P 0 |P| r=1 |Zr | − Nint Z1 , . . . , Z|P| fn ψn , (2.5) = ... Z1 ⊂X1 , Z1 =∅

Z|P| ⊂X|P| , Z|P| =∅

0 where Zj ⊂Xj is a sum over all subsets Zj ⊂ Xj of the set Xj . Then in view of equality (2.5) for group (2.1) we obtain 1 1 At (f ) n − fn ψn = lim A|P| (t, YP ) f|Xi | (Xi ) − fn (Y ) ψn lim t→0 t t→0 t K P: Y =

1 = lim A1 (t, Y )fn − fn ψn + t→0 t = − Nn fn (Y )ψn +

K P:Y = i Xi , |P|>1

K P:Y = i Xi , |P|>1

Xi ⊂P

i Xi

1 lim A|P|(t, YP ) f|Xi | (Xi )ψn t→0 t Xi ⊂P

− N int (YP ) f|Xi | (Xi )ψn . Xi ⊂P

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Thus for f ∈ L10 (FH ) ⊂ D(N) ⊂ L1 (FH ) in the sense of the norm convergence L1 (Hn ) we finally have > 1 > At (f ) n − fn − N(f ) n >L1 (Hn ) = 0. lim > t→0 t Example. We give an example which illustrates the structure of expansion (2.1). For the correlation operators f = 0, f1 (1), 0, . . . that is interpreted as satisfying the “chaos” property [2], we have n At (f ) n = An (t, 1, . . . , n) f1 (i),

n ≥ 1,

i=1

i.e., if at the initial instant there are no correlations in a system, the correlations generated by the dynamics of a system are completely governed by cumulants of groups (1.1). We now consider the structure of infinitesimal generator (2.2) for a two-body interaction potential N(f ) n (Y ) = −Nn (Y )fn (Y ) (2) − Nint (i1 , i2 ) f|X1 | (X1 )f|X2 | (X2 ), + P: Y =X1

where the symbol

K

0

X2 i1 ∈{X1 } i2 ∈{X2 }

P: Y =X1

K

X2

means summation over all partitions of the set (2)

Y into two nonempty parts X1 and X2 and the operator Nint is defined by formula (1.11). For classical systems this generator is an equivalent notation of the generator of the Liouville hierarchy [20] formulated in [21].

3. Group of operators for the quantum BBGKY hierarchy The evolution of all possible states both finitely and infinitely many quantum particles is described by the initial-value problem to the BBGKY hierarchy for marginal density operators [11], [24]. For finitely many particles this hierarchy of equations is an equivalent to the von Neumann equation. We will use notations from the previous section. Since YP ≡ (X1 , . . . , X|P| ) then Y1 is the set consisting of one element of the partition P (|P| = 1) of the set Y ≡ (1, . . . , s). In this case for n ≥ 0 the (1 + n)th-order cumulant of operators (1.1) is defined by the formula A1+n (t, Y1 , X\Y ) := (−1)|P|−1 (|P| − 1)! G|Xi | (−t, Xi ), (3.1) 0

P: {Y1 ,X\Y }=

K

i Xi

Xi ⊂P

where P is the sum over all possible partitions P of the set {Y1 , X \ Y } = {Y1 , s+1, . . . , s+n} into |P| nonempty mutually disjoint subsets Xi ⊂ {Y1 , X \Y }. On the space L1α (FH ) a solution of the initial-value problem to the BBGKY hierarchy is defined by a one-parametric mapping [24] with the following properties.

Groups of Operators of Quantum Many-particle Systems

349

Theorem 3.1. If f ∈ L1α (FH ) and α > e, then the one-parametric mapping R1 t → (U (t)f )s (Y ) :=

∞ 1 Trs+1,...,s+n A1+n (t, Y1 , X\Y )fs+n (X) n! n=0

(3.2)

is a C0 -group. On the subspace L1α,0 ⊂ L1α (FH ) the infinitesimal generator B = M∞ n=0 Bn of group (3.2) is defined by the operator (s ≥ 1) (Bf )s (Y ) := −Ns (Y )fs (Y ) (3.3) s s ∞ 1 1 (k+n) Trs+1,...,s+n − Nint (i1 , . . . , ik , X\Y )fs+n (X), + k! n! n=1 k=1

i1 =···=ik =1

(k+n)

where on L10 (Hs+n ) ⊂ L1 (Hs+n ) the operator Nint

is defined by formula (1.11).

Proof. If f ∈ L1α (FH ) mapping (3.2) is defined provided that α > e and the following estimate holds [24] U (t)f L1α (FH ) ≤ cα f L1α (FH ) , where cα = e2 (1 − αe )−1 is a constant. Similar to (2.4) this estimate comes out from the inequality for cumulant (3.1) A1+n (t)fs+n L1 (Hs+n ) ≤ n!en+2 fs+n L1 (Hs+n ) . The strong continuity property of the group U (t) over the parameter t ∈ R1 is a consequence of the strong continuity of group (1.1) of the von Neumann equation. We now construct an infinitesimal generator of group (3.2). Taking into account that for fn ∈ L10 (Hn ) equality (1.4) holds, we differentiate the expression of cumulant (3.1) in the sense of the point-by-point convergence. According to equality (2.5) for n ≥ 1, we derive 1 lim A1+n (t, Y1 , X\Y )fs+n ψs+n t (|Z|+n) = − Nint (Z, s + 1, . . . , s + n)fs+n ψs+n t→0

Z ⊂ Y, Z = ∅

=

|Y | 1 k!

k=1

where

0 Z⊂X

|Y |

(k+n)

− Nint

(i1 , . . . , ik , X\Y )fs+n ψs+n ,

i1 =···=ik =1

is a sum over all subsets Z ⊂ X of the set X.

(3.4)

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V.I. Gerasimenko

Then taking into account formula (1.4) for n = 1 and equality (3.4) for group (3.2) we obtain 1 U (t)f s − fs ψs t ∞ 1 1 1 Trs+1,...,s+n lim A1+n (t, Y1 , X\Y )fs+n ψs = lim A1 (t, Y )fs − fs ψs + t→0 t t→0 t n! n=1

lim

t→0

= −Ns fs ψs +

s ∞ 1 1 Trs+1,...,s+n n! k! n=1 k=1

s

(k+n)

− Nint

(i1 , . . . , ik , X\Y )ψs .

i1 =··· ···=ik =1

Thus if L1α,0 ⊂ D(B) ⊂ L1α (FH ) we finally have in the sense of the norm convergence > >1 lim > U (t)f − f − Bf >L1 (FH ) = 0, α t→0 t where the operator B on L1α,0 is given by formula (3.3).

Example. We give an example which illustrates the structure of infinitesimal generator (3.3). In the case of two-body interaction potential (1.3) operator (3.3) has the form (s ≥ 1) (Bf )s (Y ) = −Ns (Y )fs (Y ) +

s

(2) Trs+1 − Nint (i, s + 1)fs+1 (Y, s + 1),

(3.5)

i=1 (2)

where the operator Nint is defined on L10 (Hs+1 ) ⊂ L1 (Hs+1 ) by formula (1.11) for n = 2. For H = L2 (Rν ) in the framework of kernels of operators fs (s-particle density matrix or marginal distributions [11]) operator (3.5) takes a canonical form of a generator of the quantum BBGKY hierarchy [2, 24]

(Bf )s (q1 , . . . , qs ; q1 , . . . , qs ) s s (2) i 2 − =− Φ (qi − qj ) − Φ(2) (qi − qj ) (Δqi − Δq ) + i 2 i=1 i<j=1

i i=1 s

−

× fs (q1 , . . . , qs ; q1 , . . . , qs ) dqs+1 Φ(2) (qi − qs+1 ) − Φ(2) (qi − qs+1 )

× fs+1 (q1 , . . . qs , qs+1 ; q1 , . . . , qs , qs+1 ). We remark that in [24] we discuss other possible representations of the evolution group for the BBGKY hierarchy on the space L1α (FH ).


351

4. Group of operators for the quantum dual BBGKY hierarchy The evolution of marginal observables of both finitely and infinitely many quantum particles is described by the initial-value problem to the dual BBGKY hierarchy. This hierarchy of equations is dual to the quantum BBGKY hierarchy in the sense of bilinear form (1.5) and for finitely many particles one is an equivalent to the Heisenberg equation (the dual von Neumann equation). For systems of classical particles the dual BBGKY hierarchy was examined in [2], [27], [25]. In this section we will use such abridged notations: Y ≡ (1, . . . , s), X ≡ Y \{j1 , . . . , js−n }. According to notations of section 2, the set (Y \X)1 consists of one element Y \X = (j1 , . . . , js−n ), i.e., the set (j1 , . . . , js−n ) is connected subset of the partition P (|P| = 1). In the case under consideration the dual cumulants A+ 1+n (t), n ≥ 0, of groups (1.6) are defined by the formula (−1)|P|−1 (|P| − 1)! G|Xi | (t, Xi ), A+ 1+n t, (Y \X)1 , X := P: {(Y \X)1 ,X}=

K

i Xi

Xi ⊂P

(4.1) 0 where P is the sum over all possible partitions P of the set {(Y \X)1 , j1 , . . . , js−n } into |P| nonempty mutually disjoint subsets Xi ⊂ {(Y \X)1 , X}. On the space Lγ (FH ) a solution of the initial-value problem to the dual BBGKY hierarchy is defined by a one-parametric mapping (the adjoint mapping to (3.2) in the sense of bilinear form (1.5)) with the following properties. Theorem 4.1. If g ∈ Lγ (FH ) and γ < e−1 , then the one-parametric mapping (4.2) R1 t → U + (t)g s (Y ) :=

s n=0

1 (s − n)!

s

A+ 1+n t, (Y \X)1 , X gs−n (Y \X), s ≥ 1

j1 =···=js−n =1

M∞ is a C0∗ -group. The infinitesimal generator B+ = n=0 B+ n of this group of operators is a closed operator for the ∗-weak topology and on the domain of the definition D(B+ ) ⊂ Lγ (FH ) which is the everywhere dense set for the ∗-weak topology of the space Lγ (FH ) it is defined by the operator (B+ g)s (Y ) := Ns (Y )gs (Y ) +

s

1 n! n=1

s k=n+1

1 (k − n)!

(4.3) s

(k)

Nint (j1 , . . . , jk )gs−n (Y \{j1 , . . . , jn }),

j1 =···=jk =1

(k)

where the operator Nint is given by formula (1.11). Proof. If g ∈ Lγ (FH ) mapping (4.2) is defined provided that γ < e−1 and the following estimate holds U + (t)g Lγ (FH ) ≤ e2 (1 − γe)−1 g Lγ (FH ) .

352

V.I. Gerasimenko

Similar to (2.4) this estimate comes out from the inequality (U + (t)g)s L(Hs ) s 1 ≤ (s − n)! n=0

s

j1 =···=js−n =1 P:{Y \X)1 ,X}=

≤

s

gs−n L(Hs−n )

n=0

(|P| − 1)! gs−n L(Hs−n )

K i

Xi

n+1 s! s(n + 1, k)(k − 1)!, n!(s − n)! k=1

where s(n + 1, k) is the Stirling number of the second kind. On the space Lγ (FH ) the ∗-weak continuity property of the group U + (t) over the parameter t ∈ R1 is a consequence of the ∗-weak continuity of group (1.6) of the Heisenberg equation [10]. To construct an infinitesimal generator of the group {U + (t)}t∈R we firstly differentiate the nth-term of expansion (4.2) in the sense of the point-by-point convergence of the space Lγ . If g ∈ D(N ) ⊂ Lγ (FH ) similar to equality (1.12) for (1 + n)th-order dual cumulant (4.1), n ≥ 1, we derive 1 (|Z|+n) t, (Y \X)1 , X gs−n (Y \X)ψs= Nint (Z, X)gs−n (Y \X)ψs lim A+ 1+n t→0 t Z ⊂ Y \X, Z = ∅

=

s−n k=1

1 k!

(k+n)

Nint

(i1 , . . . , ik , X)gs−n (Y \X)ψs .

(4.4)

i1 =···=ik ∈{j1 ,...,js−n }

Then according to equalities (1.7) and (4.4) for group (4.2) we obtain 1 + 1 U (t)g s − gs ψs = lim A+ lim 1 (t)gs − gs ψs t→0 t t→0 t s s 1 1 + lim A+ 1+n t, (Y \X)1 , X gs−n (Y \X)ψs t→0 t (s − n)! n=1 j1 =···=js−n =1

s s s 1 1 (k) = Ns gs ψs + Nint (j1 , . . . , jk )gs−n (Y \{j1 , . . . , jn })ψs , n! (k − n)! n=1 k=n+1

j1 =··· ···=jk =1

where we used the identity s 1 n! n=0

s j1 =···=jn =1

gs−n (Y \{j1 , . . . , jn }) =

s 1 n! n=0

s

gn (j1 , . . . , jn ) (4.5)

j1 =···=jn =1

which is valid in view of the Maxwell-Boltzmann statistics symmetry property. Thus if g ∈ D(B+ ) ⊂ Lγ (FH ) in the sense of the ∗-weak convergence of the space Lγ (FH ) we finally have 1 + U (t)g − g − B+ g = 0, w∗ − lim t→0 t


353

M + where the generator B+ = ∞ n=0 Bn of group (4.2) is given by formula (4.3) (the dual operator to generator (3.3)). Example. We now give examples of expansion (4.2) and infinitesimal generator (4.3). The sequence g = 0, g1 (1), 0, . . . corresponds to the additive-type observable [27] and in this case expansion (4.2) for the group U + (t) get the form s (U + (t)g)s (Y ) = A+ g1 (j), s ≥ 1. s (t, 1, . . . , s) j=1

In the case of two-body interaction potential (1.3) operator (4.3) has the form s (2) (B+ g)s (Y ) = Ns (Y )gs (Y ) + Nint (j1 , j2 )gs−1 (Y \{j1 }), s ≥ 1, (4.6) j1 =j2 =1 (2)

where the operator Nint is defined by (1.11) for n = 2. If H = L2 (Rν ) in terms of kernels of operators gs , s ≥ 1, for expression (4.6) we have (B+ g)s (q1 , . . . , qs ; q1 , . . . , qs ) s s (2) i 2 − =− Φ (qi − qj ) − Φ(2) (qi − qj ) (−Δqi + Δqi ) + 2 i=1 1=i<j × gs (q1 , . . . , qs ; q1 , . . . , qs ) −

s i (2) Φ (qi − qj ) − Φ(2) (qi − qj ) 1=i=j

j

j

× gs−1 (q1 , . . . , ∨, . . . , qs ; q1 , . . . , ∨, . . . , qs ),

j

where (q1 , . . . , ∨, . . . , qs ) ≡ (q1 , . . . , qj−1 , qj+1 , . . . , qs ). This expression for a system of classical particles is defined as a generator of the dual BBGKY hierarchy stated in [2], [27].

5. Conclusion The concept of cumulants (1.9) of groups (1.1) of the von Neumann equations forms the basis of group expansions for quantum evolution equations, namely, the von Neumann hierarchy for correlation operators [23], as well as the BBGKY hierarchy for s-particle density operators [24] and the dual BBGKY hierarchy [25]. In the case of quantum systems of particles obeying Fermi or Bose statistics groups (2.1), (3.2) and (4.2) have different structures. The analysis of these cases will be given in a separate paper. In the paper [24] we discuss other representations of a group for the BBGKY hierarchy on the space L1α (FH ) and in [25] of a group for the dual BBGKY hierarchy of classical systems of particles. We have stated the properties of groups (2.1) and (3.2) on the space L1α (FH ) and dual group (4.2) on Lγ (FH ). To describe the evolution of infinitely many

354

V.I. Gerasimenko

particles [2] it is necessary to define the one-parametric family of operators (3.2) on more general spaces than L1α (FH ), for example, on the space of sequences of bounded operators containing the equilibrium states [22]. For dual group (4.2) the problem lies in the definition of functional (1.5) for operators from the corresponding spaces. In both these cases every term of the corresponding expansions contains the divergent traces [2], [24], [27] and the analysis of such a question for quantum systems remains an open problem. On the space Lγ (FH ) one-parametric mapping (4.2) is not a strong continuous group. The group {U + (t)}t∈R of operators (4.2) defined on the space Lγ (FH ) is dual to the strong continuous group {U (t)}t∈R of operators (3.2) for the BBGKY hierarchy defined on the space L1α (FH ) and the fact that one is a C0∗ -group follows also from general theorems about properties of the dual semigroup [13], [17]. As mentioned above the group {G(−t)}t∈R of operators (1.1) preserves positivity [16], [18]. The same property must be valid for the group {U (t)}t∈R of operators (3.2) for the BBGKY hierarchy, but how to prove this property one is an open problem. We have constructed infinitesimal generators (2.2), (3.3) on the subspace L1α,0 ⊂ L1α (FH ) and generator (4.3) on D(B+ ) ⊂ Lγ (FH ). The question of how to define the domains of the definition D(N), D(B) and D(B+ ) of corresponding generators (2.2), (3.3) and (4.3) is an open problem [13], [18].

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[9] J. Fr¨ ohlich and E. Lenzmann, Mean-Field Limit of Quantum Bose Gases and Non´ linear Hartree Equation. Séminaire Equations aux Dérivées Partielles. 2003–2004, ´ Exp. No. XIX, Ecole Polytech., Palaiseau, (2004). [10] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. 5, Springer-Verlag, 1992. [11] D.Ya. Petrina, Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems. Kluwer, 1995. [12] F.A. Berezin and M.A. Shoubin, Schr¨ odinger Equation. Kluwer, 1991. [13] O. Bratelli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 1, Springer-Verlag, 1979. [14] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1995. [15] A. Bellini-Morante and A.C. McBride, Applied Nonlinear Semigroups. John Wiley and Sons, 1998. [16] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications. Springer, 2006. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, 1983. [18] R. Aliki and K. Lendi, Quantum dynamical semigroups and applications. Lect. Notes in Phys. 286, Springer, 1987. [19] P.A. Markowich, On the equivalence of the Schr¨ odinger and the quantum Liouville equations. Math. Meth. Appl. Sci. 11 (1989), 459–469. [20] V.O. Shtyk, On the solutions of the nonlinear Liouville hierarchy. J. Phys. A: Math. Theor. 40 (2007), 9733–9742. [21] M.S. Green, Boltzmann equation from the statistical mechanical point of view. J. Chem. Phys. 25 (5) (1956), 836–855. [22] J. Ginibre, Some applications of functional integrations in statistical mechanics. In Statistical Mechanics and Quantum Field Theory. (Eds. S. De Witt and R. Stora, Gordon and Breach), (1971), 329–427. [23] V.I. Gerasimenko and V.O. Shtyk, Evolution of correlations of quantum manyparticle systems. J. Stat. Mech. 3 (2008), P03007, 24p. [24] V.I. Gerasimenko and V.O. Shtyk, Initial-value problem for the Bogolyubov hierarchy for quantum systems of particles. Ukrain. Math. J. 58 (9) (2006), 1329–1346. [25] V.I. Gerasimenko and T.V. Ryabukha, Cumulant representation of solutions of the BBGKY hierarchy of equations. Ukrain. Math. J. 54 (10) (2002), 1583–1601. [26] V.I. Gerasimenko, T.V. Ryabukha and M.O. Stashenko, On the structure of expansions for the BBGKY hierarchy solutions. J. Phys. A: Math. Gen. 37 (2004), 9861–9872. [27] G. Borgioli and V. Gerasimenko, The dual BBGKY hierarchy for the evolution of observables. Riv. Mat. Univ. Parma. 4 (2001), 251–267. V.I. Gerasimenko Institute of Mathematics, National Academy of Science of Ukraine 3 Tereshchenkivs’ka St., 01601 Kyiv, Ukraine e-mail: [email protected]



Box-like Shells with Longitudinal Cracks V.A. Grishin, V.V. Reut and E.V. Reut Abstract. The problem of how to determine the stress state of an infinite boxlike shell of rectangular profile is solved. Two cracks are located on opposite sides of the shell and parallel to its edges. On applying a Fourier transform, the problem can be reduced to a system of two integral equations with respect to jumps at the corner of rotation and normal displacements of the crack edges. The system of integral equations is solved by the method of orthogonal polynomials. Dependence of the stress intensity factor on the length of cracks and the geometrical dimensions of the cross-sections of the shell is demonstrated. Mathematics Subject Classification (2000). Primary 74K25; Secondary 74R10. Keywords. Box shell, plate, crack, stress intensity factor.

1. Preamble Thin-walled shells of a rectangular structure are used widely in construction, shipbuilding and mechanical engineering. In order to minimize the tedious details of research into plate construction as force elements of building mechanics, researchers have made various assumptions depending on types of loads and conditions of their fastenings. Among the first papers in this direction, the intense condition of thin-walled cores of open and closed structures, we refer the reader to the related works of Vlasov, Ganilidze, Panovko, Kan, and Reyssner. Papkovich has applied the methods of plane elasticity theory to the study of box constructions. Thus he assumed that each plate is in a flat intense condition and cooperates with adjoining plates only by tangential efforts. Contrary to Papkovich, in papers of Smotrov and Fleyshman the problems were solved with only the basic assumption that the construction edges do not bend and play a role of rigid support. In a general statement or with the use of a minimum quantity of simplifying assumptions, problems on plate construction were solved by various numerical methods, among which are the following ones: variational-difference

358

V.A. Grishin, V.V. Reut and E.V. Reut

method (method of conjugated gradients), method of finite elements, and the variational method of Kantorovich-Vlasov. The more difficult problems for compound shell constructions, in view of actual conditions of their interaction, were solved by Mossakovsky and his disciples by the homogeneous solutions method. So in a paper of Musiyaki and Poshivalova, the matrix-vector method (based on a method of homogeneous solutions) is offered to analyze the constructions of plate design, and the problem of a folded-plate construction is solved. In a paper of Mossakovsky and Poshivalova the results are given of a comparison of solutions for the problem of a thin-walled bar under constrained torsion to a method of homogeneous solution with Vlasovs results. In a paper of Mossakovsky and Kulikov the method of homogeneous solutions was applied to problems with dynamic loading. In paper [1] an account of the method of box-like shell constructions was offered, and it reduced the problem to one about the joint planar-bend stress condition for a plate with defects, which role is played by the edges of a shell. The advantage of this method consists in 1) the number of necessary differential equations and conditions of the joint is twice reduced, and in 2) the solution methods for planar and bending problems for plates with defects now are well developed, and one can find bibliographies in [5, 7]. In papers [2, 3] the problems of inclusion setting in box-like shells are solved by these methods. In paper [4] the problem of the stress state of a boxed shell with a crack on an shell edge is solved. In the present paper the problem of a longitudinal crack is studied.

2. The problem statement 2a

-a c

y

0

-c

2b x z Fig. 1 Let us consider a problem of the stress state of a box-like shell of infinite length and rectangular structure, weakened by a pair of symmetric cracks (Fig. 1). We suppose that all plates of which the shell is made are of one material and have identical thickness h, Poisson factor ν, elasticity module E, and cylindrical rigidity D. Crack edges are loaded by the bending moments m (y) and planar stretching loadings σ (y). Loadings that influence the shell are symmetric with the planes

Box-like Shells with Longitudinal Cracks

359

of symmetry of the shell and are such that the crack edges are not closed. By the method stated in [1], the problem is reduced to searching for a differential equations system solution: Δ2 w (x, y) = 0 Δ2 σx (x, y) = 0,

−a < x < b, x = 0, |y| < ∞

(2.1)

which satisfy the conditions on the shell edge v = τxy = ϕxy = Mxy = 0 u = −(w+ + w− ); w = u+ + u− σx = −h−1 [(Vx )+ + (Vy )− ]

(2.2)

Vx = h [(σx )+ + (σy )− ] and boundary conditions Vx = τxy = 0; Mx = m (y) ; σx = σ (y) ; x = −a, |y| < c Vx = ϕx = u = τxy = 0; x = −a, |y| > c Vx = ϕx = u = τxy = 0; x = b. Here u, v, w are the displacements along the axes with respect to x, y, z; ϕx , Mx , Vx , σx , τxy – the angle of turn, bending moment, generalized cross force, normal and tangential stresses. It is convenient to present the boundary conditions as: Vx = τxy = 0; ϕx = χ(y); u = μ(y), x = −a (2.3) Vx = ϕx = u = τxy = 0, x = b, where χ(y) and μ(y) – unknown functions on an interval |y| < c, equal to zero outside this interval– represent by themselves an angle of inclination and normal displacements of the crack edges. Without loss of generality, it is possible to consider that the necessary variable change in x results in both y and c = 1.After application of a Fourier transformation to elastic unknown values and loadings, similar to the way it was done in [1], and also to unknown functions χ(y) and μ(y), we get 1 χα =

1 χ(y)e

iαy

μ(y)eiαy dy.

dy, μα =

−1

−1

The problem (2.1)–(2.3) is reduced to the system of two integral equations 1 d2 π dy 2

1 −1

1 K11 μ(η) dη + ln |y − η| χ(η) K21

−1

K12 K22

σ∗ (y) μ(η) dη = m∗ (y) χ(η)

(2.4)

360


where m∗ = 2(Dγ)−1 m(y); σ∗ = −2σ(y) ∞ 1 Kij (y, η) = kij (α)eiα(η−y) dα 2π

−∞ + + Ω0 (−a) + f0μ Ω3 (−a); k11 (α) = p11 (α) − f3μ + + k12 (α) = −f3χ Ω0 (−a) + f0χ Ω3 (−a) − − k22 (α) = p22 (α) − f3χ M0 (−a) + f0χ M3 (−a); − − k21 (α) = −γ −f3μ M0 (−a) + f3μ M3 (−a) −1 p11 (α) exp|α|(a+b) = Gα (−a, b) − 0, 5 (a + b) |α| LGα (−a, b)+

(2.5) (2.6)

(2.7) (2.8)

p22 (α)γ exp|α|(a+b) = −α4 Gα (−a, b) + 0, 5α(a + b) − 2(1 − ν)−1 α2 LGα (−a, b) Ωk (x) = Tk+ Gα (x, t); Mk (x) = R2− Tk− Gα (x, t); γ = (3 + ν)/(1 − ν). Here the following differential operators were used: ∂kf ; k = 0, 1; R2± f = L + (1 ± ν) α2 f k ∂x d2 f ∂ L − (1 ± ν) α2 f ; Lf = 2 − α2 f R3± f = dx ∂x Sf = T − − T + f ; Hf = T − + T + f ; T ±f = f (±0) Sk± f = S Rk± f ; Hk± f = H Rk± f ; Tk± f = T Rk± f

Rk± f =

and G(x, ξ) – Green function of the boundary problem L2 u(x) = 0, x ∈ (a, b); u = u = 0, x = −a, b. + + − − + + − − , f3μ , f0μ , f3μ ; Fχ = f0χ , f3χ , f0χ , f3χ are the solution of Vectors Fμ = f0μ the linear algebraic equation system AF = B for the right-hand parts B = Hμ and B = Hχ correspondingly, where Text is missing!!! This solution is obtained by solving the one-dimensional discontinuous boundary problems system L2 fα± = 0; R3− fα− R1− fα− R3± fα± Sj± fα± S3+ fα+ S0+ fα+

=

−a < x < b, x = 0

R1+ fα+

=

0; R3+ fα+

(2.9)

= −α Eμα ; 4

= χα , x = −a =

R1± fα±

(2.10)

= 0, x = b

=0

j = 1, 2 = α4 E H0− fα− ; H3+ fα+ = α4 E S0− fα− = Dh−1 H3− fα− ; H0+ fα+ = −Dh−1 S3− fα− .

(2.11)


361

Then we can write the solution of problem (2.9)–(2.11) in the form fα− (x) = χα R1− Gα (x, −a) + fq− − f3− T0− Gα + f0− T3− Gα ; fα+ (x) = μ(−α4 E)Gα (x, −a) + fq+ − f3+ T0+ Gα + f0+ T3+ Gα .

(2.12)

So, the stated problem is reduced to a system of integral equations (2.4) containing unknown functions χ(y) and μ(y), which represent the angle of inclination and normal displacements of the crack edges.

3. Construction of the approximate solution of the integral equations system Let us take advantage of the method of orthogonal polynomials [5] and search for a solution as the expansion of unknown functions into a series about some Chebychev polynomials of the second kind Uk (η) with the unknown coefficients

∞ μk μ(η) 2 = 1−η Uk (η), |η| < 1. (3.1) χ(η) χk k=0

Let us substitute (3.1) into (2.4), and multiply each equation of this system by 1 − y 2 Un (y) and integrate by y on the interval (−1, 1). We take into consideration the spectral correspondence [5]: 1 d2 π dx2

1 ln

−1

1

1 − y 2 Un (y)dy = −(n + 1)Un (x) |y − x|

and orthogonal correspondence [6] 1

π 1 − y 2 Um (y)Un (y)dy = δmn 2

−1

and formulas [6]: 1

sin αx (1 − x2 ) cos αx −1

: : π(2n + 3/2 ± 1/2 J2n+2 (α) U2n+1 (x) dx = (−1)n U2n (x) J2n+1 (α) α

. Then after simple transformations and permutation of the integration order in which we get expressions for Kij (y, η), we pass to an infinite system of linear algebraic equations of the second kind which, by Poincaré-Koch, are normal with respect to the coefficients of expansion: ∞ μn μk σn (n + 1) + = , n = 0, ∞ A(k,n) (3.2) χn χk mn k=0

362


where components of a matrix A(k,n) and coefficients of the right-hand parts are (2k,2n) Aij

∞ n+k

= 4(−1)

(2n + 1)(2k + 1) ×

kij (α)α−2 J2n+1 (α)J2k+1 (α)dα;

0 (2k+1,2n+1) Aij

∞ n+k+1

= 4(−1)

(2n + 2)(2k + 2) ×

kij (α)α−2 J2n+2 (α)J2k+2 (α)dα;

0 (2k+1,2n) Aij

=

(2k,2n+1) Aij

1

σn mn

=

2 π

−1

= 0;

i, j = 1, 2

σ∗ (y)

1 − y 2 Un (y)dy. m∗ (y)

Thus kij (α), σ∗ (y), m∗ (y) are determined in (2.5)–(2.8) and Jk (α) is a Bessel function. Let us note that the procedure using the components of matrices A(k,n) is simpler in essential ways owing to block symmetry, which is easily seen by replacing n with k or vice versa. The calculation of integrals with respect to α is also simpler owing to an exponential decrease of the function under integration. The solution of infinite algebraic system (3.2) allows us to determine all elastic unknown values, using the solution of a problem in transformations (2.9)–(2.11) in the form (2.12) and the convolution theorem, and also to estimate the intensity factor of plane k+ and bend k− stresses. Following [7], we shall understand the stress intensity factor k± to be the factor through which the main parts of stresses near the crack ends are expressed. By the main parts of stresses we mean the coefficients of the singularities for stresses near to the crack ends. To obtain these main parts formulas it is enough to find a limit with y → ±1 (|y| > 1) of the integrals ⎞ ⎛ 1 2

1 d ϕ+ μ(η) ln |y − η| dη ⎠ lim ⎝ y 2 − 1 2 = χ(η) ϕ− π y→±1 dy −1

and to use the correspondence [5] 1 d2 π dy 2

1 ln −1

1 Un (η)dη

|y − η| 1 − y 2

|y| Un (y) 2 1 =

+ y − 1 · Un (y)sgn y − (n + 1)Un (y), |y| > 1. 2 2 y −1 As a result, after obtaining the integral main parts in the form ∞ ϕ+ μk = · Uk (±1), |y| > 1 ϕ− χk k=0


363

we can find the stress intensity factors values and the main parts of the elastic values. The numerical solution of the stated problem (2.1)–(2.3), which was reduced to an infinite system of linear algebraic equations (3.2), was obtained by a reduction method that eliminated four members of expansion for μ(y) and χ(y) in (3.1). And the loading, which influences the shell, undertook the role Of the bending moment of intensity m = const and planar normal stresses of intensity σ = const that applied to crack edges. Thus the stress intensity factors in both crack vertexes have identical values k± (±1) = k± and are connected with dimensionless coefficients m σ k± , k± , which were calculated, by the following correspondences √ √ m 6m c σ ; k± = k± σ c. (3.3) k± = k± 2 h In Table 1 the values of stress intensity factors of plane and bend stresses (3.3) with a/b = 0, 5 for a different ratio c/a are shown. Table 1 b/a

2

c/a σ k+ m k−

1

σ k−

· 10

m k+

· 10

3 3

σ k+ m k−

0.5

σ k−

· 10

m k+

· 10

3 3

σ k+ m k− σ k−

· 10

m k+

· 10

3 3

0.1

0.4

0.8

1

1.2

1.5

1.8

2

1.010

1.130

1.270 1.610 1.800 2.090 2.390 2.590

0.999

0.991

0.980 0.953 0.937 0.913 0.889 0.874

0.004 0.0703 0.155 0.343 0.408 0.443 0.398 0.324 0.021

0.346

0.759 1.710 2.120 2.590 2.900 3.050

1.010

1.130

1.270 1.610 1.790 2.080 2.380 2.580

0.992

0.988

0.974 0.935 0.912 0.875 0.838 0.814

0.003

0.050

0.109 0.219 0.243 0.230 0.163 0.092

0.015

0.246

0.533 1.130 1.340 1.560 1.680 1.730

1.010

1.130

1.270 1.620 1.820 2.120 2.450 2.670

0.999

0.983

0.962 0.906 0.873 0.823 0.776 0.746

0.018

0.307

0.655 1.220 1.300 1.130 0.667 0.200

0.089

0.153

0.322 0.650 0.763 0.877 0.942 0.970

The results of calculations show that, under the action of bending loadings, m the intensity factors k− of bend stresses of some orders exceed the intensity factors σ of plane stresses k− ; and under the action of plane loadings the intensity factors m σ k+ of bend stresses on some orders are lower than the intensity factors k+ of plane stresses.

364


References [1] V.A. Grishin, G.Ya. Popov, V.V. Reut, Analysis of box-like shells of rectangular cross-section. J. Appl. Math. Mech. 54, No. 4 (1990), 501–507. [2] V.A. Grishin, V.V. Reut, The stressed state of a box-like shell reinforced by a pair of symmetric inclusions parallel to the edge of the shell. J. Appl. Math. Mech. 59, No. 5 (1995), 817–820. [3] V.A. Grishin, V.V. Reut, The definition of inclusions deflection in the box shell having square section (Russian) Teoret. i Prikl. Mech. (Donetsk, Ukraine) No. 41 (2005), 198–202. [4] V.I. Migdalsky, V.V. Reut, An arbitrary oriented crack in the box shell. Differential operators and related topics. Proceedings of the Mark Krein international conference on operator theory and applications, Odessa, Ukraine, August 18–22, 1997. Volume I. Basel: Birkh¨ auser. Oper. Theory, Adv. Appl. 117 (2000), 261–266. [5] G.Ya. Popov, Elastic stress concentration near stamps, cuts, thin inclusions and supports. (Kontsentratsiya uprugikh napryazhenij vozle shtampov, razrezov, tonkikh vklyuchenij i podkreplenij). (Russian) Moskva, Nauka, 1982. [6] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series. Elementary functions. (Integraly i ryady. Ehlementarnye funktsii). (Russian) Moskva, Nauka, 1981. [7] L.T. Berejnitskij, M.V. Delyavkij, V.V. Panasyk, The bending of thin plates with crack-like defects (Russian) Kiev, Naukova Mysl, 1979. V.A. Grishin, V.V. Reut and E.V. Reut Dvoryanskaya str., 2 The Institute of Mathematics, Economics and Mechanics Odessa National University named after I.I. Mechnikov 65082 Odesa, Ukraine e-mail: [email protected]


Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces of Two Anticommuting Variables Oksana Ye. Hentosh Abstract. The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of super-integro-differential operators of two anticommuting variables coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is obtained by means of a specially constructed Lie-Backlund transformation. The Hamiltonian description for the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies to Lax-integrable (2|2 + 1)dimensional nonlinear dynamical systems and their triple Lax-type linearizations is analyzed. Mathematics Subject Classification (2000). Primary 35Q53, 35Q58, 37K05, 37K10, 37K30, 37K35; Secondary 58A50. Keywords. Operator Lie algebra, Lax-type flows, Backlund transformation, “ghost” symmetries, Davey-Stewartson system.

1. Introduction Since M. Adler’s paper [1], it has been understood that the Lax-type representations [2] for integrable (1+1)-dimensional nonlinear dynamical system hierarchies [3, 4, 5, 6, 7] on functional manifolds and their superanalogs [8, 9, 10] can be interpreted as Hamiltonian flows on a dual space to the Lie algebra of integrodifferential operators. Their Hamiltonian structures are given by the R-deformed canonical Lie-Poisson bracket and the corresponding Casimir functionals as Hamiltonians (see [1, 3, 11, 12, 10, 13]). Every Hamiltonian flow of this type on a dual space to the operator Lie algebra can be written as the compatibility condition for the spectral relationship for the corresponding integro-differential operator and the suitable eigenfunction evolution. If the above spectral relationship admits of a finite set of eigenvalues, then

366

O.Ye. Hentosh

the important problem naturally arises how to find the Hamiltonian representation for the hierarchy of Lax-type coupled with the evolutions of eigenfunctions and appropriate adjoint eigenfunctions. It was partly solved in papers [14, 15, 16, 17, 18] for the Lie algebra of integro-differential operators [1] and its supergeneralization [9, 10] for one anticommuting variable by means of the properties of variational Casimir functionals under some Lie-Backlund transformation. Section 2 deals with a general Lie-algebraic scheme for constructing the hierarchy of Lax-type flows as Hamiltonian ones on a dual space to the Lie algebra of super-integro-differential operators [10] of two anticommuting variables. In Section 3, the Hamiltonian structure for the related coupled Lax-type hierarchy is obtained by means of the Lie-Backlund transformation technique developed in [14, 15, 16, 17]. In Section 4, the corresponding hierarchies of additional, so-called “ghost”, symmetries [14, 19, 20] for the coupled Lax-type flows are established to be Hamiltonian as well. It is proved that the additional hierarchy of Hamiltonian flows is generated by the Poisson structure being obtained from the tensor product of the R-deformed canonical Lie-Poisson bracket with the standard Poisson bracket on the related eigenfunctions and adjoint eigenfunctions superspace [14, 16, 17], and the corresponding natural powers of a suitable eigenvalue are their Hamiltonians. The relation of these hierarchies to Lax integrable (2|2 + 1)-dimensional nonlinear dynamical systems and their triple Lax-type linearizations is analyzed in Section 5.

2. The general Lie-algebraic scheme Let G be the Lie algebra of super-integro-differential operators [10] of two anticommuting variables θ1 and θ2 , for which θi2 = 0, i = 1, 2: aj ∂ j ∈ G, j ∈ Z, m ∈ N, (2.1) l := ∂ m + j<m−1

aj := aj (x, θ1 , θ2 ; Dθ1 , Dθ2 ) := a0,j + a1,j Dθ1 + a3,j Dθ2 + a2,j Dθ1 Dθ2 , where θ1 , θ2 ∈ Λ1 , Λ := Λ0 ⊕ Λ1 is the Grassmann algebra over C, Λ0 ⊃ C, the symbol ∂ := ∂/∂x designates the differentiation with respect to the independent variable x ∈ R/2πZ , S, ap,j := ap,j (x, θ1 , θ2 ) = a0p,j (x) + θ1 a1p,j (x) + θ2 a3p,j (x) + θ1 θ2 a2p,j (x), p = 0, 3, are smooth superfield functions (superfunctions), a0,j , a2,j ∈ C ∞ (S × Λ21 ; C1|0 ) and a1,j , a3,j ∈ C ∞ (S × Λ21 ; C0|1 ), j ∈ Z, j < m, and the superderivatives Dθi := ∂/∂θi + θi ∂/∂x, such as Dθ21 = Dθ22 = ∂/∂x, satisfy the following relationship for any smooth superfield functions u and v: Dθi (uv) = (Dθi u)v + (−1)p(u) u(Dθi v),

i = 1, 2.

Here p(u) is the parity of u, equal to 0 for an even u and 1 for an odd u.

Lax Integrable Supersymmetric Hierarchies

367

The standard Lie commutator on G is defined for all a, b ∈ G as [a, b] := ab − ba, where the associative product of the super-integro-differential operators (2.1) takes the form [21] α νi α νi 1 ∂ ∂ a ∂ ∂ b ab := − , νi α α! ∂ξ ∂Θi ∂xα ∂θiνi α∈Z+ , νi =0,1

with ξ := ∂ and Θi := Dθi , i = 1, 2. On the Lie algebra G, there exists the ad-invariant nondegenerated symmetric bilinear form 2π

(a, b) :=

dx 0

dθ1 dθ2 resDθ1 Dθ2 (ab),

(2.2)

where the resDθ1 Dθ2 -operation is given for all a ∈ G by the expression resDθ1 Dθ2 a := a2,−1 . With the use of the scalar product (2.2), the Lie algebra G is transformed into a metrizable one. As a consequence, its dual linear space of super-integro-differential operators G ∗ is identified with the Lie algebra G, that is, G ∗ , G. The linear subspaces G+ ⊂ G and G− ⊂ G such as ⎧ ⎫ m−1 ⎨ ⎬ aj ∂ j : j = 0, m − 1 , G+ := a := ∂ m + ⎩ ⎭ j=0 + N ∞ bk ∂ −k : k ∈ N , (2.3) G− := b := k>0

where aj , j = 0, m − 1, and bk , k ∈ N, are smooth superfunctions of two anticommuting variables θi , i = 1, 2, form Lie subalgebras in G and G = G+ ⊕ G− . Due to the splitting of G into the direct sum of its Lie subalgebras (2.3), one can construct a Lie-Poisson structure [1, 3, 12, 13] on G ∗ by using a special linear endomorphism R of the linear space G: R := (P+ − P− )/2 ,

P± G := G± , P± G∓ = 0.

For any Fréchet-smooth functionals γ, μ ∈ D(G ∗ ), the Lie-Poisson bracket on G ∗ is given by the expression {γ, μ}R (l) = (l, [∇γ(l), ∇μ(l)]R ) ,

(2.4)

where l ∈ G ∗ and for all a, b ∈ G, the R-deformed commutator [1, 3, 4, 12, 13] [a, b]R := [Ra, b] + [a, Rb] satisfies the modified Yang-Baxter relationship: 1 [a, b]. 4 The linear space G with the commutator (2.5) also becomes a Lie algebra. R[a, b]R − [Ra, Rb] =

(2.5)

368

O.Ye. Hentosh

The gradient ∇γ(l) ∈ G of some functional γ ∈ D(G ∗ ) at a point l ∈ G ∗ with respect to the scalar product (2.2) is naturally defined as δγ(l) := (∇γ(l), δl) , the linear space isomorphism G , G ∗ being taken into account. Every Casimir functional γ ∈ I(G ∗ ), being invariant with respect to the Ad∗ action of the abstract Lie group G corresponding to the Lie algebra G, obeys the following condition at a point l ∈ G ∗ : [l, ∇γ(l)] = 0.

(2.6)

The relationship (2.6) is satisfied by the hierarchy of functionals γn ∈ I(G ∗ ), n ∈ Z+ , taking the form 1 (l1/m , ln/m ). (2.7) n+1 The Lie-Poisson bracket (2.4) generates the following hierarchy of Hamiltonian dynamical systems on G ∗ : γn (l) =

dl/dtn := [R∇γn (l), l] = [(∇γn (l))+ , l],

(2.8)

where the subscript “+” denotes the differential part of the corresponding superintegro-differential operator, and the Casimir functionals (2.7) are their Hamiltonians. The latter equation is equivalent to the commutator Lax-type representation. It is easy to verify that for every n ∈ Z+ the relationship (2.8) is the compatibility condition for such linear super-integro-differential equations: lf = λf,

(2.9)

df /dtn = (∇γn (l))+ f,

(2.10)

and where λ ∈ C is a spectral parameter, f ∈ W 1|0 := L∞ (S × Λ1 ; C1|0 ) if f is an even superfunction, and f ∈ W 0|1 := L∞ (S × Λ1 ; C0|1 ) if f is an odd one. The related to (2.10) dynamical system for the adjoint superfunction f ∗ takes the form df ∗ /dtn = −(∇γn (l))∗+ f ∗ , (2.11) where either (f, f ∗ ) ∈ W 2|0 or (f, f ∗ ) ∈ W 0|2 , and the superfunction f ∗ is a solution of the adjoint spectral problem l∗ f ∗ = νf ∗ with a spectral parameter ν ∈ C. Further, we will assume that the spectral relationship (2.9) admits of N ∈ N different eigenvalues λi ∈ C, i = 1, N, and will study the Lie-algebraic properties of the equation (2.8) combined with 2N ∈ N copies of (2.10), dfi /dtn = (∇γn (l))+ fi , dΦi /dtn = (∇γn (l))+ Φi ,

(2.12)


369

for the even fi ∈ W 1|0 and odd Φi ∈ W 0|1 eigenfunctions, and with the same number of copies of (2.11), dfi∗ /dtn = −(∇γn (l))∗+ fi∗ , dΦ∗i /dtn = −(∇γn (l))∗+ Φ∗i ,

(2.13)

for the suitable even fi∗ ∈ W 0|1 and odd Φ∗i ∈ W 1|0 adjoint eigenfunctions related to N ∈ N different eigenvalues νi ∈ C, i = 1, N. The equations (2.8), (2.12), and (2.13) are considered as a coupled evolution system on the extended phase space G ∗ ⊕ W 2N |2N .

3. The Poisson bracket on the extended phase space To give the description below a compact form, we will designate the left gradient ˜ i, Φ ˜ ∗ ) ∈ vector of any smooth functional γ ∈ D(G ∗ ⊕W 2N |2N ) at a point (˜l, f˜i , f˜i∗ , Φ i ∗ 2N |2N , i = 1, N, as G ⊕W δγ δγ δγ δγ δγ ∗ ∗ ˜ i, Φ ˜ i ) := ∇γ(˜l, f˜i , f˜i , Φ , , , . , ˜ i δΦ ˜∗ δ˜ l δ f˜i δ f˜∗ δ Φ i

i

On the spaces G ∗ and W N ⊕ W ∗N , there exist, respectively, the Lie-Poisson structure [1, 3, 12, 13] such as ˜ δγ δγ ϑ δγ/δ l˜ :→ ˜l, − ˜l, , (3.1) δ˜ l + δ ˜l + where ϑ˜ : T ∗ (G ∗ ) → T (G ∗ ) is an implectic operator related to the bracket (2.4) at a point ˜l ∈ G ∗ , and the canonical Poisson structure [22, 23, 24] such as ⎛ ⎛ ⎞ ⎞ δγ δγ ⎜ δ f˜∗ ⎟ ⎜ δ f˜ ⎟ i ⎟ ⎜ ⎜ ⎟ i ⎜ ⎜ δγ ⎟ ⎟ ⎜ − δγ ⎟ ⎜ ⎟ ⎜ δ f˜∗ ⎟ J˜ ⎜ δ f˜i ⎟ ⎜ ⎜ ⎟, i ⎟ (3.2) ⎜ δγ ⎟ :→ ⎜ ⎟ ⎜ − δγ ⎟ ⎜ ⎟ ⎜ δΦ ⎜ δΦ ⎟ ˜∗ ⎟ ⎜ ⎜ ˜i ⎟ i ⎟ ⎝ ⎝ δγ ⎠ δγ ⎠ − ˜∗ ˜i δΦ δΦ i where J˜ : T ∗ (W 2N |2N ) → T (W 2N |2N ) is an implectic operator related to the even 0N ˜ ∗ ∧ dΦ ˜ i ) at a point (f˜i , f˜∗ , Φ, ˜ Φ ˜ ∗ ) ∈ symplectic form ω (2) = i=1 (df˜i∗ ∧ df˜i − dΦ i i i W 2N |2N , i = 1, N . It should be noted that the Poisson structure (3.1) generates the equation (2.8) for any Casimir functional γ ∈ I(G ∗ ). Thus, on the extended phase space G ∗ ⊕ W 2N |2N , one can obtain a Poisson structure as the tensor product L˜ := ϑ˜ ⊗ J˜ of (3.1) and (3.2).

370

O.Ye. Hentosh Consider the Backlund ⎛ ˜ l ⎜ f˜i ⎜ ⎜ f˜∗ ⎜ i ⎝ Φ ˜i ˜∗ Φ i

transformation ⎛ ⎞ ˜ i, Φ ˜ ∗) l(˜l, f˜i , f˜i∗ , Φ i ⎜ ⎟ fi ⎟ B ⎜ ⎟ :→ ⎜ fi∗ ⎜ ⎟ ⎝ ⎠ Φi Φ∗i

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(3.3)

which generates on G ∗ ⊕W 2N |2N a Poisson structure L with respect to the variables (l, fi , fi∗ , Φi , Φ∗i ), i = 1, N, of the coupled evolution equations (2.8), (2.12) and (2.13). The main condition imposed on the mapping (3.3) is the coincidence of the resulting dynamical system ⎛ ⎞ l ⎜ fi ⎟ ⎟ d ⎜ ⎜ fi∗ ⎟ := −L∇γ n (l, fi , fi∗ , Φi , Φ∗i ) (3.4) ⎜ dtn ⎝ Φ ⎟ i ⎠ Φ∗i with the equations (2.8), (2.12) and (2.13) in the case where γ n ∈ I(G ∗ ), n ∈ N, are independent of the variables (fi , fi∗ , Φi , Φ∗i ) ∈ W 2N |2N , i = 1, N. To satisfy this condition, we find the variation of some Casimir functional γ n := γn |l=l(˜l,f˜i ,f˜∗ ,Φ˜ i ,Φ˜ ∗ ) ∈ D(G ∗ × W 2N |2N ), n ∈ N, under the constraint δ ˜l = 0, i i taking into account the evolutions (2.12), (2.13) and the Backlund transformation (3.3). We have: ˜ i, Φ ˜ ∗ , ) δγ n (˜l, f˜i , f˜i∗ , Φ i δ˜ l=0 P O B C N δγ δγ n n ∗ δ f˜i , + δ f˜i , ∗ = δ f˜i δ f˜i i=1 P C O B δγ n ∗ δγ n ˜ ˜ + δ Φi , ∗ + δ Φi , ˜i ˜ δΦ δΦ i P O P O N ∗ ˜ ˜i d f d f i ∗ = + δ f˜i , δ f˜i , − dtn dtn i=1 P O P O ˜∗ ˜ i dΦ dΦ ∗ i ˜ ˜ + δ Φi , − + δ Φi , − ˜ fi = fi , f˜i∗ = fi∗ , dtn dtn ˜ ∗ = Φ∗ ˜ i = Φ, Φ Φ i i N 7 8 δfi , (∇γn (l))∗+ fi∗ + δfi∗ , (∇γn (l))+ fi = i=1

8 7 + δΦ∗i , −(∇γn (l))+ Φi + δΦi , (∇γn (l))∗+ Φ∗i


=

371

N (∇γn (l))+ δfi , fi∗ + (∇γn (l))+ fi , δfi∗ i=1

+

(∇γn (l))+ (δΦi ), Φ∗i

+

−(∇γn (l))+ Φi , δΦ∗i

N −1 ∗ −1 ∗ (∇γn (l), δ(fi Dθ1 Dθ2 fi )) + (∇γn (l), δ(Φi Dθ1 Dθ2 Φi )) = i=1

N −1 ∗ −1 ∗ = ∇γn (l), δ (fi Dθ1 Dθ2 fi + Φi Dθ1 Dθ2 Φi ) i=1

:= (∇γn (l), δl),

(3.5)

∗

∗

where γn ∈ I(G ), n ∈ N, at a point l ∈ G , and the brackets ., . denote the scalar product on the space W 1|1 . As a consequence of the expression (3.5), one obtains the relation δl|δ˜l=0 = δ

N

(fi Dθ1 Dθ−1 fi∗ + Φi Dθ1 Dθ−1 Φ∗i ). 2 2

(3.6)

i=1

Having assumed the linear dependence of l on ˜l ∈ G ∗ , one gets immediately from (3.6) that N l=˜ l+ (fi Dθ1 Dθ−1 fi∗ + Φi Dθ1 Dθ−1 Φ∗i ). (3.7) 2 2 i=1

Thus, the Backlund transformation (3.3) can be written as ⎛ N ⎞ ⎛ ˜ ˜l l = l + (fi Dθ1 Dθ−1 fi∗ + Φi Dθ1 Dθ−1 Φ∗i ) ⎜ 2 2 ⎜ ⎜ fi ⎟ i=1 ⎟ B ⎜ ⎜ ⎜ f˜i = fi ⎜ f ∗ ⎟ :→ ⎜ ⎜ i ⎟ ⎜ ˜ fi∗ = fi∗ ⎝ Φi ⎠ ⎜ ⎝ ˜ i = Φi Φ Φ∗i ˜ ∗ = Φ∗ Φ i i

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(3.8)

where i = 1, N. The expression (3.8) generalizes the results obtained in papers [15, 16, 17] for the Lie algebra of integro-differential operators and for its supergeneralization [14] for one anticommuting variable. The existence of the Backlund transformation (3.8) ensures the validity of the following theorem. Theorem 3.1. Under the Backlund transformation (3.8), the dynamical system (3.4) on G ∗ ⊕ W 2N |2N is equivalent to the following system of evolution equations: # $ # $ d˜l = (∇γ n (˜l))+ , ˜l − ∇γ n (˜l), ˜l , dtn + df˜i δγ n df˜i∗ δγ n =− ∗ , = , dtn dtn δ f˜i δ f˜i

372

O.Ye. Hentosh ˜i dΦ δγ = n∗ , ˜ dtn δΦ i

˜∗ dΦ δγ i = n , ˜i dtn δΦ

i = 1, N,

where γ n := γn |l=l(˜l,f˜i ,f˜∗ ,Φ˜ i ,Φ˜ ∗ ) ∈ D(G ∗ × W 2N |2N ) and γn ∈ I(G ∗ ) is a Casimir i i functional at the point l ∈ G ∗ for every n ∈ N. Now, by means of simple calculations via the formula ˜ ∗, L = B LB

where B : T (G ∗ ⊕ W 2N |2N ) → T (G ∗ ⊕ W 2N |2N ) is the Fréchet derivative of (3.8), one finds easily the following form of the Poisson structure L on G ∗ ⊕ W 2N |2N : ⎛ ⎞ δγ δγ l, − l, ⎜ ⎟ δl + δl + ⎜ ⎟ ⎜ ⎟ N ⎜ ⎟ δγ ⎜ −1 δγ −1 ∗ ⎟ −fi Dθ1 Dθ2 + ∗ Dθ 1 Dθ 2 fi ⎟ ⎜ + ⎜ ⎟ δfi δfi i=1 ⎜ ⎟ ⎜ ⎟ ⎜ −Φi Dθ1 D−1 δγ − δγ Dθ1 D−1 Φ∗i ⎟ θ2 θ2 ∗ ⎜ ⎟ δΦi δΦ ⎜ ⎟ i ⎟ ∗ ∗ L ⎜ δγ δγ ∇γ(l, fi , fi , Φi , Φi ) → ⎜ ⎟ , (3.9) + f i ⎜ ⎟ ∗ δf δl ⎜ ⎟ i +∗ ⎜ ⎟ δγ δγ ⎜ ⎟ ∗ − − fi ⎜ ⎟ δfi δl ⎜ ⎟ + ⎜ ⎟ δγ δγ ⎜ ⎟ ⎜ ⎟ − ∗+ Φi ⎜ ⎟ δΦi δl + ⎜ ⎟ ∗ ⎝ ⎠ δγ δγ ∗ − − Φ δΦi δl + i where i = 1, N, (l, fi , fi∗ , Φi , Φ∗i ) ∈ G ∗ ⊕ W 2N |2N , and γ ∈ D(G ∗ ⊕ W 2N |2N ) is an arbitrary smooth functional. Thereby, one can formulate the following theorem. Theorem 3.2. The hierarchy of dynamical systems (2.8), (2.12) and (2.13) is a Hamiltonian one with respect to the Poisson structure L in the form (3.9) and the functionals γ n := γn ∈ I(G ∗ ), n ∈ N, which are Casimir invariants on G ∗ . Based on the expression (3.4), one can construct a new hierarchy of Hamiltonian evolution equations describing commutative flows generated on the extended phase space G ∗ ⊕ W 2N |2N by Casimir invariants γn ∈ I(G ∗ ), n ∈ N, involutive with respect to the Lie-Poisson bracket (2.4). The evolution equation hierarchy of this type associated with a super-integro-differential operator of two anticommuting variables in the form (2.1) with m = 1 was obtained in [25].

4. The additional symmetry hierarchies The hierarchy (2.8), (2.12) and (2.13) of evolution equations possesses another natural set of invariants that includes all higher powers of the eigenvalues λk ,


373

k = 1, N, m ∈ N. The latter can be considered as Fréchet-smooth functionals on the space G ∗ ⊕ W 2N |2N due to the evident representation λsk = fk∗ , ls fk + Φ∗k , ls Φk ,

(4.1)

s ∈ N, which holds under the normalizing constraints fk∗ , fk + Φ∗k , Φk = 1. In the case of the Backlund transformation (3.7), where N l := l+ + (fi Dθ1 Dθ−1 fi∗ + Φi Dθ1 Dθ−1 Φ∗i ), 2 2 i=1

the formula (4.1) gives rise to the following variation of the functionals λsk ∈ D(G ∗ ⊕ W 2N |2N ), k = 1, N: δλsk = δfk∗ , ls fk + fk∗ , ls (δfk ) + δΦ∗k , ls Φk + −Φ∗k , ls (δΦk ) + fk∗ , (δls )fk + −Φ∗k , (δls )Φk N 7 8 s δfi , (−Mks + δki ls )∗ fi∗ = (δl+ , Mk ) + i=1

8 7 8 7 + δfi∗ , (−Mks + δki ls )fi + δΦi , (Mks − δki ls )∗ Φ∗i 8 7 ∗ s i s + δΦi , (−Mk + δk l )Φi , where δki is the Kronecker delta and the operators Mks , s ∈ N, are determined as Mks :=

s−1

(lp fk )Dθ1 Dθ−1 (l∗(s−1−p) fk∗ ) + (lp Φk )Dθ1 Dθ−1 (l∗(s−1−p) Φ∗k ) . 2 2

p=0

Thus, one obtains the exact form of the gradients for the functionals λsk ∈ D(Gˆ∗ ⊕ W 2N |2N ), k = 1, N: ⎛ ⎞ Mks ⎜ (−M s + δ i ls )∗ f ∗ ⎟ i ⎟ k k ⎜ s ∗ ∗ s i s ⎟ (4.2) ∇λk (l+ , fi , fi , Φi , Φi ) = ⎜ k + δk l )fi ⎟ , ⎜ (−M ⎝ (Mks − δki ls )∗ Φ∗i ⎠ (−Mks + δki ls )Φi where i = 1, N. By means of the expression (4.2), the tensor product L˜ of the Poisson structures (3.1) and (3.2) generates a new hierarchy of coupled evolution equations on G ∗ ⊕ W 2N |2N : dl+ /dτs,k = −[Mks , ˆl+ ]+ ,

(4.3)

374

O.Ye. Hentosh dfi /dτs,k = (−Mks + δki ls )fi , dfi∗ /dτs,k = (Mks − δki ls )∗ fi∗ , dΦi /dτs,k = (−Mks + δki ls )Φi , dΦ∗i /dτs,k = (Mks − δki ls )∗ Φ∗i ,

(4.4)

where i = 1, N and τs,k , s = 1, N, k = 1, N , are evolution parameters. Owning to the Backlund transformation (3.8), the equation (4.3) can be rewritten as the following equivalent commutator relation: dl/dτs,k = −[Mks , l] = −

s−1

s−1 λpk νks−p−1 [Mk1 , l] = λpk νks−p−1 dl/dτ1,k . (4.5)

p=0

p=0

Thereby, one can formulate the following theorem: Theorem 4.1. For every k = 1, N and s ∈ N, the dynamical systems (4.5) and (4.4) are Hamiltonian ones with respect to the Poisson structure L in the form (3.9) and the invariant functionals γ s := λsk ∈ D(G ∗ ⊕ W 2N |2N ). Theorem 4.2. The dynamical systems (4.5) and (4.4) describe flows on G ∗ ⊕ W 2N |2N commuting both with each other and with the hierarchy of Lax-type dynamical systems (2.8), (2.12) and (2.13). Proof. To prove the latter theorem, it is sufficient to show that [d/dtn , d/dτ1,k ] = 0,

[d/dτ1,k , d/dτ1,q ] = 0,

(4.6)

where k, q = 1, N and n ∈ N. The first equality in the formula (4.6) follows from the identities d(∇γn (l))+ /dτ1,k = [(∇γn (l))+ , M11 ]+ , dM 1 /dtn = [(∇γn (ˆl))+ , M 1 ]− , 1

1

and the second one is a consequence of the relationship dMk1 /dτ1,q − dMq1 /dτ1,k = [Mk1 , Mq1 ].

Thus, for every k = 1, N and all s ∈ N, the dynamical systems (4.5) and (4.4) on G ∗ ⊕ W 2N |2N form a hierarchy of additional homogeneous symmetries for the Lax-type flows (2.8), (2.12) and (2.13) on G ∗ ⊕ W 2N |2N . It was the work [19] where, for the first time, the additional symmetry hierarchies for Lax integrable (1|1 + 1)-dimensional nonlinear dynamical systems associated with the Lie algebra of super-integro-differential operators of one anticommuting variable were described as commutator-type flows. They were also used to construct integrable (2|1 + 1)-dimensional dynamical systems in [14, 20]. The additional symmetry hierarchies for the integrable (2 + 1)-dimensional nonlinear dynamical systems on the dual space to the centrally extended Lie algebra of matrix integro-differential operators were obtained in [18].


375

5. The integrable (2|2 + 1)-dimensional Davey-Stewartson system If N ≥ 2, then one can obtain a new class of nontrivial independent Hamiltonian 0K flows d/dTn := d/dtn + k=1 d/dτn,k , K = 1, [N/2], n ∈ N, on G ∗ ⊕ W 2N |2N in the Lax-type forms by using the above-considered invariants for the Lie algebra G. Acting on the eigenfunctions (fi , fi∗ , Φi , Φ∗i ) ∈ W 2N |2N , i = 1, N, these flows generate Lax integrable (1|2+1)-dimensional supersymmetric nonlinear dynamical systems. For example, in the case of the element l := ∂ +

2

(fi Dθ1 Dθ−1 fi∗ + +Φi Dθ1 Dθ−1 Φ∗i ) ∈ G ∗ 2 2

(5.1)

i=1

with (f1 , f2 , f1∗ , f2∗ , Φ1 , Φ2 , Φ∗1 , Φ∗2 ) ∈ W 4|4 , the flows d/dτ := d/dτ1,1 and d/dT := d/dT2 = d/dt2 + d/dτ2,1 on G ∗ ⊕ W 4|4 , which act on the functions fi , fi∗ , Φi , Φ∗i , i = 1, 2, give rise to these supersymmetric nonlinear dynamical systems: ∗ ∗ f1,τ = f1,x +u ¯1 f2∗ + α ¯ 1 Φ∗2 ,

f1,τ = f1,x + u1 f2 − α1 Φ2 ,

¯ 2 f2∗ + u ¯2 Φ∗2 , Φ1,τ = Φ1,x + α2 f2 − u2 Φ2 , Φ∗1,τ = Φ∗1,x + α u 1 f1 + α ¯ 2 Φ1 , f2,τ = −¯

∗ f2,τ = −u1 f1∗ − α2 Φ∗1 ,

¯ 1 f1 − u ¯ 2 Φ1 , Φ2,τ = −α

Φ∗2,τ = −α1 f1∗ + u2 Φ∗1 ,

(5.2) (5.3)

and f1,T = f1,xx + f1,τ τ + w2 (Dθ1 Dθ2 f1 ) + w1 (Dθ1 f1 ) + w3 (Dθ2 Φ1 ) + (w0 + 2v1,τ − (f1 f1∗ + Φ1 Φ∗1 )2 )f1 − 2βτ Φ1 , ∗ ∗ ∗ ∗ ∗ = −f1,xx − f1,τ f1,T τ − w2 (Dθ1 Dθ2 f1 ) + (w1 + (Dθ2 w2 ))(Dθ1 f1 )

+ (w3 − (Dθ1 w2 ))(Dθ2 f1∗ ) − (w0 + (Dθ1 w1 ) + (Dθ2 w3 ) + 2v1,τ − (f1 f ∗ + Φ1 Φ∗ )2 )f ∗ − 2β¯τ Φ∗ , 1

1

1

1

Φ1,T = Φ1,xx + Φ1,τ τ + w2 (Dθ1 Dθ2 Φ1 ) + w1 (Dθ1 Φ1 ) + w3 (Dθ2 Φ1 ) + (w0 − 2v2,τ − (f1 f ∗ + Φ1 Φ∗ )2 )Φ1 + 2β¯τ f1 , 1

1

Φ∗1,T = −Φ∗1,xx − Φ∗1,τ τ − w2 (Dθ1 Dθ2 f1∗ ) + (w1 + (Dθ2 w2 ))(Dθ1 Φ∗1 ) + (w3 − (Dθ1 w2 ))(Dθ2 Φ∗1 ) − (w0 + (Dθ1 w1 ) + (Dθ2 w3 ) − 2v2,τ − (f1 f1∗ + Φ1 Φ∗1 )2 )Φ∗1 − 2βτ f1∗ ,

(5.4)

f2,T = f2,xx + f2,τ τ + w2 (Dθ1 Dθ2 f2 ) + w1 (Dθ1 f2 ) + w3 (Dθ2 f2 ) + (w0 − (f1 f1∗ + Φ1 Φ∗1 )2 )f2 − u ¯1 f1,τ − α ¯ 2 Φ1,τ + u ¯1,τ f1 − α ¯ 2,τ Φ1 , ∗ ∗ ∗ ∗ ∗ = −f2,xx − f2,τ f2,T τ − w2 (Dθ1 Dθ2 f2 ) + (w1 + (Dθ2 w2 ))(Dθ1 f2 )

+ (w3 − (Dθ1 w2 ))(Dθ2 f2∗ ) − (w0 + (Dθ1 w1 ) + (Dθ2 w3 ) ∗ − (f1 f1∗ + Φ1 Φ∗1 )2 )f2∗ + u1 f1,τ + α2 Φ∗1,τ − u1,τ f1∗ − α2,τ Φ∗1 ,

Φ2,T = Φ2,xx + Φ2,τ τ + w2 (Dθ1 Dθ2 Φ2 ) + w1 (Dθ1 Φ2 ) + w3 (Dθ2 Φ2 ) + (w0 − (f1 f1∗ + Φ1 Φ∗1 )2 )Φ2 − α ¯ 1 f1,τ − u ¯2 Φ1,τ + α ¯ 1,τ f1 + u ¯2,τ Φ1 ,

376

O.Ye. Hentosh Φ∗2,T = −Φ∗2,xx − Φ∗2,τ τ − w2 (Dθ1 Dθ2 Φ∗2 ) + (w1 + (Dθ2 w2 ))(Dθ1 Φ∗2 ) + (w3 − (Dθ1 w2 ))(Dθ2 Φ∗2 ) − (w0 + (Dθ1 w1 ) + (Dθ2 w3 ) ∗ − (f1 f1∗ + Φ1 Φ∗1 )2 )Φ∗2 − α1 f1,τ − u2 Φ∗1,τ − α1,τ f1∗ + u2,τ Φ∗1 ,

(5.5)

where (Dθ1 v1 ) = (Dθ2 f1 f1∗ ) , (Dθ1 β) = (Dθ2 f1 Φ∗1 ) ,

(Dθ1 v2 ) = (Dθ2 Φ1 Φ∗1 ), ¯ = (Dθ f ∗ Φ1 ), (Dθ β)

(Dθ1 u1 ) = (Dθ2 f1 f2∗ ) ,

(Dθ1 u2 ) = (Dθ2 Φ1 Φ∗2 ),

(Dθ2 f1∗ f2 ) , (Dθ2 f1 Φ∗2 ) , (Dθ2 f1∗ Φ2 ) ,

(Dθ1 u ¯2 ) = (Dθ2 Φ∗1 Φ2 ),

¯1 ) = (Dθ1 u (Dθ1 α1 ) = ¯1) = (Dθ1 α

1

2

(5.6)

1

(Dθ1 α2 ) = (Dθ2 Φ1 f2∗ ), (Dθ1 α ¯ 2 ) = (Dθ2 Φ∗1 f2 ).

(5.7)

In the above, (∇γ2 (l))+ := ∂ 2 + w0 + w1 Dθ1 + w3 Dθ1 + w2 Dθ1 Dθ1 for some functions w0 , w2 ∈ C ∞ (S × Λ1 ; C1|0 ) and w1 , w3 ∈ C ∞ (S × Λ1 ; C0|1 ) that depend parametrically on the variables τ, T ∈ R. The Lax-type flow d/dt2 associated with the element (5.1) was first constructed in [25]. Together, the systems (5.2)–(5.7) represent a Lax integrable (2|2 + 1)-dimensional nonlinear dynamical system with an infinite sequence of local conservation laws in the form (2.7). Its Lax-type linearization is given by the spectral problem (2.9) and the following evolution equations for an arbitrary eigenfunction f ∈ W 1|0 or f ∈ W 0|1 : fτ = −M11 f,

(5.8)

fT = ((∇γ2 (l))+ − M12 )f, .

(5.9)

The relations (5.8) and (5.9) give rise to the additional nonlinear constraints: w0,τ = 2w1 (f1 (Dθ2 f1∗ ) − Φ1 (Dθ2 Φ∗1 )) − 2w3 (f1 (Dθ1 f1∗ ) − Φ1 (Dθ1 Φ∗1 )) − 2w2 ((Dθ2 f1 )(Dθ2 f1∗ ) − (Dθ2 Φ1 )(Dθ2 Φ∗1 )) ∗ ) + Φ1 Φ∗1,x )) + (−w2,x − (Dθ1 w1 ) − 2w2 (f1 f1,x

+ (Dθ1 w1 ))(f1 f1∗ + Φ1 Φ∗1 ) + 2(f1 (Dθ1 Dθ2 f1∗ ) + Φ1 (Dθ1 Dθ2 Φ∗1 ))x , w2,τ = 2(f1 f1∗ + Φ1 Φ∗1 )x , w1,τ = (−2w3 + Dθ1 )(f1 f1∗ + Φ1 Φ∗1 ) − w2 ((Dθ1 f1 )f1∗ + (Dθ1 Φ1 ))Φ∗1 ) + w2 (f1 (Dθ1 f1∗ ) − Φ1 (Dθ1 Φ∗1 )) − 2(f1 (Dθ2 f1∗ ) − Φ1 (Dθ2 Φ∗1 ))x , w3,τ = (2w1 + (Dθ2 w2 ))(f1 f1∗ + Φ1 Φ∗1 ) − w2 ((Dθ2 f1 )f1∗ + (Dθ2 Φ1 ))Φ∗1 ) + w2 (f1 (Dθ2 f1∗ ) − Φ1 (Dθ2 Φ∗1 )) + 2(f1 (Dθ1 f1∗ ) − Φ1 (Dθ1 Φ∗1 ))x .

(5.10)


377

When f1 := ψ, f1∗ := −θ1 θ2 ψ ∗ , f2 = f2∗ = 0, and Φ1 = Φ∗1 = Φ2 = Φ∗2 = 0, the equations (5.2), (5.4), (5.6), and (5.10) are reduced to the Lax integrable (2 + 1)dimensional Davey-Stewartson system [26, 3]: ψT = ψxx + ψτ τ + 2(S − 2ψψ ∗ )ψ, ∗ − ψτ∗τ − 2(S − 2ψψ ∗ )ψ ∗ , ψT∗ = −ψxx

Sxτ = (∂/∂x + ∂/∂τ )2 ψψ ∗ , 0 0 + 4ψψ ∗ , w0 := w00 , v1,τ := v1,τ , and ψ, ψ ∗ ∈ L∞ (S1 ; C). where 2S := w00 + 2v1,τ Therefore, the Lax integrable (2|2 + 1)-dimensional supersymmetric generalization of the Davey-Stewartson system [3, 26] is obtained above by using additional symmetry hierarchies for the Lax integrable (1|2 + 1)-dimensional nonlinear dynamical systems associated with the Lie algebra of super-integro-differential operators of two anticommuting variables. The method of additional symmetries, developed above, is efficient in constructing a wide class of (2|2+1)-dimensional supersymmetric nonlinear dynamical systems with triple Lax-type linearization. The latter makes it possible to find soliton type solutions of constructed systems via the binary Darboux-Backlund transformations [27, 28].

6. Conclusion The above-constructed Lie-Backlund transformation (3.8) on the dual space G ∗ to the Lie algebra G of super-integro-differential operators of two anticommuting variables allows one to establish that the coupled dynamical systems (2.8), (2.12) and (2.13) on the extended phase space G ∗ ⊕W 2N |2N are Hamiltonian with respect to the Poisson structure obtained from the tensor product of two canonical Poisson structures and the corresponding Casimir functionals are their Hamiltonians. By means of the Lie-Backlund transformation (3.8), it is shown that the obtained Poisson structure and natural powers of suitable eigenvalues of associated spectral and adjoint spectral problems generate a set of additional symmetry hierarchies for the dynamical systems (2.8), (2.12) and (2.13). It should be noted that the structure of the Lie-Backlund transformation (3.8) strongly depends on the ad-invariant scalar product chosen for an operator Lie algebra G and on the Lie algebra decomposition [10]. Since there exist other possibilities of choosing ad-invariant scalar products on G, such decompositions naturally give rise to other Lie-Backlund transformations. A new method for constructing integrable (2|2 + 1)-dimensional dynamical systems with triple Lax-type linearizations arising as Hamiltonian flows on the extended phase space G ∗ ⊕ W 2N |2N is represented. Due to the triple Lax-type linearizations, their soliton type solutions can be found by means of the DarbouxBacklund transformations [27, 28]. For the multi-dimensional supersymmetric dynamical systems of this type, the reduction procedure [3, 5, 29] upon nonlocal invariant subspaces can be developed as well.

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O.Ye. Hentosh

Acknowledgment The author would like to thank Prof. A.K. Prykarpatsky for useful discussions.

References [1] M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structures of a Korteweg-de Vries Equation. Invent. Math. 50 (1979), 219–248. [2] P.D. Lax, Periodic solutions of the KdV equation. Commun. Pure and Appl. Math. 28 (1975), 141–188. [3] M. Blaszak, Multi-Hamiltonian theory of dynamical systems, Springer Verlag, BerlinHeidelberg, 1998. [4] L.D. Faddeev, L.A. Takhtadjan, Hamiltonian methods in the theory of solitons. Springer, New York-Berlin-Heidelberg, 1987. [5] O. Hentosh, M. Prytula, A. Prykarpatsky, Differential-geometric and Lie-algebraic foundations of studying integrable nonlinear dynamical systems on functional manifolds. Lviv National University, Lviv, 2006. (In Ukrainian) [6] S.P. Novikov (ed.), Soliton theory: method of the inverse problem. Nauka, Moscow, 1980. (In Russian) [7] A.K. Prykarpatsky, I.V. Mykytiuk, Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects. Kluwer Academic Publishers, Dordrecht-Boston-London, 1998. [8] B.A. Kupershmidt, Elements of superintegrable systems. Basic technique and results. D. Reidel Publishing Company, Dordrecht, 1987. [9] Yu.I. Manin, A.O. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys. 28 (1985), 65–77. [10] W. Oevel, Z. Popowicz, The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems. Commun. Math. Phys. 139 (1991), 441–460. [11] Feng Yu, Bi-Hamiltonian structure of super KP hierarchy. J. Math. Phys. 33 (1992), 3180–3189. [12] W. Oevel, R-Structures, Yang-Baxter equations and related involution theorems. J. Math. Phys. 30 (1989), 1140–1149. [13] M.A. Semenov-Tian-Shansky, What is the R-matrix. Funct. Analysis and its Appl. 17 (1983), no. 4, 17–33. (In Russian) [14] O.Ye. Hentosh, Lax integrable supersymmetric hierarchies on extended phase spaces. Symmetry, Integrability and Geometry: Methods and Applications 2 (2006), 11 pp.; nlin.SI/0601007. [15] W. Oevel, W. Strampp, Constrained KP hierarchy and bi-Hamiltonian structures. Commun. Math. Phys. 157 (1993), 51–81. [16] A.K. Prykarpatsky, O.Ye. Hentosh, The Lie-algebraic structure of (2+1)-dimensional Lax type integrable nonlinear dynamical systems. Ukrainian Math. J.56 (2004), 939– 946.


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[17] Y.A. Prykarpatsky, The structure of integrable Lax type flows on nonlocal manifolds: dynamical systems with sources. Math. Methods and Phys.-Mech. Fields 40 (1997), no. 4, 106–115. (In Ukrainian) [18] O.Ye. Hentosh, A.K. Prykarpatsky, Integrable three-dimensional coupled nonlinear dynamical systems related with centrally extended operator Lie algebras. Opuscula Mathematica 27 (2007), no. 2, 231–244. [19] H. Aratyn, E. Nissimov, S. Pacheva, Supersymmetric KP hierarchy: “ghost” symmetry structure, reductions and Darboux-Backlund solutions. J. Math. Phys. 40 (1999), 2922–2933. [20] E. Nissimov, S. Pacheva, Symmetries of supersymmetric integrable hierarchies of KP type. J. Math. Phys. 43 (2002), 2547–2586. [21] A.O. Radul, Lie algebras of differential operators, their central extensions and W algebras. Funct. Analysis and its Appl., 25 (1991), no. 1, 33–49. (In Russian) [22] F.A. Berezin, Introduction to algebra and analysis with anticommuting variables. Moscow University Publisher, Moscow, 1983. (In Russian) [23] V.N. Shander, Analogues of the Frobenius and Darboux theorems. Reports of Bulgarian Academy of Sciences 36 (1983), 309–311. [24] V.N. Shander, About the complete integrability of ordinary differential equations on supermanifolds. Funct. Analysis and its Appl. 17 (1983), no. 1, 89–90. (In Russian) [25] Z. Popowicz, The extended supersymmetrization of the multicomponent KadomtsevPetviashvili hierarchy. J. Phys. A: Math. Gen. 29 (1996), 1281–1291. [26] M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform. SIAM, Philadelphia, 1981. [27] V.B. Matveev, M.I. Salle, Darboux-Backlund transformations and applications. Springer, New York, 1993. [28] A.M. Samoilenko, A.K. Prykarpatsky, Y.A. Prykarpatsky, The spectral and differential-geometric aspects of a generalized de Rham-Hodge theory related with Delsarte transmutation operators in multi-dimension and its applications to spectral and soliton problems. Nonlinear Analysis 65 (2006), 395–432. [29] O.E. Hentosh, Hamiltonian finite-dimensional oscillator-type reductions of Lax integrable superconfomal hierarchies. Nonlinear Oscillations 9 (2006), no. 1, 13–27. Oksana Ye. Hentosh 3B Naukova St. 79060 Lviv, Ukraine e-mail: [email protected]



On the Modified Spectral Stefan Problem and Its Abstract Generalizations N.D. Kopachevsky and V.I. Voytitsky The paper is dedicated to brothers Mark and Selim Kreins

Abstract. The aim of this work is to study the properties of the multi-component spectral problem generated by the linearized modified Stefan problem. On the basis of the abstract Green’s formula for a triple of Hilbert spaces, proved by N.Kopachevsky and S.Krein, an abstract generalization of the spectral problem is considered. Studying auxiliary abstract boundary value problems and properties of the corresponding operators, we prove that the spectrum consists of real normal eigenvalues and that the system of eigenelements forms an orthonormal basis in some Hilbert space. Mathematics Subject Classification (2000). Primary 35P05; Secondary 35P10. Keywords. Spectral problem, abstract Green’s formula, Hilbert space, embedding of spaces, compact self-adjoint operator, normal eigenvalues, orthonormal basis.

1. Introduction The Stefan problem is a mathematical model for description of phase transitions from one aggregate state to another. The problem belongs to a class of nonlinear boundary value problems where the temperature of the substance and the form of the unknown dynamic interphase boundary are to be found. In the classical statement of the Stefan problem, the boundary is determined by the Stefan condition and the fact that the temperature of the substance is equal to the melting temperature (see, e.g., [1]). In the late 80ies of the last century, a more precise statement of the mathematical model for phase transitions was considered in a number of works (see, e.g., [2]–[4]). The kinetic, or so-called Hibbs-Thomson, law for the interphase boundary was used. The corresponding nonlinear problem came to be called the modified Stefan problem.

382

N.D. Kopachevsky and V.I. Voytitsky

In nonlinear problems, the method of reduction of the problem to the study of model initial-boundary value problems is usually used. At the same time, the properties of a nonlinear boundary value problem are essentially determined by the properties of solutions of the model problems. This fact stimulates us to consider new linear spectral problems generated by the modified Stefan problem. In this paper, we study the abstract spectral problem generated by the modified Stefan problem with the Hibbs-Thomson conditions from [5] (see also [6] and [7]). Using the abstract Green’s formula for a mixed boundary value problem, proved by N. Kopachevsky (see [8], [9]), and studying the operators of auxiliary boundary value problems, we prove the basis property of the system of eigenelements and some properties of the spectrum. More precisely, we show that: the spectrum consists of real normal eigenvalues; there exists a branch of positive eigenvalues with a unique limit point +∞, a finite or an infinite number of negative eigenvalues and, possibly, zero-eigenvalue. The present paper continues the investigations [10], [11] and [9].

2. On the modified and model Stefan problems In [5] (see also [6]), the following statement of the modified Stefan problem is considered. It is necessary to find the unknown fields of temperature uj (x, t) (j = 1, 2), given in time-varying domains Ωj (t) ⊂ Rm (Ω1 (t) ⊂ Ω2 (t)), and the unknown interphase boundary Γ(t) if these functions satisfy the equations ∂uj − ∇(Aj (x, t)∇uj ) = fj ∂t the boundary conditions

(in Ωj (t)),

l0 Vn = A2 ∇u2 · nt − A1 ∇u1 · nt

j = 1, 2,

(on Γ(t));

(2.1)

(2.2)

uj = −σ(x, t)K(x, t) − (x, t)σ(x, t)Vn

(on Γ(t)),

u2 = g

(on ∂Ω2 (t) \ Γ(t)),

j = 1, 2;

(2.3) (2.4)

and the initial values uj |t=0 = ϕ0j

(in Ωj (0)), j = 1, 2.

(2.5)

Here: l0 > 0 is the latent heat of crystallization; nt is the unit normal to Γ(t) (outward to the Ω1 (t)); Vn is the velocity of Γ(t) in the direction of nt ; σ(x, t) > 0 is the surface tension; K(x, t) is the mean curvature of Γ(t) at point (x, t); (x, t) > 0 or ≡ 0 is a relaxation parameter. We also assume that the coefficients ajkl (x, t) of the matrices Aj (x, t) satisfy the condition of ellipticity: 0 < c0j ≤

m

ajkl (x, t)ξk ξl ≤ c−1 0j ,

t ∈ [0, T0 ], ξ ∈ Rm , |ξ| = 1,

j = 1, 2.

k,l=1

(2.6) In the above statement, (2.2) is the Stefan condition, and (2.3) is determined by the Hibbs-Thomson law.

On the Modified Spectral Stefan Problem

383

By using some additional conditions, it is proved in [5] that problem (2.1)– (2.5) has a unique classical solution on a sufficiently small [0, T ], T < T0 . This fact is established by reducing the problem to answering the question whether the model linear problem ∂uj − aj (P )Δuj = 0 ∂t 2 ∂ζ ∂uj + (−1)j aj (P ) = g1 l0 ∂t j=1 ∂n1 ∂ζ = g2 ∂t ζ|t=0 = 0,

uj − σ(P )ΔΓ0 ζ + β(P )ζ + α(P ) uj |t=0 = 0,

(in Rjm ),

j = 1, 2;

(2.7)

(on Γ0 );

(2.8)

(on Γ0 );

(2.9) (2.10)

has a unique solution for each point P ∈ Γ(0). In [5], problem (2.7)–(2.10) is considered in the half-spaces Rjm := {x ∈ Rm : (−1)j xm > 0}, j = 1, 2, divided by the hyperplane Γ0 := {x ∈ Rm : xm = 0}. Here, besides the functions uj (t, x), there is a new unknown function ζ(t, x), x ∈ Γ0 , in the boundary conditions. It determines small movements of Γ(0). For each P ∈ Γ(0), the values aj (P ) and σ(P ) are positive constants, the constant α(P ) := (P )σ(P ) ≥ 0, and β(P ) :=

m−1 ∂ 2 ϕ0j (P ) − σ(P )K0k (P ) ∂n0

(2.11)

k=1

is of R. In (2.11), K0k (P ), k = 1, . . . , m − 1, are the principal curvatures of Γ(0) at the point P . In [5] (see also [6]), it is proved that in the case β(P ) > 0 for any P ∈ Γ(0), problem (2.7)–(2.10) is a well-posed Cauchy problem. The properties of corresponding spectral problem, considered in this paper, support this assertion.

3. Statement of the multi-component modified spectral Stefan problem Let us consider problem (2.7)–(2.10) for a set of bounded jointed domains Ωj ∈ Rm (j = 1, q := 1, . . . , q) with Lipshitz boundaries Γj := ∂Ωj . We also assume that aj (P ) ≡ σ(P ) ≡ 1, l0 = 1. Denote by Γjj (j = 1, q) the free part of Γj , and by Γjk (k = 1, q) that part of the boundary of Ωj , that is jointed with the boundary of Ωk (k = j). Obviously, Γjk = Γkj . If Ωj and Ωk are not jointed, then we assume that mes Γjk = 0. Suppose that Γjk is a smooth (m − 1)-dimensional measurable manifold with boundary.

384

N.D. Kopachevsky and V.I. Voytitsky So, we can formulate the following linear multi-component transmission prob-

lem: ∂uj − Δuj = fj ∂t ∂uj ∂ζjk ∂uk − − = hjk ∂t ∂njk ∂nkj γjk uj = γkj uk ∂ζjk = gjk ∂t γjj uj = 0

γjk uj − ΔΓjk ζjk + βjk ζjk + αjk

ζjk = 0, uj (0) =

u0j

(in Ωj ),

ζjk (0) =

0 ζjk

(in Ωj );

(3.1)

(on Γjk );

(3.2)

(on Γjk );

(3.3)

(on Γjk );

(3.4)

(on Γjj );

(3.5)

(on ∂Γjk );

(3.6)

(on Γjk ).

(3.7)

Here: uj (x, t), x ∈ Ωj , and ζjk (x, t), x ∈ Γjk , are unknown functions; γjk uj := uj Γ , k > j; jk

(3.8)

∂/∂njk are normal derivatives; ΔΓjk are the Laplace-Beltrami operators acting in the smooth manifolds Γjk ; αjk ≥ 0 and βjk ≥ 0 are given constants; u0j = 0 0 = ζjk (x), x ∈ Γjk , are given functions. We also suppose u0j (x), x ∈ Ωj , and ζjk that mes Γjj > 0. Let us find normal solutions to the homogeneous problem corresponding to (3.1)–(3.7): uj (x, t) = uj (x)e−λt , ζjk (x, t) = ζjk (x)e

−λt

x ∈ Ωj ; , x ∈ Γjk .

(3.9) (3.10)

We get the problem −Δuj = λuj ∂uk ∂uj + = −λζjk ∂njk ∂nkj γjk uj = γkj uk γjk uj + Bjk ζjk = λαjk ζjk γjj uj = 0

(in Ωj );

(3.11)

(on Γjk );

(3.12)

(on Γjk );

(3.13)

(on Γjk );

(3.14)

(on Γjj ).

(3.15)

We will call this problem the multi-component modified spectral Stefan problem. Here Bjk are linear self-adjoint operators acting in L2 (Γjk ): Bjk ζjk := −ΔΓjk ζjk + βjk ζjk ,

D(Bjk ) = {ζjk (x) ∈ H 2 (Γjk ) : ζjk |∂Γjk = 0}. (3.16) It is known that the spectrum of any Bjk consists of real normal eigenvalues with a limit point +∞. For any βjk ∈ R, the operator Bjk is bounded below. It can have no more than a finite number of non-positive eigenvalues (taking into account their multiplicities).


385

4. Statement of the abstract spectral problem Let us consider the abstract generalization of problem (3.11)–(3.15) on the basis of the abstract Green’s formula for mixed boundary value problems (see [8] and [9]). So, let Ej , Fj and Gj , j = 1, q, be given Hilbert spaces satisfying the following conditions: (i) any Fj is boundedly imbedded into Ej (notation: Fj ⊂→ Ej ); (ii) for every Fj , there exists an abstract trace operator γj : Fj → Gj such that R(γj ) =: (G+ )j ⊂→ Gj . Then it is proved in [12]–[16] (see also [17], [18]) that there exist the unique operators Lj : D(Lj ) = Fj → Fj∗ ⊃ Ej ; ∂j : D(∂j ) = Fj →

(G+ )∗j

(4.1)

=: (G− )j ⊃ Gj ,

(4.2)

such that the abstract Green’s formula holds: ηj , Lj uj Ej = (ηj , uj )Fj − γj ηj , ∂j uj Gj ,

∀ηj , uj ∈ Fj , j = 1, q.

(4.3)

Further constructions are taken from articles [8] and [9]. Assume that there exist resolutions Gj =

q Q

Gjk ,

(G+ )j =

k=1

q

˙ (+)(G + )jk ,

j = 1, q,

(4.4)

k=1

where (G+ )jk ⊂→ Gjk ⊂→ (G+ )∗jk , ∀j, k = 1, q, and (G+ )jk = (G+ )kj , Gjk = Gkj , (G+ )∗jk = (G+ )∗kj . Let ρjk : (G+ )j → (G+ )jk and ωjk : (G+ )jk → (G+ )j be the abstract restriction operators and the operators of extension by zero respectively, Ijk := ρjk ωjk being the identity operators in (G+ )jk . Then any pjk := ωjk ρjk : (G+ )j → (G+ )j is a projection in (G+ )j . Suppose that pjk are continuous projections; then (4.3) implies ηj , Lj uj Ej = (ηj , uj )Fj −

q

γjk ηj , ∂jk uj Gjk ,

∀ηj , uj ∈ Fj , j = 1, q, (4.5)

k=1

where γjk := ρjk γj : Fj → (G+ )jk ,

∗ ∂jk := ωjk ∂j : Fj → (G+ )∗jk .

(4.6)

Here γjk are the abstract bounded operators that generalize the bounded trace operators (3.8) acting on a part of the boundary, and ∂jk generalize the normal derivatives defined on a part of the boundary.

386


According to the previous constructions, we can formulate the following abstract spectral transmission problem: Lj uj = λuj ∂jk uj + ∂kj uk = (−λ)ζjk γjk uj = γkj uk γjk uj + Vjk ζjk = λRjk ζjk γjj uj = 0

(in Ej );

(4.7)

(in Gjk );

(4.8)

(in Gjk );

(4.9)

(in Gjk );

(4.10)

(in Gjj ).

(4.11)

∗ ∈ We suppose here that: uj ∈ Fj and ζjk ∈ Gjk are unknown elements; Rjk = Rjk L(Gjk ), Rjk ≥ 0; the spectrum of every self-adjoint operator Vjk , acting in Gjk , is discrete, i.e., consists of normal eigenvalues. Let us denote by dVjk < ∞ the dimension of Ker Vjk and by κVjk , 0 ≤ κVjk ≤ ∞, the number (taking into account their multiplicities) of negative eigenvalues of Vjk . We will refer to (4.7)–(4.11) as the abstract modified spectral Stefan problem. Note that problem (3.11)–(3.15) is a special case of problem (4.7)–(4.11), where Mq Ej = L2(Ωj ), Fj = H 1 (Ωj ) ⊂→ L2 (Ωj ), Gj = L2 (Γj ) = k=1 L2 (Γjk ) and γjk u = uΓ , γjk : Fj → (G+ )jk = H 1/2 (Γjk ) ⊂→ L2 (Γjk ). jk In this case, the restriction operators ρjk ϕj := ϕj Γ , ∀ϕj ∈ H 1/2 (Γj ), (4.12) jk

associate any function ϕj ∈ H (Γj ) with its part ϕjk , defined on Γjk ⊂ Γj . The operators of extension by zero ωjk : H 1/2 (Γjk ) → H 1/2 (Γj ) are defined as follows: x ∈ Γjk , ϕjk , ωjk ϕjk := (4.13) 0, x ∈ Γj \ Γjk . 1/2

Definitions (4.12) and (4.13) imply that ρjk ωjk are identities in H 1/2 (Γjk ). Hence, the operators pjk = ωjk ρjk : H 1/2 (Γj ) → H 1/2 (Γj ) (4.14) have the property p2jk = pjk , i.e., they are projections. It is proved in [8] and [9] that these operators pjk are bounded projections in H 1/2 (Γjk ) since ωjk and ρjk are bounded (under some additional assumptions on boundaries Γj ). Introduce the Hilbert spaces E :=

q Q j=1

Ej ,

F :=

q Q j=1

Fj ,

G :=

q

⊕Gjk

(4.15)

j=1 k>j

with corresponding scalar products. Let F0 be the subspace of all u ∈ F that satisfy the main (in the sense of calculus of variations) boundary conditions: , F0 := u = (u1 , . . . , uq ) ∈ F : γjk uj = γkj uk , γjj uj = 0, j = 1, q . (4.16) Since any γjk : Fj → (G+ )jk is bounded, F0 is a subspace of F . We assume that · F0 := · F .


387

Introduce these operators, acting on u ∈ F0 : Lu := (L1 u1 , . . . , Lq uq ) ∈ F0∗ ;

(4.17)

q ˙ γu := γjk uj j=1,q,k>j ∈ G+ := (+)(G + )jk ;

(4.18)

j=1 k>j q ∗ ˙ (+)(G ∂u := ∂jk uj + ∂kj uk j=1,q,k>j ∈ (G+ )∗ := + )jk .

(4.19)

j=1 k>j

Assume also that N := Ker γ ⊂ F0 and M := F0 ( N are infinite-dimensional subspaces of the space F0 . Further constructions are based on the assumption that N is dense in E (N = E) with respect to the norm of E. This property holds for many problems of mathematical physics. For example, if q = 1, F = H 1 (Ω), E = L2 (Ω), and γ is a trace operator on Γ = ∂Ω, Ω ⊂ Rm , then N := Ker γ = H01 (Ω),

H01 (Ω) = L2 (Ω).

(4.20)

Since for all j, k = 1, q we have Fj ⊂→ Ej , (G+ )jk ⊂→ Gjk ; F0 ⊃ N and N is dense in E, then F0 ⊂→ E and G+ ⊂→ G. Obviously, the operator γ : F0 → G+ is bounded, so, there exists some abstract Green’s formula corresponding to the triple of Hilbert spaces E, F0 , G and the abstract trace operator γ. This formula will be contain the operators L and ∂, given in (4.17) and (4.19) respectively. Indeed, summing up the Green’s formulas (4.5) over j from 1 to q, we obtain q

ηj , Lj uj Ej =

j=1

q

(ηj , uj )Fj −

j=1

q (γjk ηj , ∂jk uj Gjk j=1 k>j q

+ γkj ηk , ∂kj uk Gjk ) +

γjj ηj , ∂jj uj Gjj ,

∀ηj , uj ∈ Fj . (4.21)

j=1

If now u = (u1 , . . . , uq ) ∈ F0 , η = (η1 , . . . , ηq ) ∈ F0 , then (4.21) implies q j=1

ηj , Lj uj Ej =

q j=1

(ηj , uj )Fj −

q

γjk ηj , ∂jk uj + ∂kj uk Gjk ,

∀η, u ∈ F0 .

j=1 k>j

(4.22) By definitions (4.17)–(4.19) of the operators L, ∂, γ, we have an equivalent form of the Green’s formula (4.22): η, LuE = (η, u)F − γη, ∂uG ,

∀η, u ∈ F0 .

(4.23)

On the basis of the previous constructions, the abstract modified spectral Stefan problem (4.7)–(4.11) can be rewritten as Lu = λu

(in E);

(4.24)

∂u = (−λ)ζ

(in G),

(4.25)

(in G).

(4.26)

γu + V ζ = λRζ

388


Here u ∈ F0 and ζ ∈ G are unknown elements. In the resolution of G (see (4.15)), we have V := diag (Vjk )j=1,q,k>j and R := diag (Rjk )j=1,q,k>j . So, the properties of the operators V and R are similar to the corresponding properties of Vjk and Rjk . More precisely, R = R∗ ∈ L(G), R ≥ 0, and the spectrum of the self-adjoint operator V consists of real normal eigenvalues. Let us denote by κV , 0 ≤ κV ≤ ∞, the number (taking account of their multiplicities) 0 0 0 of negative eigenvalues 0of V and let dV := dim Ker V < ∞. Then dV = qj=1 k>j dVjk , κV = qj=1 k>j κVjk . We will further suppose that the operator (V GKerV )−1 exists and is compact. It should be noted that abstract Green’s formula (4.23) allows us to consider general problem (4.7)–(4.11). Then L and ∂ are any operators appearing in some abstract Green’s formula constructed by arbitrary E, F, G, and γ that satisfy (i) and (ii). Also, the operators V and R can be arbitrary ones that satisfy the above-mentioned conditions. Problem (4.24)–(4.26) (when V is sign definite and R ≡ 0) was studied earlier in [10] and [11]. In the present, work we suggest another approach to problem (4.7)–(4.11) and consider the general case.

5. The operator-matrix form of the abstract spectral problem Now, we get the operator form of problem (4.24)–(4.26). For this purpose, we consider so-called auxiliary boundary value problems of S.Krein and the operators corresponding to them (see [14], Section 1.3). So, let a solution to (4.24)–(4.26) be represented in the form u = v + w where v and w are solutions to the following problems: Lv = f (in E), ∂v = 0 (in G); (5.1) Lw = 0 (in E),

∂w = ψ (in G).

(5.2)

Here f := λu ∈ E, ψ := −λζ ∈ G. It is proved in [13] (see also [14]) that (5.1) and (5.2) have the unique solutions v = A−1 f = λA−1 u,

w = T ψ = (−λ)T ζ,

(5.3)

where A : D(A) → E is a self-adjoint positive definite operator of the Hilbert pair (F0 ; E), and T : G → M ⊂ F0 is a bounded operator determined by the identity (γw, ϕ)G = (w, T ϕ)F ,

∀w ∈ F0 , ∀ϕ ∈ G.

(5.4)

From (5.3) we have: u = v + w = λA−1 u − λT ζ.

(5.5)

Suppose further that F0 is compact embedded in E and G+ is compact embedded in G (notations: F0 ⊂→ ⊂→ E, G+ ⊂→ ⊂→ G). Then the operators A−1 : E → E and γ : F0 → G are compact. Introduce Q := γA−1/2 : E → G,

Q∗ := A1/2 T : G → E.

(5.6)


389

They are self-conjugate and compact since by (5.4) (γA−1/2 w, ϕ)G = (A−1/2 w, T ϕ)F = (A1/2 A−1/2 w, A1/2 T ϕ)E = (w, A1/2 T ϕ)E ,

∀w ∈ E, ∀ϕ ∈ G.

It follows from (5.5) that problem (4.24)–(4.26) is equivalent to the problem u = λA−1 u − λA−1/2 Q∗ ζ, γu + V ζ = λRζ.

(5.7) (5.8)

Substituting u from (5.7) into (5.8) we obtain: u = λA−1 u − λA−1/2 Q∗ ζ, −1/2

λQA

∗

u − λQQ ζ + V ζ = λRζ.

The matrix form of (5.9)–(5.10) is A−1 u I 0 =λ Bu ¯ := ζ 0 V −QA−1/2

−A−1/2 Q∗ QQ∗ + R

(5.9) (5.10)

u =: λA¯ u. ζ

(5.11)

Here u ¯ := (u; ζ)t is an element of the Hilbert space H := E ⊕ G with the corresponding norm ¯ u 2H := u 2E + ζ 2G . Obviously, the properties of the operators B and V are equivalent, i.e., the spectrum of B consists of real normal eigenvalues, with the multiplicity of the eigenvalue λ = 0 equal to dV < ∞. Lemma 5.1. The operator A is a bounded self-adjoint positive operator acting in H. Proof. The structure (5.11) of the operator-matrix A implies its self-adjointness and boundedness. Let us prove that A is non-negative and Ker A = {¯0}. Indeed, A−1 −A−1/2 Q∗ 0 −A−1/2 −1/2 ∗ + 0 = A= ≥ 0. −A Q 0 R Q −QA−1/2 QQ∗ + R ¯ then A−1 u − A−1/2 Q∗ ζ = 0 or A−1 u = T ζ =: v ∈ M. For elements If A¯ u = 0, v ∈ M , we have Lv = 0 (see (5.2)). On the other hand, v = A−1 u implies v ∈ D(A), i.e., ∂v = 0. Since problem (5.2) has a unique solution, we have v = 0. Then u = 0 and ζ = 0 (Ker T = {0}). Therefore, Ker A = {¯0}. So, problem (4.24)–(4.26) is reduced to problem (5.11), which is considered, for example, in [14], Section 1.5. Such problems usually arise in studying oscillation problems for mechanical systems with an infinite number of degrees of freedom. Consider the special case of problem (5.11) when Ker B = {¯0}. In this case, λ = 0 is an eigenvalue and 0 < dim Ker V =: dim G0 = dV < ∞. Introduce ˜ where G ˜ := G ( G0 . Denote by P˜ and P0 the corresponding ˜ := H ( G0 = E ⊕ G H

390


˜ and G0 respectively. ortho-projections from G on G (5.11) we have: ⎛ ⎞⎛ ⎞ ⎛ −1 I 0 0 u A1 A ⎝0 V˜ 0⎠ ⎝ ζ˜ ⎠ = λ ⎝ A∗1 K1 A∗2 K2∗ 0 0 0 ζ0

Since A is self-adjoint, from ⎞⎛ ⎞ u A2 ⎠ ⎝ K2 ζ˜ ⎠ . K3 ζ0

(5.12)

Here V˜ := P˜ V P˜ , ζ˜ := P˜ ζ, ζ0 := P0 ζ, A1 := −A−1/2 Q∗ P˜ , A2 := −A−1/2 Q∗ P0 , ˜ ⊕ G0 . and Ki are blocks of the operator-matrix QQ∗ + R in the resolution G = G Obviously, to λ = 0 there corresponds the orthonormal system consisting of dV < ∞ elements of the form u¯ = (0; 0; ζ0 ) ∈ Ker B, ζ0 ∈ G0 . If λ = 0, then (5.12) implies A∗2 u + K2∗ ζ˜ + K3 ζ0 = 0.

(5.13)

Since A is positive, the block K3 is a (dV × dV ) positive definite matrix. Hence, the matrix K3−1 exists and is positive definite, too. Then ˜ ζ0 = −K3−1 A∗2 u − K3−1 K2∗ ζ, and (5.12) implies I 0 u −A2 K3−1 A∗2 + A−1 ˜ B˜ u := =λ ˜ ˜ −K2 K3−1 A∗2 + A∗1 0 V ζ

(5.14)

u . ζ˜ (5.15) Here Ker V˜ = {0} and Ker B˜ = {˜ 0}. The matrix in the right-hand side of (5.15) ˜ = E ⊕ G. ˜ It is significant that the is a self-adjoint positive operator acting in H block −A2 K3−1 A∗2 + A−1 of this matrix is a compact operator since in (5.12) the block (A1 ; A2 ) = −A−1/2 Q∗ is compact. So, in the case Ker B = {¯ 0} problem (5.11) can be reduced to (5.15) defined ˜ ⊂ H. This problem has the same form as (5.11), but now all on the subspace H the operators have trivial kernels. −A2 K3−1 K2∗ + A1 −K2 K3−1 K2∗ + K1

6. The main result Consider the I B˜u˜ = 0

general problem (see (5.11)) u u 0 A11 A12 ˜u, =: λA˜ = λ A21 A22 ζ˜ V˜ ζ˜

where Ker B˜ = {˜ 0} and A˜ is a positive bounded Ker B = {0}, we can omit all “tildes” in (6.1)). Let V˜ = J|V˜ | where J = J ∗ = J −1 and positive definite. Then they commute with J therefore ˜ 1/2 J |B| ˜ 1/2 , J = J ∗ = J −1 = I B˜ = |B| 0

˜ = E ⊕ G, ˜ u ˜∈H

(6.1)

operator having A11 ∈ S∞ (E) (if the operators |V˜ | and |V˜ |1/2 are (see, e.g., [14], Section 1.5) and 0 , J

˜ = |B|

I 0

0 . |V˜ |

(6.2)


391

˜ and J have the same number of By their constructions, the operators V˜ , J, B, negative eigenvalues. Taking into account their multiplicities, it is equal to κV . Making the substitutions ˜ η := |V˜ |1/2 ζ˜ ∈ G, ˜ 1/2 u ˜ u ˆ := (u; η)t = |B| ˜ ∈ H, we can transform problem (6.1) to the problem u A12 |V˜ |−1/2 A11 = λC u ˆ, Ju ˆ = λ ˜ −1/2 η |V | A21 |V˜ |−1/2 A22 |V˜ |−1/2

(6.3) (6.4)

˜ −1/2 A| ˜ B| ˜ −1/2 . C := |B| (6.5)

˜ Lemma 6.1. Operator C is a self-adjoint positive compact operator acting in H. Proof. The self-adjointness of C is evident. Let us prove its compactness. By the assumption, the operator (V GKerV )−1 exists and is compact. So, |V˜ |−1/2 is compact. Also, we have A11 ∈ S∞ (E). Therefore, the operator-matrix C consists of only compact operators, i.e., it itself is a compact operator. Since λ = 0 is not an eigenvalue of (6.1), from (6.5) we have: J Cu ˆ = μˆ u,

μ := 1/λ.

(6.6)

Problem (6.6) is an eigenvalue problem for a positive compact or J -positive compact operator acting (subject to J ) in the Hilbert (κV = 0, J = I), or Pontryagin ˜ J := E ⊕ G ˜ with the metrics (κV < ∞), or Krein (κV = ∞) space H [ˆ u, vˆ] := (J u ˆ, vˆ)H˜ .

(6.7)

˜ J is a Hilbert space, we can apply the theorem of Hilbert-Schmidt. Otherwise, If H we can apply the theorems concerning the properties of J -self-adjoint compact operators acting in J -space (see, e.g., [19]). So, we can prove the discreteness and reality of the spectrum and the basis property of eigenelements in some J -space. But in the particular problem (6.6) we always have the basis property of eigenelements in some Hilbert space. Indeed, Theorem 6.1 implies that the operator C 1/2 exists and is positive and compact. Substituting ˜ ˆ ∈ H, ηˆ := C 1/2 u

(6.8)

into (6.5) and applying the operator C 1/2 to the left-hand side of (6.5), we have η, Kˆ η := C 1/2 J C 1/2 ηˆ = μˆ

μ := 1/λ.

(6.9)

Obviously, the operator K is a compact self-adjoint operator acting in the Hilbert ˜ Ker K = {0}. The property R(C) = H ˜ and (6.9) imply that the form space H, (Kˆ η , ηˆ)H˜ takes positive values on the subspace of infinite dimension and negative values on the subspace of dimension κV . So, the theorem of Hilbert-Schmidt implies that the system of eigenelements ˜ of K forms an orthonormal basis {ˆ ηn }∞ n=1 in H and the spectrum consists of positive

392


− κV normal eigenvalues μ+ k → +0 (k → ∞) and κV negative eigenvalues {μk }k=1 , − where κV = ∞ implies μk → −0 (k → ∞). Making the inverse substitutions into (6.8) and (6.4), we obtain the following result.

Theorem 6.2 (on spectral properties). Problem (6.1) has a system of eigenelements ˜ −1/2 u ˜ −1/2 C −1/2 ηˆn that is complete in Hilbert space u ñ := (un ; ζñ )t = |B| ˆn = |B| ˜ ˜ = H ( G0 , G0 = Ker V. Here, {ˆ ηn }∞ H n=1 is an orthonormal basis in H consisting of eigenelements of K. The system {˜ un }∞ also forms an orthonormal basis in n=1 ˜ For these elements the identities ˜ ˜ of the operator A. the energetic space H A ˜un , u ˜m )H˜ = (C u ˆn , u ˆm )H˜ = (ˆ ηn , ηˆm )H˜ = δnm (˜ un , u˜m )H˜ ˜ = (A˜ A

(6.10)

hold. The spectrum of problem (6.1) ((5.11) and (4.24)–(4.26)) is real. It consists of normal eigenvalues. If dV = dim Ker V = 0, then the spectrum of (6.1) consists of positive + ∞ eigenvalues {λ+ k = 1/μk }k=1 with a limit point +∞ and κV negative eigenval− − κV ues {λk = 1/μk }k=1 , where κV = ∞ implies λ− k → −∞ (k → ∞). In this case, ∞ ∞ {˜ u n }∞ = {¯ u } = {(u ; ζ )} is a system of eigenelements of problems n n=1 n n n=1 n=1 ˜ (5.11) and (4.24)–(4.26) in the space H = H = E ⊕ G. If 0 < dV < ∞, then λ = 0 is an eigenvalue of finite multiplicity dV . The ˜ system {(˜ un ; 0)}∞ n=1 of eigenelements in H = H⊕G0 can be completed to a complete system in H by a finite number of elements (0; ζ0n ), ζ0n ∈ G0 , corresponding to λ = 0 ({ζ0n } forms an orthonormal basis in G0 ). Theorem 6.2 can be applied to abstract modified spectral Stefan problem 0the0 (4.7)–(4.11). In the resolution G = qj=1 k>j ⊕Gjk we have V = diag (Vjk )j=1,q,k>j , ⎛ ⎞ ⎛ ⎞ q q H=E⊕G=⎝ ⊕Ej ⎠ ⊕ ⎝ ⊕Gjk ⎠ . (6.11) 0q

j=1

0q

j=1 k>j

0 Obviously, dV = j=1 k>j dVjk and κV = j=1 k>j κVjk . The multi-component modified spectral Stefan problem (3.11)–(3.15) is also a special case of (4.7)–(4.11). So, it has a complete system of eigenelements in the space ⎛ ⎞ ⎛ ⎞ q q H=E⊕G=⎝ ⊕L2 (Ωj )⎠ ⊕ ⎝ ⊕L2 (Γjk )⎠ (6.12) 0q

0

0

j=1

j=1 k>j

˜ for dV = k>j dBjk = 0, V = diag (Bjk )j=1,q,k>j , or in the space H = j=1 H ( Ker V if dV > 0. Since Bjk = −ΔΓjk + βjk I, problem (3.11)–(3.15) has no negative eigenvalues iff ∀j, k = 1, q : βjk ≥ −λ1 (−ΔΓjk ). Otherwise, it has (taking 0 0 into account their multiplicities) κV = qj=1 k>j κBjk negative eigenvalues. The presence of negative eigenvalues corresponds to instability in the process of phase transfer. It means that the corresponding initial-boundary value problem is not a well-posed Cauchy problem. So, the spectral results that were obtained


393

here allow us to make the results from [5], where it was proved that the modified Stefan problem is solvable whenever every βjk > 0, more precise.

References [1] Meyrmanov A. M., Stefan Problem. Novosibirsk, “Nauka”, 1986. (In Russian) [2] Caroli B., Caroli C., Misbah C., Roulet B., The Hibbs-Thomson Law. J. Phys. 48, 1987. [3] Xie W., Stefan Problem with a Kinetic Condition at the Free Boundary. Ann. mat. pura ed appl. 21 (1990), no. 2, 362–373. [4] Luckhaus S., The Stefan Problem with Hibbs-Thomson Law. Sezione di Analisi Matematica e Probabilit` a, Universit` a di Pisa, 2(591), no. 75, 1991. [5] Radkevich E.V., On the Existence Conditions of Classical Solution of the Modified Stefan Problem (the Hibbs-Tomson Law). Math. Sbornik. 183 (1992), no. 2, 77–101. (In Russian) [6] Radkevich E.V., Modifications of Hibbs-Thomson and the Existence of Classical Solution of the Modified Stefan Problem. Proc. of USSR’s Academy of Sciences 315 (1990), no. 6, 1311–1315. (In Russian) [7] Basaliy B.V., Degtyar’ov S.P., On the Stefan Problem with Kinetic and Classical Condition on a Free Boundary. Ukr. Math. Journal 44 (1992), no. 2, 155–166. (In Russian) [8] Kopachevsky N.D., The Abstract Green’s Formula for Mixed Boundary Problems, Scientific Notes of Taurida National University. Series “Mathematics. Mechanics. Informatics and Cybernetics” 20(59) (2007), no. 2, 3–12. (In Russian) [9] Kopachevsky N.D., Starkov P.A., Voytitsky V.I., Multicomponent Transmission Problems and Auxiliary Abstract Boundary Value Problems. Modern Math. Fundamental Directions, 2009. (To appear, in Russian) [10] Voytitsky V.I., The Abstract Spectral Stefan Problem. Scientific Notes of Taurida National University. Series “Mathematics. Mechanics. Informatics and Cybernetics” 19(58) (2006), no. 2, 20–28. (In Russian) [11] Voytitsky V.I., On the Spectral Problems Generated by the Linearized Stefan Problem with Hibbs-Thomson Law. Nonlinear Boundary Value Problems 17 (2007), 31–49. (In Russian) [12] Kopachevsky N.D., On the Abstract Green’s Formula for a Triple Hilbert spaces and its Applications to Stokes Problem. Taurida Bulletin of Inform. and Math. 2 (2004), 52–80. (In Russian) [13] Kopachevsky N.D., Krein S.G., The Abstract Green’s Formula for a Triple of Hilbert Spaces, Abstract Boundary Value and Spectral Abstract Problems. Ukr. Math. Bulletin, 1 (2004), no. 1, 69–97. (In Russian) [14] Kopachevsky N.D., Krein S.G., Ngo Zuy Kan, Operator Methods in Linear Problems of Hydrodynamics. Evolution and Spectral Problems. Moscow, “Nauka”, 1989. (In Russian)

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[15] Kopachevsky N.D., Krein S.G., Operator Approach to Linear Problems of Hydrodynamics. Vol. 1: Self-adjoint Problems for an Ideal Fluid. Birkh¨ auser Verlag, Basel, Boston, Berlin, 2001. (Operator Theory: Advances and Applications, Vol. 128.) [16] Kopachevsky, N.D., Krein, S.G., Operator Approach to Linear Problems of Hydrodynamics. Vol. 2: Nonselfadjoint Problems for Viscous Fluid. Birkh¨ auser Verlag, Basel, Boston, Berlin, 2003. (Operator Theory: Advances and Applications, Vol. 146.) [17] Aubin J.-P., Approximate Solution of Elliptic Boundary Value Problems. Moscow, “Mir”, 1977. (In Russian) [18] Showalter R., Hilbert Space Methods for Partial Differential Equations. Electronic Journal of Differential Equations, 1994. [19] Azizov T.Ya., Iohvidov I.S. Principles of Linear Operator Theory in Spaces with an Indefinite Metric. Moscow, “Nauka”, 1986. (In Russian) N.D. Kopachevsky 28 Prospekt Pobedy, Apt. 16 95034 Simferopol, Ukraine e-mail: [email protected] V.I. Voytitsky 9 Dm. Ulyanov Str., Apt. 2 95013 Simferopol, Ukraine e-mail: [email protected]


The Discontinuous Solution for the Piece-homogeneous Transversal Isotropic Medium Oleksandr Kryvyy Abstract. The method of the reduction of the problems about the inter-phase defects in the piece-homogeneous transversal isotropic medium to the systems of the 2D singular integral equations is proposed. The method is based on the proposed way of solving the boundary Riemann problem in the space of the generalized functions of the slow growth in part of the variables and the discontinuous solution of the equations of the inhomogeneous transversal isotropic elasticity obtained with the help of this method. Mathematics Subject Classification (2000). 74G70; 74B05. Keywords. Medium, generalized functions, inter-phase defect, potential, transversal isotropy, discontinuous solution, boundary Riemann problem, singular integral equation.

Introduction The discontinuous solution for the isotropic medium allowing us to reduce the problems about thin defects of arbitrary nature in the plane z = 0 to the 2D systems of singular integral equations (SIE) is built in the article [1]. The stated method is generalized for the case of the piece-homogeneous isotropic medium and the problems about the circular inclusions with different conditions on the edges of the inclusions are considered in the article [2]. In both cases the problems were considered in the classes of the piece-differential functions which laid the appropriate limits on the loadings and complicated the substantiations of the constructions. In the present article the problem about the inter-phase defects of the arbitrary nature in the compound transversal isotropic medium is formulated in the form of the boundary problem for the differential equations in the space of the generalized 3 functions of the slow growth S (R ) and is reduced to the Riemann problem in 3 S (R ). The method of solution of this problem summarizing the results of the

396

O. Kryvyy

articles [3, 4] is proposed. The stated method allowed us to find the discontinuous solution for the piece-homogeneous transversal isotropic medium. As a result the singular integral correlations which connect the jumps and sums of stresses and displacements in the plane of the connection of the semi-mediums which enable us to reduce the problems about the inter-phase defects of arbitrary nature directly to the systems of singular integral equations are built.

1. The formulation of the problems about the defects and the reduction of the problem to the Riemann problem in S (R3 ) Let the common nature defects (crack, exfoliated and non-exfoliated inclusions) occupying the area Ω be in the plane of connection of two different transversal isotropic semi-mediums z = 0. On the edges of the semi-mediums depending on the type of the defects six of the following values are considered known: 6 6 {ζk± } = {vk } , (x, y) ∈ Ω; z=±0

v = {v k (x , y, z)} k=1,6 = {σ z , τyz , τxz , u , v , w } .

(1.1)

Let us determine the integral correlations in the plane z = 0 to find the rest of the functions from (1.1) which connect differences (jumps) and sums 6 χ ± = {χ± k (x, y)} , + − ± 2 χ± k = χk (x, y) = ζk (x, y) ± ζk (x, y), (x, y) ∈ R ;

(1.2)

of the components of the vector displacements and the tensor of stresses. Let us consider the stresses to be disappearing at infinity, as otherwise in view of linearity the problem can be reduced [1] to the considered formulation if the solution of the appropriate problem without crack is built. Let us express the components of vector v in terms of the jumps (1.2). Let’s name this solution in accordance with [1] the discontinuous solution for the piecehomogeneous transversal isotropic medium. We can obtain the following system of differential equations with z = 0 in the class of differential functions coming from the equations of equilibrium and the generalized Hooke law [5] concerning the components of the vector of the displacements u = {uj }3 = {vj+3 }3 . D[∂1 , ∂2 , ∂3 ]u = 0, z = 0,

(∂1 ≡ ∂/∂x, ∂2 ≡ ∂/∂y , ∂3 ≡ ∂/∂z ).

(1.3)

D[∂1 , ∂2 , ∂3 ] = {Lkj }3 , L11 = a11 ∂12 + a66 ∂22 + a44 ∂32 , L22 = a66 ∂12 + a11 ∂22 + a44 ∂32 , 2 2 L33 = a44 ∂12 + a44 ∂22 + a33 ∂32 , L12 = (a66 + a12 )∂12 , L13 = (a44 + a13 )∂13 , − 2 L23 = (a66 + a13 )∂23 , Ljk = Lkj , akj = θ(z)a+ kj + θ(−z)akj ,

a66 =

1 (a11 − a12 ). 2

The Discontinuous Solution

397

The rest of the components of vector v can be found by the formulas v1 = a11 (∂1 u1 + ∂2 u2 ) + a33 ∂3 u3 , v2 = a44 (∂2 u3 + ∂3 u2 ),

(1.4)

v3 = a44 (∂3 u1 + ∂1 u3 ). Let us present the solution of equation (1.3) coming from the presentations of the Lame equations for the homogeneous transversal isotropic medium [6, 7] in the following way uj = ∂j (ψ1 + ψ2 ) + (−1)j ∂2−j ψ3 , j = 1, 2;

u3 = ∂3 (κ1 ψ1 + κ2 ψ2 ), z = 0, (1.5)

functions ψj (j = 1, 3) satisfy the equations Pj [∂1 , ∂2 , ∂3 ]ψj = 0, z = 0,

(Pj = ∂32 + ξj2 (∂12 + ∂22 )).

(1.6)

Piece-constants ξj2 = θ(z)(ξj+ )2 + (ξj− )2 θ(−z), (j = 1, 3) are the solutions of the equations a33 a44 ξ 4 + (a13 (a13 + 2a44 ) − a11 a33 )ξ 2 + a11 a44 = 0;

a44 ξ 2 − a66 = 0;

− the rest κj = θ(z)κ+ j + κj θ(−z), (j = 1, 2) allow the presentation

κj = (a11 − a44 ξj2 )(ξj2 (a13 + a44 ))−1 = (a11 + a44 )(ξj2 a33 − a44 )−1 . Let Sp (R ) be the subspace of the generalized functions of the slow growth 3 g(x, y, z) ∈ S (R ) for which cz (g) ≤ p; cz (g) is the order of singularity [8] in variable z. Then coming from equation (1.6) concerning functions ψj (j = 1, 3) in the 3 subspace S1 (R ) we can obtain differential equations with discontinuous coefficients 1 8− 7 2 Pj [∂1 , ∂2 , ∂3 ]ψj = fj , fj = δ (k) (z) ∂31−k ψj , ψj ∈ S1 (R ). (1.7) 3

k=0

Let Hm (R ) be a class of functions fω± (α,β) ∈ Sp (R ) which is analytical in parameter ω = γ + iγi in each finite point of the complex plane with the exception of, maybe, lines Imω = 0 and satisfying with |Im ω| > ε > 0 and with some integer m the estimation 2

2

|fω (α, β)| ≤ Aε (1 + |ω|)m , (Aε < ∞) .

(1.8)

The function fγ (α, β) ∈ S (R ) allows the analytical presentation in variable γ 2 if the function fω (α,β) ∈ Hm (R ) exists that (in the sense of convergence in the 2 space S(R )) 3

lim (fγ+iε (α, β) − fγ−iε (α, β)) = fγ+ (α, β) − fγ− (α, β) = f (α, β) .

ε→0

(1.9)

Let Ωm (R ) be the subspace of generalized functions f ∈ S (R ), for which the functions allowing analytical presentation (1.9) in variable γ belong to the class 2 3 3 Hm (R ). Let Ω ±,m± (R ) be the subspace of functions f± ∈ Ωm± (R ), for which 3

3

398

O. Kryvyy

functions fω± (α,β) ∈ Hm± (R2 ) allowing analytical presentation (1.9) accordingly with ±Im ω < 0 have the form fω± (α, β) = Mm± Mm (ω, α, β) =

m

ω k Φk (α,β), Φk ∈ S (R ), (Mm ≡ 0, m < 0) . 2

(1.10)

k=0

The following statements are true. Theorem 1.1. Let g(x, y, z) ∈ Sp (R ), 3

g(x, y, z) = ∂1n1 ∂2n2 ∂3p g0 (x, y, z);

(1.11)

g0 is the continuous function of the slow growth allowing the presentation g0 = 3 z n 0 g∗ (x, y, z), (n0 ≥ 0, g∗ (x, y, 0) = 0), then f = F3 [g] ∈ Ωp−n0 −1 (R ) , where Fn is the n-dimensional Fourier transform operator. Proof. Let’s introduce the Karleman-Fourier [9] transform of the functions g ∈ 3 Sp (R ) taking into account (1.11) in the following way s = sign(Imω)) p ∗ ˆ (z)eiωz dz, F[g; ω] = (−iω) s θ(sz)z n0 fα,β R ∗ fα,β (z)

= (−iα)n1 (−iβ)n2 F2 [g∗ ]. (1.12) ˆ ω] is analytical with Imω = 0 and allows According to [9] the function F[g; the analytical presentation for f = F3 [g]. Let’s prove the estimation (1.8) taking into account for definiteness Imω > ε > 0, then according to (1.12) we can write ∞ ∞ p p n ∗ iωz 0 ˆ ω] = (−iω) z fα,β (z)e dz = (−iω) z n0 fε (z)e(iω+ε)z dz. F[g; 0

0

∗ From which because of the limitation of the function fε (z) = fα,β (z)e−εz with ε > 0 we shall obtain the estimation (Aε = n! max fε (z)) z∈(0;+∞)

∞ ˆ p n0 (iω+ε)z z fε (z)e dz F[g; ω] ≤ (−iω) 0 (−iω)p = Aε ≤ Aε (1 + |ω|)p−n0 −1 , (iω + ε)n0 +1

The theorem is proved. Theorem 1.2. Let g± ∈ S±,p (R ) = S± (R ) ∩ Sp ± (R ) where ± & % 3 3 S± (R ) = g ± ∈ S (R ) supp g± = R2 × R± . 3

3

Then f± = F3 [g± ] ∈ Ω ± , m± (R ), m± = p± − 1. 3

3


399

Proof. From the condition g± ∈ Sp ± (R3 ) the possibility of the presentation of the the possibility of form (1.11) follows and, hence, taking into account of g± ∈ S± the presentation g± = ∂3p (∂1n1 ∂2n2 g0± +

p± −1

z j ϕj (x, y)); ϕj ∈ S (R2 )

(1.13)

j=1

g0± (x, y, z) is the continuous function with bearer: supp g0± = R2 × R± . The proˆ ± ; ω] ∈ Hp± (R2 ) and Theorem 1.1. complete the proof. The theorem is perty F[g 0 proved. Taking into account these theorems and applying the 3D Fourier transform to the equations (1.7) we shall obtain the Riemann problem in parameter γ for the definition of the transformations of the functions ψj (j = 1, 3) in subspaces Ω ±,1 (R3 ) + − − p+ j Ψj = −pj Ψj + Qj , (j = 1, 3) Ψ± j = F3 [θ( ± z)ψj ] ∈ Ω ±,0 (R )). 3

p± j Qj =

1

(1.14)

= θ(±z)P[−iα, −iβ, −iγ],

0 0 (−iγ)k νjk (α, β), νjk = F2

#7 8− $ ∂31−k ψj .

k=0

We shall implement the solution of the problems (1.14) being guided by the 3 given below method of the solution of the scalar problem in space S (R ).

2. About one method of the solution of the boundary Riemann problem in one variable in space S (R3 ) The boundary Riemann problem in the scalar formulation in one variable in space S (R3 ) consists in the following: it is necessary to find two functions f± ∈ 3 Ω ±, m± (R ) which 3 (f+ ,ϕ) = (f− , G (α, β, γ) ϕ) + (q, ϕ) , (q ∈ S (R ), ϕ ∈ S R3 ) , (2.1) q is the determined function, where g = F−1 3 [q] ∈ Sn (R ). G ∈ Θμ ,G = 0, Θμ is 3 the class of the H¨ older multipliers in parameter γ, ((α,β) ∈ R2 ) in S(R ). It is easy to note that in the considered spaces the statements, proved in [3], allow the generalization of the statement. 3

Theorem 2.1. If f (α, β, γ) ∈ Ωp (R ), then the presentation 3

f = f+ − f− , f± ∈ Ω ±,p (R ) , 3

(2.2)

is true. f± are determined up to the functions of the form Mp (α, β, γ) from (1.10).

400

O. Kryvyy

Theorem 2.2. Let f± (α, β, γ) ∈ Ω ± , m± (R3 ) , then if (f+ , ϕ) = (f− , ϕ) , ϕ ∈ S(R3 )

(2.3)

then f+ = f− = Mp (α, β, γ) , p ≤ min {m+ , m− }, Mp (α, β, γ) is the function of the form (1.9). Proof. Let’s pass to the solution of the problem (2.1). Let the index of the coefficient be bounded: Indγ G = k < ∞, then according to [10, 11], the presentation k X+ γ−i , X± (α, β, γ) = lim X(α, β, ω); (2.4) G(α, β, γ) = ω→α±i0 γ+i X− X(α, β, ω) = eKαβ (ω) ; +∞ −k 1 τ −i dτ . Kα,β (ω) = ln G (α, β, τ ) 2πi τ +i τ −ω −∞

is true. X± ∈ Θμ are the boundary values of the functions, bounded at infinity, analytical in variable γ, accordingly in the upper and lower half-spaces of the complex plane ω. The presentation (2.4), Theorems 1.2, 2.1 and also that (γ + i)k ∈ Θμ allow us to write the condition (2.1) in the form 0 0 k 3 −1 0 = f± (α, β, γ) (γ ± i) X± − qk± , ϕ ∈ S(R ) (2.5) f+ , ϕ = f− , ϕ , f± −1 qk = (γ + i) k qX+ = qk+ − qk− ,qk± ∈ Ω ±,n+k−1 (R ) . 3

The belonging of the functions f 0± according to the subspaces Ω ± (R ) is obvious. 3 0 Let k ≥ 0, then if m ≥ n − 1 (m = min {m+ , m− } ), then f± ∈ Ω ±,m+k (R ) and, 0 hence, on the basis of Theorem 2.2: f± = Mm+k (α, β, γ), where Mm+k are the functions of the form (1.10). The solution of the problem (2.1) in this case will take the following form: 3

f± (α, β, γ) = (γ ± i)

−k

X± (Mm+k + qk± ) ∈ Ω ±,m (R ) 3

(2.6)

If m < n − 1, then in accordance with Theorems 1.1 and 2.1 for the existence of 3 the solution of the problem (2.1) in the subspaces Ω ±,m (R ) it is necessary that in the relation gk = ∂1n1 ∂2n2 ∂3k+n g0 , (gk = F−1 3 [qk ]) the function g0 should allow the presentation g0 (x, y, z) = z n ∗ g∗ (x, y, z) , g∗ (x, y, 0) = 0, 3

(2.7) n∗ = n − m − 1, g∗ ∈ S(R ). In this case the solution of the problem (2.1) is also determined by the relations (2.6). So the following statement has been proved. 3 Theorem 2.3. Let Indγ G (α, β, γ) = k ≥ 0, g = F−1 3 [q] ∈ Sn (R ), then if m ≥ n − 1 (m = min {m+ , m− } ), then the common solution of the problem (2.1) exists in 3 the subspaces Ω ±,m (R ) and is determined by the relations (2.6). If m < n−1, then for the existence of the solution (2.10) in Ω ±,m (R3 ) it is necessary and sufficient


401

that the conditions (2.7) should be satisfied. Similarly the following statement is determined on the basis of the theorems 2.1, 2.2, 1.2. Theorem 2.4. Let Indγ G (α, β, γ) = k < 0, then if m ≥ n − 1 − k (m = min {m+ , m− } ), the common solution of the problem (2.1) exists in the subspaces 3 Ω ±,m (R ) and is determined by the relations (2.6). If m < n − k − 1, then for 3 the existence of the solutions in Ω ±,m (R ) it is necessary and sufficient that the condition (2.7) should be satisfied, in which n∗ = n − k − m − 1. Consequence. Let m = n − 1, then if k ≥ 0, the problem (1.1) is solvable in 3 the subspaces Ω ±,m (R ), and if k < 0, then it is solvable in the determined subspaces with the condition (2.7) being satisfied, in which n∗ = −k. The common solution of the problem (2.1) is determined by the relations (2.6) and depends on m + k (m + k > 0) of the arbitrary functions from the space S (R2 ).

3. The solution of the boundary problem and the construction of integral relations The results obtained above allow us to pass to the solution of the problem (1.14). The presentation of the coefficients of the problem ± ± ± ± 2 2 2 p± j = (γ − ωj r)(γ + ωj r), r = α + β , ωj = iξj ,

and Theorem 2.1 allow us to transform the boundary condition (1.14) in the following way (j = 1, 3): + − − − gj+ Ψ+ j − Qj = −gj Ψj − Qj ,

(3.1)

Here, the notations are introduced gj±

=

γ ± ωj± γ ± ωj∓

,

Q± j

=

1 0 (−ir)k−1 (∓ωj∓ )k νjk k=0

(ωj+ + ωj− )(γ ± ωj∓ r)

.

According to [5] the condition Imωj± = 0 is true. For the sake of definiteness we will consider Imωj± > 0 (Reξj± > 0), then the functions that stand in the right3 hand and the left-hand parts of the equality (3.1) belong to the spaces Ω ± , 1 (R ) accordingly. Hence, applying the theorem 2.2 and taking into account the property lim Ψ± j = 0 let’s write the solution of the problem (1.14) in the form γ→∞

Ψ± j = ±

1 0 (−ir)k−1 (∓ωj∓ )k νjk k=0

(ωj+ + ωj− )(γ ± ωj± r)

,

(j = 1, 3).

(3.2)

0 of the functions ψj and their derivatives The solutions (3.2) contain the jumps νjk ∂3 ψj . Let’s express them in terms of the jumps of physical quantities on the plane z = 0. To achieve this, applying the presentations (1.4) and (1.5) transformed with the help of the Fourier transformation in variables x and y we will make up jumps of stresses and displacements with z = 0 and invert the obtained equalities. As a

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O. Kryvyy

0 result of the subsequent exclusion of νjk (α, β) we will get the following expressions ± for the transformations Vj (α, β, z) = θ(±z)F2 [vj ] of the physical quantities (1.1) 2 2 k,± ± ± − − − ± −1 ˜1 q11 ek − r ((−iβ)χ ˜2 + (−iα)χ ˜3 ) qk,± V1 = θ(±z) χ 12 ek k=1

+((−iα)χ ˜− 4

+

k=1

(−iβ)χ ˜− 5)

2

± qk,± 13 ek

−

rχ ˜− 6

k=1

2

± qk,± 14 ek

: ,

(3.3)

k=1

2 ± = θ(±z) r−1 (−iβ)χ ˜− qk,± 1 21 ek

V2±

k=1

2 k,± ± −2 − 2 2 3,± ± +r χ ˜2 (−iβ) q22 ek + (−iα) q22 e3 k=1

˜− −(−iα)(−iβ)r−2 χ 3 ˜− −(−iα)(−iβ)r−1 χ 4

2

2

± qk,± 22 ek

−

± q3,± 22 e3

k=1 ± 2 3,± ± qk,± 23 ek + (−iα) q23 e3

k=1

2 ˜− −r−1 χ 5 (−iβ)

2

: 2 k,± ± ± − 2 3,± ± + (−iβ) χ ˜ qk,± e − (−iα) q e q e 23 k 23 3 24 k , 6

k=1

V4±

k=1

2 ± = θ(±z) −r−2 (−iα)χ ˜− qk,± 31 ek 1

−(−iα)(−iβ)r−3 χ ˜− 2

2

k=1 3,± ± ± qk,± 32 ek + q32 e3

k=1

2 k,± ± −3 − 2 2 3,± ± ˜3 (−iα) q32 ek − (−iβ) q32 e3 −r χ ˜− r−2 χ 4 ˜− +(−iα)(−iβ)r−2 χ 5

k=1

2

(−iα)

2

2

± qk,± 33 ek

+

k=1 ± qk,± 33 ek

−

± q3,± 33 e3

± (−iβ)2 q3,± 33 e3

−

(−iα)r−1 χ ˜− 6

k=1

V6±

2

± qk,± 34 ek

k=1

: ,

2 2 ± ± −2 = θ(±z) r−1 χ ˜− qk,± ((−iβ)χ ˜− ˜− qk,± 1 2 + (−iα)χ 3) 41 ek + r 42 ek k=1

−r−1 ((−iα)χ ˜− ˜− 4 + (−iβ)χ 5)

k=1

2 k=1

± qk,± ˜− 6 43 ek + χ

2 k=1

: ± qk,± e 44 k .


403

Here, the following notations are introduced ±

k−1 ±iωk rz e± , qj,± 1,k = ( − i) k = e

1

1,± ± ± 2,± s02j−n,k (a± 13 bj,1−n ∓ ωj a33 bj,1−n ),

n=0 k−1 qj,± 3,k = i

1

s0j+n,k b1,± j,1−n ,

n=0 k ± qj,± 2,k = i a44

1

± 1,± s02j−n,k (b2,± j,1−n ∓ ωj bj,1−n ),

n=0 k qj,± 4,k = i

1

s0j+n,k b2,± j,1−n , (j = 1, 2; k = 1, 4),

n=0 ± 3−m k+2−m (±a± q3,± 44 ω3 ) m,k = i

1

s∗n+1,k b1,± 3,n ,(m = 2, 3; k = 2, 3),

n=0 −1

S

=

{s0jk }4 , S

= {sjk }4 ,

± k ± ± n−1 ∗ 2 (ωj+ + ωj− )−1 , S−1 bn,± ∗ = {sjk } , S∗ = {sjk }j=2,3;k=5,6 , j,k = −i(∓ωj ) (∓κj ωj )

s1,k = h1k−1,1 , k = 1, 2, s1,k = h1k−3,2 , k = 3, 4, s2,k = h2k−1,1 , k = 1, 2, s2,k = h2k−3,2 , k = 3, 4, s2,k = h2k−5,3 , k = 5, 6, s3,k = h3k−1,1 , k = 1, 2, s3,k = h3k−3,2 , k = 3, 4, s3,k = h3k−5,3 , k = 5, 6, n,+ n,− s4,k = h4k−1,1 , k = 1, 2, s4,k = h4k−3,2 , k = 3, 4, hn+2 j,k = bj,k −bj,k , n = 1, 2, 2,+ 2,− + 1,+ − − 1,− 2 + + 1,+ − − 1,− h23,k = a+ 44 ω3 b3,k + a44 ω3 b3,k , hj,k = a44 (bj,k − ωj bj,k ) − a44 (bj,k + ωj bj,k ), 1,+ + + 2,+ + 1,+ + + 2,+ h1j,k = a+ 13 bj,k − a33 ωj bj,k − a13 bj,k + a33 ωj bj,k , j = 1, 2, k = 0, 1.

The formulae for V3± , V5± are obtained from the formulas accordingly for V2± , V4± , ˜± ˜− ˜− by means of the permutation χ ˜− 2 ⇔χ 3, χ 4 ⇔χ 5 , α ⇔ β. Inverting the formulae (3.3) let’s use the formula (3.1.26) [1]), then the solution will contain the linear combinations of the operators in the form of j,k Ln [f ] = f (x, τ )Knj,k (x − t, y − τ, z)dxdy (3.4) R2

Kj,k n

1 j k ∂ ∂ = 2π 1 2

∞

± e−ξm |z|ρ

2 + (y − τ )2 ) dρ, ρ J (x − t) 0 ρn

(3.5)

0

J0 (x) is the Bessel function. The kernels (3.5) satisfy the conditions n−(j +k) ≤ 0, ± Reξm > 0, hence, the components of the vector v don’t go out of the subspace 3 S1 (R ) and the bearer of their singularity is found on the plane z = 0. Using the

404

O. Kryvyy

table from [11] it is easy to escape quadratures in the kernels of the operator (3.4), as a result, if we denote

R∗j,± = Rj,± + ξj± |z| , Rj,± = (x − t)2 + (y − τ )2 + (ξj± z)2 , then, the expression for the discontinuous solution can be written in the following form vj (x, y, z) = θ(z)vj+ (x, y, z) + θ(z)vj− (x, y, z); (3.6) 2 2 j,± q11 1 1 1 1 v1± = ∓χ− χ− − qj,± + χ− 1 2 ∂2 3 ∂1 12 ± ∂3 2π Rj,± j=1 Rj,± Rj,± ξ j=1 j ±

R2 2 j=1

−

3

χ− k

k=2

+ −

5 k=4

χ− k

2

: 2 qj,± 1 1 qj,± − 2 − 2 − 2 1 13 14 χ − χ dtdτ ∂ + χ ∂ ∂ 3 4 13 5 23 6 Rj,± Rj,± ξj± (ξj± )2 Rj,± j=1 2 1 1 ± v2 = χ− qj,± 1 21 ∂2 2π Rj,± j=1 2

R2

y−τ x−t + (−1)k q3,± 22 ∂k−1 ∗ R R R R∗3,± j,± 3,± j,± j=1 2 1 x−t v4± = χ− qj,± 1 31 ∂2 2π Rj,± R∗j,± j=1

3

χ− k

k=2

qj,± 33 ∂k−3

:

qj,± 22 ∂4−k

2

R2

x−t qj,± 32 ∂4−k R∗j,± j=1

+

y−τ (−1)k q3,± 32 ∂k−1 ∗ R3,±

: 2 x−t y−τ 1 j,± − +χ +(−1)k q3,± ∂ q ∂ 6 33 6−k 43 1 Rj,± Rj,±∗ R3,± R∗3,± Rj,± j=1

k=1

2 2 qj,± 1 j,± 1 − − 42 v6± = χ− q − (y − τ )χ + (x − t)χ 1 2 3 41 2π R R R∗j,± j,± j,± j=1 j=1 R2

+

2

qj,± 43

j=1 for v3± ,

5

χ− k ∂k−3

k=4

: 2 1 qj,± 1 − 44 dtdτ. ∓ χ6 ∂ Rj,± ξ ± 3 Rj,± j=1 j

v5± can be obtained from the v2± , v4± accordingly by The formulae − − − means of the permutation χ− 2 ⇔ χ3 , χ4 ⇔ χ5 , x ⇔ y, ∂1 ⇔ ∂2 . Let’s research the conduct of the operators entering in (3.6) with z → ±0, as the bearer of singularity of discontinuous solutions concentrates on the plane z = 0. The stated operators are generalization of potentials of the simple and double layers and their derivatives in case of transversal isotropic medium. Let’s consider the following of them: 1 ± Φ1,k [f ] (z) = θ (±z) f (t, τ ) ∂3k dtdτ , k = 1, 2, Rj,± R2

The Discontinuous Solution Φ± 2,k

[f ] (z) = θ (±z)

f (t, τ ) ∂k R2

(y − t)k−1 (x − τ )2−k dtdτ , k = 1, 2 ∗ Rj,± Rj,±

405

(3.7)

Theorem 3.1. Let f (x, y) be the function that is integrated on the plane z = 0, then ± ± (3.8) lim Φ± 1,k [f ](z) = ∓2πξj δ1,k f (x, y) + δ2,k Φ1,k [f ](0); z→±0

± lim Φ± 2,k [f ](z) = πf (x, y) + Φ2,k [f ](0).

z→±0

Proof. By integrating directly and passing to the polar coordinates it is easy to determine the properties: ± ± Φ± 1,k [1](z) = ∓2πξj δ1,k , Φ1,k [1](z) = π, ± Φ± 1,k [f ](0) = 0, Φn,k [1](0) = 0 (n, k = 1, 2).

(3.9)

δ1,k is the Kronecker symbol. By using the presentations (3.9) and applying the method of paper [12] we will obtain the properties (3.8). The theorem is proved. ± Thus, the operators Φ± 1,1 [f ](z), Φ2,k [f ](z) will have a jump when leaving the appropriate subspaces and getting into the plane z = 0. For the operators ± Φ± 2,k [f ](z) the value of this jump will coincide. For the operators Φ1,2 [f ](z) and the other operators from (3.6) the stated leaving will be continuous. Taking into account the latter and (3.1) we will pass in (3.8) to the limit: z → ±0 and make up the sums of physical quantities (1.1): : 5 χ− y−τ − y−t − 1 − − 6 χ+ q + q dtdτ = q χ −q ∂ χ + χ + χ 11 1 13 j−3 j 11 14 3 1 2π r03 2 r03 3 r0 j=4 R2

y−τ − x−t x−t − − − q21 3 χ1 + q22 χ2 ∂1 2 − χ3 ∂2 2 r0 r0 r0 2 R : 1 + − 2 1 2 1 −q23 χ− dtdτ + q24 ∂2 χ− χ4 ∂23 3 − q23 3 − q+ ∂ 23 1 6, r0 r0 r0 5 x−t − x−t 1 y−τ 1 1 − − − + χ+ q q χ− = − χ + χ + q χ ∂ + q − q ∂ 33 4 31 32 4 32 2 2 32 2 3 2 2π r02 1 r0 r0 r0 R2 : x−t x−t 1 − − − − χ + q dtdτ, −q− ∂ + χ ∂ χ ∂ 1 2 1 33 4 5 34 6 r20 r20 r0 q41 − y−τ − x−t − 1 + − χ − q42 χ + 2 χ3 χ6 = q44 χ6 + 2π r0 1 r02 2 r0 2 R 1 1 +q43 χ− dtdτ + χ− 4 ∂1 5 ∂2 r0 r0 where j,− ± 1 2 3 qjkn = qj,+ kn + qkn , qkn = qkn + qkn , qkn = qkn ± qkn . χ+ 2

1 1 = − q21 χ− 2 + 2 2π

406

χ+ 4

O. Kryvyy + + Formulas for χ+ 3 , χ5 can be obtained from the formulae accordingly for χ2 , − − − − by means of the permutation: χ2 ⇔ χ3 , χ4 ⇔ χ5 , x ⇔ y, ∂1 ⇔ ∂2 .

Conclusions. Thus, the discontinuous solution for the piece-homogeneous transversal isotropic medium (3.6) and integral relations (3.10) are obtained. The latter generalize the relations obtained in [1, 2] for isotropic medium and allow us to reduce the problems about the inter-phase defects in the piece-homogeneous transversal isotropic medium to the 2D integral equations or their systems.

References [1] G.Ya. Popov, The stress concentration near punches, sections, thin inclusions and supports. M.: Nauka, 1982. [2] V.V. Yefimov, A.F. Kryvyy, G.Ya. Popov, The problems about the stress concentration near the circular defect in the compound elastic medium. Izvestiya Rossijskoi Akademii Nauk. Mehanika tverdogo tela. 2 (1998), 42–58. [3] O.F. Kryvyy, The tunnels inclusions in piece-homogeneous anisotropic medium. Math. methods and phys.-mech. fields. 50 2, (2007), 55–65. [4] A.F. Kryvyy, The fundamental solution for the four-component anisotropic plane. Visnyk Odeskogo derzhavnogo universytetu. Phys.-math. sciences. v. 8 2, (2003), 140–149. [5] S.G. Lekhnitskyy, The theory of elasticity of anisotropic solid. M.: Nauka, 1977. [6] H.A. Elliot, Axial symmetric stress distribution in aelotropic hexagonal crystals. The problem of the plane and related problems. Proc. Cambridge Phil. Soc. 45 (1949), 621–630. [7] H.C. Hu, On the three-dimensional problems of the theory of elasticity of a transversely isotropic body. Deta Sci. Sinica. 2 (1953), 145–151. [8] Yu.A. Brychkov, About the smoothness concerning of variables solutions of the linear differential equations with partial derivatives. Differential equations. v. 10 2, (1974), 281–289. [9] G. Bremerman, The distributions, complex variables and Fourier transforms. Mir, 1983. [10] D.F. Gakhov, The boundary problems. M: Nauka, 1977. [11] Yu.A. Brychkov, A.P. Prudnikow, The integral transformations of generalized functions. M: Nauka, 1977. [12] N.M. Gyunter, The theory of potential and its application to the basic problems of mathematical physics. M: Gos. Izd. Tekhniko-teoreticheskoy literatury, 1953. Oleksandr Kryvyy P.O. Box 65029 Didrikhson st. 8 Odessa, Ukraine e-mail: [email protected]


On the Carleman Ultradifferentiability of Weak Solutions of an Abstract Evolution Equation Marat V. Markin To M.G. Krein in honor of his centenary

Abstract. For the evolution equation y (t) = Ay(t) with a normal operator A in a Hilbert space, conditions on A are found that are necessary and sufficient for all weak solutions of the equation on [0, ∞) to belong to a certain Carleman class of strongly ultradifferentiable vector functions. Mathematics Subject Classification (2000). Primary 34G10; Secondary 47B15, 47D06. Keywords. Weak solution, normal operator, Carleman ultradifferentiability.

1. Introduction Consider the evolution equation y (t) = Ay(t),

(1.1)

with a normal operator A in a complex Hilbert space H with an inner product (·, ·). We are to find conditions necessary and sufficient for all weak solutions of the equation on [0, ∞) to belong to a certain Carleman class of strongly ultradifferentiabile vector functions. With our goal attained, all the principal results of paper [17] obtain their natural generalization. In defining a weak solution of equation (1.1) on an interval [0, T ) (0 < T ≤ ∞), we follow [1], i.e., it’s understood to be a vector function y : [0, T ) → H such that

408

M.V. Markin

(i) y(·) is strongly continuous on [0, T ). (ii) For any g ∈ D(A∗ ) (D(·) is the domain of an operator and A∗ is the operator adjoint to A): d (y(t), g) = (y(t), A∗ g), t ∈ [0, T ). dt Remark 1.1. Note that the weak solutions defined in such a manner need not be strongly differentiable anywhere on [0, T ). As was shown in [16], the general weak solution of equation (1.1) on [0, T ) (0 < T ≤ ∞) is as follows: ; y(t) = etA f, t ∈ [0, T ), f ∈ D(etA ), 0≤t 0, ∃c > 0 : max g (n) (t) ≤ cαn mn , n = 0, 1, 2, . . . a≤t≤b

and

def , C(mn ) (I, X) = g(·) ∈ C ∞ (I, X) ∀[a, b] ⊆ I, ∀α > 0 ∃c > 0 :

max g (n) (t) ≤ cαn mn , n = 0, 1, 2, . . .

a≤t≤b

are called the Carleman of strongly ultradifferentiable functions correspond-∞ , classes ing to the sequence mn n=0 of Roumieu’s and Beurling’s types, respectively (for numeric functions, see [2, 14, 13]). Obviously, C(mn ) (I, X) ⊆ C{mn } (I, X). Remark 2.1. Observe that, for mn := [n!]β or, due to Stirling’s formula, mn := nβn , n = 0, 1, 2, . . ., 0 ≤ β < ∞, we obtain the well-known Gevrey classes,

The Carleman Ultradifferentiability of Weak Solutions

409

E {β} (I, X) and E (β) (I, X) (for numeric functions, see [7]). In particular, E {1} (I, X) and E (1) (I, X) are the classes of real analytic and entire vector functions, respectively (for numeric functions, see [14]). 2.2. Carleman classes of vectors Let C ∞ (A) =

def

∞ ;

D(An ).

n=0

The vector sets def , C{mn } (A) = f ∈ C ∞ (A) ∃α > 0, ∃c > 0 : An f ≤ cαn mn , n = 0, 1, 2, . . . and def , C(mn ) (A) = f ∈ C ∞ (A) ∀α > 0 ∃c > 0 : An f ≤ cαn mn , n = 0, 1, 2, . . . are , called -∞ the Carleman classes of the operator A corresponding to the sequence mn n=0 of Roumie’s and Beurling’s types, respectively. As is easily seen, C(mn ) (A) ⊆ C{mn } (A). Remark 2.2. For mn := [n!]β or mn := nβn , n = 0, 1, 2, . . ., 0 ≤ β < ∞, the above are the Gevrey classes of the operator A, E {β} (A) and E (β) (A) (see, e.g., [10, 9, 11]). In particular, E {1} (A) and E (1) (A) are the celebrated classes of analytic and entire vectors, respectively [20, 8]. , -∞ 2.3. Conditions on the sequence mn n=0 , -∞ The sequence mn n=0 being subject to the condition (WGR)

For any α > 0, there exist such a c = c(α) > 0 that cαn ≤ mn ,

n = 0, 1, 2, . . . ,

the scalar function def

T (λ) = m0

∞ λn , 0 ≤ λ < ∞, (00 := 1), m n n=0

first introduced by S. Mandelbrojt [14] is well defined (see also [11]). The function T (·) is, evidently, continuous, increasing, and T (0) = 1. Let def

M (λ) = ln T (λ),

0 ≤ λ < ∞.

The function M (·) is also continuous, increasing, and M (0) = 0. Its inverse M −1 (·) defined on [0, ∞) inherits all the aforementioned properties of M (·).

410

M.V. Markin

We shall need the following regions in the complex plane C associated with the function M (·): def , Mb+ = λ ∈ C Re λ ≥ b+ M | Im λ| and

def , Mb− ,b+ = λ ∈ C Re λ ≤ −b− M | Im λ| or Re λ ≥ b+ M | Im λ| ,

where b+ and b− are positive constants. According to [10] (see also [9, 11]),,for -a normal operator A in a complex ∞ Hilbert space H and a positive sequence mn n=0 satisfying condition (WGR), < D(T (t|A|)), C{mn } (A) = t>0

C(mn ) (A) =

;

(2.1) D(T (t|A|)),

t>0

the function T (·) being replaceable by any nonnegative, continuous, and increasing function L(·) defined on [0, ∞) such that c1 L(γ1 λ) ≤ T (λ) ≤ c2 L(γ2 λ),

λ ≥ R,

with some positive γ1 , γ2 , c1 , c2 , and a nonnegative R. In particular [11], T (·) in (2.1) is replaceable by 1/2 ∞ λn λ2n def def , 0 ≤ λ < ∞, or P (λ) = m0 , 0 ≤ λ < ∞. S(λ) = m0 sup m2n n≥0 mn n=0 Remark 2.3. In [18], equalities (2.1) indispensable for our further discourse were generalized to the case of a scalar type spectral operator in a complex Banach space (see, e.g., [4, 6]). , -∞ The positive sequence mn n=0 will be subject to the condition (GR) For some α > 0 and c > 0, cαn n! ≤ mn ,

n = 0, 1, 2, . . . ,

or its stronger version (SGR) For any α > 0, there is a c = c(α) > 0 such that cαn n! ≤ mn ,

n = 0, 1, 2, . . . ,

combined with the condition (BC) For some l > 0 and h > 1, lhn ≤

n k=0

or its stronger version

mn , mk mn−k

n = 0, 1, 2, . . . ,


411

(SBC) For some l > 0, L > 0 and h > 1, H > 1, n mn lhn ≤ ≤ LH n , n = 0, 1, 2, . . . . mk mn−k k=0

Remark 2.4. Obviously, (GR)-conditions are stronger than (WGR). As for (BC)conditions, they resemble the well-known binomial coefficients identity n n = 2n , n = 0, 1, 2, . . . , k k=0

directly arrived at when mn = n!, n = 0, 1, 2, . . . . Observe also that there are sequences of positive numbers satisfying both (SGR) and (SBC), e.g., mn = [n!]β , n = 0, 1, 2, . . . , 1 < β < ∞ and that (GR)conditions and (BC)-conditions are independent (see Appendix). By condition (GR), for a certain α > 0 and a certain c > 0, ∞ ∞ −1 λn (α−1 λ)n T (λ) = m0 = m0 c−1 eα λ , ≤ m0 c−1 mn n! n=0 n=0

0 ≤ λ < ∞.

Whence, M (λ) ≤ ln(m0 c−1 ) + α−1 λ, 0 ≤ λ < ∞. Therefore, there is such an R = R(α, c) > 0 that M (λ) ≤ 2α−1 λ, Substituting M

−1

R ≤ λ < ∞.

(λ) for λ, we arrive at the following estimate:

2α−1 M −1 (λ) ≥ λ,

M (R) ≤ λ < ∞,

(2.2)

with some α > 0 and R > 0. Similarly, one can derive from condition (SGR) estimate (2.2) with an arbitrary α > 0 and some R = R(α) > 0. Condition (BC) implies that with some h > 1 and l > 0 (Cauchy’s product of series) T 2 (λ) = m20

∞ n n=0 k=0

∞

1 (hλ)n λn ≥ m20 l = m0 lT (hλ), mk mn−k mn n=0

0 ≤ λ < ∞.

Whence, M (λ) ≥ 2−1 M (hλ) + 2−1 ln(m0 l), 0 ≤ λ < ∞. Inductively, we infer that, for certain h > 1 and l > 0 and any natural n, n −n n −k M (λ) ≥ 2 M (h λ) + ln(m0 l) 2 (2.3) k=1 = 2−n M (hn λ) + [1 − 2−n ] ln(m0 l),

0 ≤ λ < ∞.

Substituting h−n λ for λ, we obtain: M (λ) ≤ 2n M (h−n λ) − [2n − 1] ln(m0 l), for some h > 1 and l > 0 and any natural n.

0 ≤ λ < ∞,

(2.4)

412

M.V. Markin

Analogously, condition (SBC) implies that, along with (2.3) and (2.4), the function M (·) satisfies the following estimates: M (λ) ≤ 2−n M (H n λ) + [1 − 2−n ] ln(m0 L),

0 ≤ λ < ∞,

(2.5)

and M (λ) ≥ 2n M (H −n λ) − [2n − 1] ln(m0 L),

0 ≤ λ < ∞,

(2.6)

with some H > 1 and L > 0 and any natural n. Hereafter, unless specified otherwise, we regard {mn }∞ n=0 to be a sequence of positive numbers, A a normal operator in a complex Hilbert space H with an inner product (·, ·) and the induced norm · , σ(A) its spectrum, and EA (·) the operator’s spectral measure (the resolution of the identity) [5, 22]. For the spectral measure of a normal operator and its operational calculus frequently referred to, we shall adopt the abbreviations s.m. and o.c., respectively.

3. Carleman ultradifferentiability of a particular weak solution Proposition 3.1. Let I be a subinterval of an interval [0, T ) (0 < T ≤ ∞). Then the restriction of a weak solution y(·) of equation (1.1) on [0, T ) to I belongs to the Carleman class C{mn } (I, H) (C(mn ) (I, H)) if and only if for any t ∈ I,

y(t) ∈ C{mn } (A) (C(mn ) (A), respectively) in which case, for any natural n, y (n) (t) = An y(t),

t ∈ I.

Proof. “Only if” part. Assume that a weak solution y(·) of (1.1) on [0, T ) (0 < T ≤ ∞) restricted to a subinterval I ⊆ [0, T ) belongs to the Carleman class C{mn } (I, H) (C(mn ) (I, H)). Then necessarily y(·) ∈ C ∞ (I, H). Whence, by [16], Corollary 4.1, y(t) ∈ C ∞ (A) for t ∈ I and, for any natural n: y (n) (t) = An y(t),

t ∈ I.

The restriction of y(·) to I belonging to the class C{mn } (I, H) (C(mn ) (I, H)), for an arbitrary t ∈ I, some (any) α > 0, and some c > 0, we also have: An y(t) = y (n) (t) ≤ cαn mn ,

n = 0, 1, . . . .

Therefore, y(t) ∈ C{mn } (A) (C(mn ) (A)),

t ∈ I.

“If” part. Let y(·) be a weak solution of equation (1.1) on the interval [0, T ) (0 < T ≤ ∞) such that, for any t ∈ I, where I is a subinterval of [0, T ), y(t) ∈ C{mn } (A) (C(mn ) (A)).


413

Hence, for arbitrary t ∈ I and some (any) α > 0, there is such a c(t, α) > 0 that An y(t) ≤ c(t, α)αn mn ,

n = 0, 1, 2, . . . .

(3.1)

By [16], Corollary 4.1, the inclusions C(mn ) (A) ⊆ C{mn } (A) ⊆ C ∞ (A) imply that y(·) ∈ C ∞ (I, H) n = 0, 1, 2, . . . , t ∈ I. E D(etA ) such that Hence (see Introduction), there is an f ∈ y (n) (t) = An y(t),

0≤t0}

|λ|2n e2a Re λ d(EA (λ)f, f )

≤ {λ∈σ(A)| Re λ≤0}

|λ|2n e2b Re λ d(EA (λ)f, f )

+ {λ∈σ(A)| Re λ>0}

≤

|λ| e

2n 2a Re λ

σ(A)

= A e

n aA

|λ|2n e2b Re λ d(EA (λ)f, f )

d(EA (λ)f, f ) + σ(A)

f + A e 2

n bA

f = A y(a) 2 + An y(b) 2 2

n

= y (n) (a) 2 + y (n) (b) 2 . Whence, by (3.1), we conclude:

n max y (n) (t) 2 ≤ [c(a, α) + c(b, α)] max α(a), α(b) mn , n = 0, 1, 2, . . . .

a≤t≤b

The latter implies that y(·) restricted to the subinterval I ⊆ [0, T ) belongs to the Carleman class C{mn } (I, H) (C(mn ) (I, H)).

414

M.V. Markin

4. Carleman ultradifferentiability of weak solutions Theorem 4.1. Let the sequence {mn }∞ n=0 satisfy conditions (GR) and (BC). Then the following statements are equivalent. (i) Every weak solution of equation (1.1) on [0, ∞) belongs to the Carleman class C(mn ) [0, ∞), H . (ii) Every weak solution of equation (1.1) on [0, ∞) belongs to the Carleman class C{mn } [0, ∞), H . (iii) There is a b+ > 0 such that the set σ(A) \ Mb+ is bounded. Proof. To prove the theorem, we are to show the validity of the closed chain of logical implications (iii) ⇒ (i) ⇒ (ii) ⇒ (iii). Observe that the implication (i) ⇒ (ii) immediately follows from the inclusion C(mn ) [0, ∞), H ⊆ C{mn } [0, ∞), H . Let’s show that (iii) ⇒ (i). Assume that for some b+ > 0, the set σ(A) \ Mb+ is bounded. Recall that (see Introduction) an arbitrary weak solution y(·) of equation (1.1) on [0, ∞) is of the form: y(t) = etA f, E D(etA ). with some f ∈ 0≤t 0 be the constant from estimate (2.2), which holds due to condition (GR). Then, for arbitrary s > 0 and t ≥ 0, 2 T (s|λ|) d(EA (λ)y(t), y(t)) = T 2 (s|λ|) d(EA (λ)etA f, etA f ) σ(A)

σ(A)

by the properties of the o.c.; = T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) = σ(A)

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)\Mb+

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f )

+ σ(A)∩Mb+ ∩{λ| Re λ<M(R)}

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) < ∞.

+ σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}


415

E E, Indeed, the λ - boundedness of the sets σ(A) \ Mb+ and σ(A) Mb+ Re λ < M (R) , the finiteness of the measure, and the continuity of the integrated function on C, imply that, for arbitrary s > 0 and t ≥ 0, T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) < ∞ σ(A)\Mb+

and

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) < ∞. σ(A)∩Mb+ ∩{λ| Re λ<M(R)}

For an arbitrary s > 0, let’s fix a sufficiently large natural N such that h−N s[2α−1 + 1] ≤ 1, where h > 1 and α > 0 are the constants from conditions (BC) and (GR), respectively, and set γ := max(1, b−1 + ). Then, for any t ≥ 0, T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

e2M(s|λ|) e2t Re λ d(EA (λ)f, f )

= σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

≤

e2M(s[| Re λ|+| Im λ|]) e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≥ M (R)}, | Im λ| ≤ M −1 (b−1 + Re λ); −1 −1 e2M(s[Re λ+M (b+ Re λ)]) e2t Re λ d(EA (λ)f, f ) ≤ σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

for λ ∈ σ(A)∩Mb+ ∩{λ| Re λ ≥ M (R)}, by (2.2), 2α−1 M −1 (Re λ) ≥ Re λ, where α > 0 is the constant from condition (GR); −1 −1 −1 −1 ≤ e2M(s[2α M (Re λ)+M (b+ Re λ)]) σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

e

2t Re λ

d(EA (λ)f, f )

since γ := max(1, b−1 ); + ≤ σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

−1

e2M(s[2α

+1]M −1 (γ Re λ)) 2t Re λ

e

d(EA (λ)f, f )

416

M.V. Markin

for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≥ M (R)}, by (2.4), M (s[2−1 α + 1]M −1 (γ Re λ)) ≤ 2N M (h−N s[2−1 α + 1]M −1 (γ Re λ)) − [2N − 1] ln(m0 l), where h > 1 and l > 0 are the constants from condition (BC); N −N −1 −1 −2[2N −1] e22 M(h s[2 α+1]M (γ Re λ)) ≤ (m0 l) σ(A)∩Mβ b ∩{λ| Re λ≥M(R)} +

e

2t Re λ

d(EA (λ)f, f )

recall that h−N s[2−1 α + 1] ≤ 1; ≤ (m0 l)−2[2

N

−1]

N

M(M −1 (γ Re λ))

N

γ+t] Re λ

e22

σ(A)∩Mβ ∩{λ| Re λ≥M(R)} b +

e2t Re λ d(EA (λ)f, f ) = (m0 l)−2[2

N

−1]

e2[2

d(EA (λ)f, f )

σ(A)

by (4.1); < ∞. Thus, for any s > 0 and t ≥ 0, T 2 (s|λ|) d(EA (λ)y(t), y(t)) < ∞, σ(A)

which, by (2.1), implies that y(t) ∈ C(mn ) (A),

0 ≤ t < ∞.

By Proposition 3.1, we infer that y(·) ∈ C(mn ) ([0, ∞), H). Let’s show now that (ii) ⇒ (iii). We shall prove this implication by contrapositive. Assume the negation of (iii), i.e., that for any b+ > 0, the set σ(A) \ Mb+ is unbounded. Then, for any n = 1, 2, . . . , the set : σ(A) \ M2−n n−1 = σ(A) ∩ λ ∈ C Re λ < 2−n n−1 M (Im λ) is unbounded. Hence, there is a sequence of points of the complex plane {λn }∞ n=1 such that λn ∈ σ(A),

n = 1, 2, . . . ;

Re λn < 2−n n−1 M (| Im λn |), n = 1, 2, . . . ; λ0 := 0, |λn | > max n, |λn−1 | , n = 1, 2, . . . . In particular, the latter implies, that the points λn are distinct.


417

As is easily seen, for n = 1, 2, . . . , , λ ∈ C Re λ < 2−n n−1 M (Im λ) and |λ| > max n, |λn−1 | is an open set in C. Then, for any natural n, there exists such an εn > 0 that, along with the point λn , this set contains the open disk centered at λn Δn := {λ ∈ C |λ − λn | < εn . Then, for any λ ∈ Δn , Re λ < 2−n n−1 M (Im λ), |λ| > max n, |λn−1 | .

(4.2)

Furthermore, since the points λn are distinct, we can regard the radii of the disks, εn , to be small enough so that 0 < εn
0 such that Re λn ≤ ω, n = 1, 2, . . . . , Then unbounded is the set σ(A) ∩ λ ∈ C Re λ ≤ ω . Whence, by [16], Theorem 5.1, we infer that there is a weak solution y(·) of equation (1.1) on [0, ∞) such that y(·) ∈ C ∞ ([0, ∞), H). Moreover, y(·) ∈ C{mn } ([0, ∞), H). Suppose now that the sequence {Re λn }∞ n=1 is unbounded above. Therefore, there is a subsequence {Re λn(k) }∞ such that k=1 Re λn(k) ≥ k,

k = 1, 2, . . . .

(4.5)

418

M.V. Markin Consider the vector f :=

∞

e−n(k) Re λn(k) en(k) ,

k=1

which, as is easily seen, is well defined. By the pairwise orthogonality of the projection operators EA (Δn(k) ), EA (Δn(k) )f = e−n(k) Re λn(k) en(k) , EA (∪∞ k=1 Δn(k) )f

k = 1, 2, . . . ,

= f.

(4.6)

For any t ≥ 0, we have: e2t Re λ d(EA (λ)f, f ) σ(A)

by (4.6);

∞ e2t Re λ d(EA (λ)EA (∪∞ k=1 Δn(k) f, ∪k=1 Δn(k) f )

= σ(A)

by the properties of the s.m. and o.c.; ∞ = e2t Re λ d(EA (λ)f, f ) = k=1Δ n(k)

∪∞ k=1 Δn(k)

=

∞

e2t Re λ d(EA (λ)f, f )

e2t Re λ d(EA (λ)EA (Δn(k) )f, EA (Δn(k) )f )

k=1Δ n(k)

by (4.6); =

∞

e−2n(k) Re λn(k)

k=1

e2t Re λ d(EA (λ)en(k) , en(k) ) Δn(k)

for λ ∈ Δn(k) , by (4.3), Re λ = Re λn(k) + (Re λ − Re λn(k) ) ≤ Re λn(k) + |λ − λn(k) | ≤ Re λn(k) + εn(k) ≤ Re λn(k) + 1; ∞ −2n(k) Re λn(k) 2t(Re λn(k) +1) e e 1 d(EA (λ)en(k) , en(k) ) ≤ k=1

= e2t

∞

Δn(k)

e−2(n(k)−t) Re λn EA (Δn(k) )en(k) 2

k=1

by (4.4); = e2t

∞ k=1

e−2(n(k)−t) Re λn(k)


419

by the comparison test; < ∞. Indeed, given an arbitrary fixed t ≥ 0, for all sufficiently large natural n’s such that n(k) ≥ t + 1, by (4.5), we have: e−2(n(k)−t) Re λn(k) ≤ e−2k . Therefore,

;

f∈

D(etA ).

(4.7)

0≤t 0, we have similarly: T 2 (s|λ|) d(EA (λ)f, f ) σ(A)

=

∞

e−2n(k) Re λn(k)

k=1

T 2 (s|λ|) d(EA (λ)en(k) , en(k) ) = ∞. Δn(k)

Indeed, for λn(k) ∈ Δn(k) , by (4.3), (4.5), and (4.2), Re λ = Re λn(k) − (Re λn(k) − Re λ) ≥ Re λn(k) − |λn(k) − λ| ≥ Re λn(k) − εn(k) ≥ Re λn(k) − 1/n(k) ≥ 0 and

Re λ < 2−n(k) n(k) Hence, for λ ∈ Δn(k) ,

−1

M (| Im λ|).

|λ| ≥ | Im λ| ≥ M −1 (2n(k) n(k) Re λ) ≥ M −1 (2n(k) n(k)[Re λn(k) − 1/n(k)]). Using this estimate, we have: e−2n(k) Re λn(k) T 2 (s|λ|) d(EA (λ)en(k) , en(k) ) Δn(k)

≥e

−2n(k) Re λn(k)

e2M(sM

−1

(2n(k) n(k)[Re λn(k) −1/n(k)]))

d(EA (λ)en(k) , en(k) )

Δn(k)

by (2.3), for some h > 1, l > 0 and any natural k; ≥ e−2n(k) Re λn(k) Δn(k)

e

2[2−n(k) M(hn(k) sM −1 (2n(k) n(k)[Re λn(k) −1/n(k)]))+[1−2−n(k) ] ln(m0 l)]

d(EA (λ)en(k) , en(k) ) for all sufficiently large natural k’s such that hn(k) s ≥ 1; −n(k)

≥ (m0 l)2[1−2

] −2n(k) Re λn(k) 2n(k)(Re λn(k) −1/n(k))

e

e

= m0 le−2 .

420

M.V. Markin Hence, by (2.1), f ∈ C{mn } (A).

By (4.7) (see Introduction), the vector function y(t) = etA f,

0 ≤ t < ∞,

is a weak solution of (1.1) on [0, ∞). However, y(0) = f ∈ C{mn } A), which, by Proposition 3.1, implies that y(·) ∈ C{mn } [0, ∞), H). All the possibilities concerning the sequence {Re λn }∞ n=1 considered, we arrive at the conclusion: the negation of (iii) makes it possible to single out a weak solution of equation (1.1) on [0, ∞), which does not belong to the Carleman class C{mn } [0, ∞), H), i.e., implies the negation of (ii). At this point, let’s consider the strong Carleman ultradifferentiability of the weak solutions on the open semi-axis, (0, ∞). We are to observe that omitting the endpoint 0 leads to essentially different results. Theorem 4.2. Let the sequence {mn }∞ n=0 satisfy conditions (GR) and (SBC). Then every weak solution of equation (1.1) on [0, ∞) belongs to the Carleman class C{mn } (0, ∞), H if and only if there are such b+ > 0 and b− > 0 that the set σ(A) \ Mb− ,b+ is bounded. Proof. “If” part. Thus, our premise is that, there are b+ > 0 and b− > 0 such that the set σ(A) \ Mb− ,b+ is bounded. Consider an arbitrary weak solution y(·) of equation (1.1) on [0, ∞), which (see Introduction) is of the form: where f ∈

E 0≤t 0, let’s fix a sufficiently large natural N such that 2−N γ ≤ t/2, where γ := max(1, b−1 + ). Set s := H −N [2α−1 + 1]−1 > 0 where α > 0 is the constant from (2.2). Note that the choice of s > 0 depends on t > 0. For any t > 0, we have: T 2 (s|λ|) d(EA (λ)y(t), y(t)) = T 2 (s|λ|) d(EA (λ)etA f, etA f ) σ(A)

σ(A)

(4.8)

The Carleman Ultradifferentiability of Weak Solutions by the properties of the o.c.; = T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) = σ(A)

421

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)\Mb− ,b+

let R > 0 be the constant from estimate (2.2); T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) + σ(A)∩Mb− ,b+ ∩{λ|−M(R) 0, the integrals T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb− ,b+ ∩{λ| Re λ≥M(R)}


and σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}

are finite as well. For the first of two integrals (4.9), for an arbitrary t > 0, we have: T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}


= σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

(4.9)

422

M.V. Markin ≤

e2M(s[| Re λ|+| Im λ|]) e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb+ ∩{λ| Re λ≥1}

for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≥ M (R)}, | Im λ| ≤ M −1 (b−1 + Re λ); −1 −1 e2M(s[Re λ+M (b+ Re λ)]) e2t Re λ d(EA (λ)f, f ) ≤ σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≥ M (R)}, by (2.2): 2α−1 M −1 (Re λ) ≥ Re λ with some α > 0; −1 −1 −1 −1 e2M(s[2α M (Re λ)+M (b+ Re λ)]) ≤ σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

e2t Re λ d(EA (λ)f, f ) since γ := max(1, b−1 ); + ≤

−1

e2M(s[2α

+1]M −1 (γ Re λ)) 2t Re λ

e

d(EA (λ)f, f )

σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≥ M (R)}, according to (2.5): M (s[2α−1 + 1]M −1 (γ Re λ)) ≤ 2−N M (H n s[2α−1 + 1]M −1 (γ Re λ)) + [1 − 2−N ] ln(m0 L) with some H > 1 and L > 0; −N −N N −1 −1 e22 M(H s[2α +1]M (γ Re λ)) = (m0 L)2[1−2 ] σ(A)∩Mb+ ∩{λ| Re λ≥M(R)}

e

2t Re λ

d(EA (λ)f, f )

for s := H −N [2α−1 + 1]−1 > 0; −N

= (m0 L)2[1−2

]

e2[2

−N

γ+t] Re λ

σ(A)∩Mb+ ∩{λ| Re λ≥M(R)} 2[1−2−N ]

≤ (m0 L)

e2[2

−N

γ+t] Re λ

d(EA (λ)f, f ) < ∞

σ(A)

by (4.8). For the second of two integrals (4.9), we have: T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}


= σ(A)∩Mb+ ∩{λ| Re λ≤−M(R)}

d(EA (λ)f, f )


423

≤

e2M(s[| Re λ|+| Im λ|]) e2t Re λ d(EA (λ)f, f ) σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}

for λ ∈ σ(A) ∩ Mb− ,b+ ∩ {λ| Re λ ≤ −M (R)}, | Im λ| ≤ M −1 (b−1 − [− Re λ]); −1 −1 e2M(s[− Re λ+M (b− [− Re λ])]) e2t Re λ ≤ σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}

d(EA (λ)f, f ) due to (2.2), for λ ∈ σ(A) ∩ Mb− ,b+ ∩ {λ| Re λ ≤ −M (R)}, 2α−1 M −1 (− Re λ) ≥ (− Re λ) with some α > 0; −1 −1 −1 −1 ≤ e2M(s[2α M (− Re λ)+M (b− [− Re λ])]) σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}

e2t Re λ d(EA (λ)f, f ) recall that γ := max(1, b−1 − ); =

−1

e2M(s[2α

+1]M −1 (γ[− Re λ])) 2t Re λ

e

σ(A)∩Mb− ,b+ ∩{λ| Re λ≤−M(R)}

d(EA (λ)f, f ) for λ ∈ σ(A) ∩ Mb+ ∩ {λ| Re λ ≤ −M (R)}, by (2.5), M (s[2α−1 + 1]M −1 (γ[− Re λ])) ≤ 2−N M (H N s[2α−1 + 1]M −1 (γ[− Re λ])) + [1 − 2−N ] ln(m0 L) with some H > 1 and L > 0; −N N −1 −1 2[1−2−N ] e22 M(H s[2α +1]M (γ[− Re λ]) = (m0 L) σ(A)∩Mb+ ∩{λ| Re λ≤−M(R)}

e

2t Re λ

d(EA (λ)f, f )

for s := H −N [2α−1 + 1]−1 > 0; −N

= (m0 L)2[1−2

]

−N

e2[t−2

γ] Re λ

d(EA (λ)f, f )

σ(A)∩Mb+ ∩{λ| Re λ≤−M(R)}

since 2

−N

γ ≤ t/2; −N

≤ (m0 L)2[1−2

]

et Re λ d(EA (λ)f, f )

σ(A)∩Mb+ ∩{λ| Re λ≤−M(R)} 2[1−2−N ]

≤ (m0 L)

σ(A)

by (4.8).

et Re λ d(EA (λ)f, f ) < ∞

424

M.V. Markin

Thus, for an arbitrary weak solution y(·) of equation (1.1) on [0, ∞) and any t > 0, there is such an s > 0 that T 2 (s|λ|) d(EA (λ)y(t), y(t)) < ∞. σ(A)

Therefore, by (2.1), y(t) ∈ C{mn } (A), t > 0. The latter, by Proposition 3.1, implies that an arbitrary weak solution y(·) of equation (1.1) on [0, ∞) belongs to the Carleman class C{mn } (0, ∞), H . “Only if” part. Let’s prove this part by contrapositive. Thus, assume that, for any pair of positive constants b+ and b− , the set σ(A) \ Mb− ,b+ is unbounded. Then for any natural n, the set σ(A) \ Mn−2 ,n−2 is unbounded. Hence, we can choose a sequence of points in the complex plane {λn }∞ n=1 such that λn ∈ σ(A),

n = 1, 2, . . . ;

M (| Im λ|) < Re λn < n−2 M (| Im λ|), n = 1, 2, . . . ; λ0 := 0, |λn | > max n, |λn−1 | , n = 1, 2, . . . . −n

−2

The latter, in particular, implies that the points λn are distinct. Since the set , λ −n−2 M (| Im λ|) < Re λ < n−2 M (| Im λ|), |λ| > max [n, |λn−1 |] is open in C for any natural n, there exists such an εn > 0 that this set, along with the point λn , contains the open disk centered at λn : Δn = {λ ∈ C |λ − λn | < εn . Then, for any λ ∈ Δn , − n−2 M (| Im λ|) < Re λ < n−2 M (| Im λ|) |λ| > max n, |λn−1 | .

(4.10)

Furthermore, since the points λn are distinct, we can regard the radii of the disks, εn , to be small enough so that 1 , n = 1, 2, . . . ; n and Δi ∩ Δj = ∅, i = j. 0 < εn
0 such that | Re λn | ≤ ω, n = 1, 2, . . . . , Therefore, the set σ(A) ∩ λ ∈ C −ω ≤ Re λ ≤ ω is unbounded. Whence, by [16], Theorem 5.2, we infer that there is a weak solution y(·) of equation (1.1) on [0, ∞) such that y(·) ∈ C ∞ ((0, ∞), H). Moreover, y(·) ∈ C{mn } ((0, ∞), H). Now, suppose that the sequence {Re λn }∞ n=1 is unbounded. Hence, there is a subsequence {Re λn(k) }∞ such that k=1 Re λn(k) → ∞

or

Re λn(k) → −∞

as k → ∞.

First, assume that Re λn(k) → −∞ as k → ∞. In this case, one can regard without restricting generality that Re λn(k) ≤ −k, Consider the vector

k = 1, 2, . . . .

(4.13)

∞ 1 en(k) , f := k n=1

2 2 which is well defined since {1/k}∞ k=1 ∈ ( stands for the space of square summable numeric sequences). By the pairwise orthogonality of the projection operators EA (Δn ), we infer:

1 en(k) , k EA (∪∞ k=1 Δn(k) )f = f.

EA (Δn(k) )f =

k = 1, 2, . . . ,

For any t ≥ 0, e2t Re λ d(EA (λ)f, f ) σ(A)

by (4.14);

= σ(A)

∞ e2t Re λ d(EA (λ)EA (∪∞ k=1 Δn(k) )f, EA (∪k=1 Δn(k) )f )

(4.14)

426

M.V. Markin

by the properties of the s.m. and o.c.; ∞ 2t Re λ = e d(EA (λ)f, f ) = k=1Δ n(k)

∪∞ k=1 Δn(k)

=

∞

e2t Re λ d(EA (λ)f, f )

e2t Re λ d(EA (λ)EA (Δn(k) )f, EA (Δn(k) )f )

k=1Δ n(k)

by (4.14);

∞ 1 = k2 k=1

e2t Re λ d(EA (λ)en(k) , en(k) )

Δn(k)

for λ ∈ Δn(k) , by (4.13) and (4.11), Re λ = Re λn(k) + (Re λ − Re λn(k) ) ≤ Re λn(k) + |λ − λn | ≤ −k + 1 ≤ 0; ∞ ∞ 1 1 ≤ 1 d(E (λ)e , e ) = EA (Δn(k) )en(k) 2 A n(k) n(k) k2 k2 k=1

k=1

Δn(k)

by (4.12); ∞ 1 < ∞. k2 k=1 E Thus, f ∈ D(etA ), which (see Introduction) implies that the vector

=

0≤t 0, we have similarly: T 2 (s|λ|) d(EA (λ)y(1)f, y(1)f ) = T 2 (s|λ|) d(EA (λ)eA f, eA f ) σ(A)

σ(A)

∞ 1 = k2 k=1

e2M(s|λ|) e2 Re λ d(EA (λ)en(k) , en(k) ) = ∞.

Δn(k)

Indeed, for all λ ∈ Δn(k) , based on (4.11), (4.13), and (4.10), we have: Re λ = Re λn(k) + (Re λ − Re λn(k) ) ≤ Re λn(k) + |λn(k) − λ| ≤ Re λn(k) + εn(k) ≤ −k + 1 ≤ 0 and −n(k)−2 M (| Im λ|) < Re λ < n(k)−2 M (| Im λ|). Therefore, for λ ∈ Δn(k) , −n(k)−2 M (| Im λ|) < Re λ ≤ −k + 1 ≤ 0.


427

Whence, for λ ∈ Δn(k) : and |λ| ≥ | Im λ| ≥ M −1 (n(k)2 [− Re λ]).

Re λ ≤ −k + 1 ≤ 0

(4.15)

For any s > 0, let’s fix a sufficiently large natural N so that hN s ≥ 1, where h > 1 is the constant from condition (SBC). Then, using the preceding estimates, we have: 1 e2M(s|λ|) e2 Re λ d(EA (λ)en(k) , en(k) ) k2 Δn(k)

≥

1 k2

e2M(sM

−1

(n(k)2 [− Re λ])) 2 Re λ

e


Δn(k)

by (2.3), M (sM −1 (n(k)2 [− Re λ])) ≥ 2−N M (hN sM −1 (n(k)2 [− Re λ])) + [1 − 2−N ] ln(m0 l); −N N −1 2 2[1−2−N ] 1 e22 M(h sM (n(k) [− Re λ])) e2 Re λ ≥ (m0 l) k2 Δn(k)

d(EA (λ)en(k) , en(k) ) recall that hN s ≥ 1; −N

≥ (m0 l)2[1−2

] 2n(k) Re λn(k)

e2(2

e

−N

n(k)2 −1)[− Re λ]


Δn(k)

for all sufficiently large natural k’s such that 2−N n(k)2 − 1 ≥ 1, by (4.15); 1 2(k−1) e2(k−1) ≥ 2e 1 d(EA (λ)en(k) , en(k) ) = → ∞ as k → ∞. k k2 Δn(k)

Hence, for the weak solution y(·) of equation (1.1) on [0, ∞), T 2 (s|λ|) d(EA (λ)y(1)f, y(1)f ) = ∞, 0 < s < ∞. σ(A)

Whence, by (2.1), y(1) ∈ C{mn } (A). By Proposition 3.1 this implies that y(·) ∈ C{mn } (0, ∞), H . There remains the last possibility to be considered: Re λn(k) → ∞ as k → ∞. In this case, we can regard without restricting generality that Re λn(k) ≥ k,

k = 1, 2, . . . .

(4.16)

428

M.V. Markin Consider the vector f :=

∞

e−n(k) Re λn(k) en(k) ,

n=1 2 which is well defined since, by (4.16), {e−n(k) Re λn(k) }∞ k=1 ∈ . By the pairwise orthogonality of EA (Δn(k) ),

EA (Δn(k) )f = e−n(k) Re λn(k) en(k) , EA (∪∞ k=1 Δn(k) )f

k = 1, 2, . . . ,

= f.

(4.17)

For any t ≥ 0, in the same manner as above e2t Re λ d(EA (λ)f, f ) σ(A)

by (4.17); =

∞

e

−2n(k) Re λn(k)

k=1

e2t Re λ d(EA (λ)en(k) , en(k) ) Δn(k)

for λ ∈ Δn(k) , by (4.11), Re λ = Re λn(k) + (Re λ − Re λn(k) ) ≤ Re λn(k) + |λ − λn(k) | ≤ Re λn(k) + εn(k) ≤ Re λn(k) + 1; ∞ ≤ e−2n(k) Re λn(k) e2t(Re λn(k) +1) 1 d(EA (λ)en(k) , en(k) ) k=1

= e2t

∞

Δn

e−2[n(k)−t] Re λn(k) EA (Δn(k) )en(k) 2

k=1

by (4.12); = e2t

∞

e−2[n(k)−t] Re λn(k) < ∞

n=1

by the comparison test. Hence, f∈

;

D(etA ),

0≤t 0, we have similarly: T 2 (s|λ|) d(EA (λ)y(1)f, y(1)f ) = T 2 (s|λ|) d(EA (λ)eA f, eA f ) σ(A)

σ(A)


=

∞

e

−2n(k) Re λn(k)

k=1

=

∞

429

T 2 (s|λ|)e2 Re λ d(EA (λ)en(k) , en(k) ) Δn(k)

e

−2n(k) Re λn(k)

k=1

e2M(s|λ|) e2 Re λ d(EA (λ)en(k) , en(k) ) = ∞. Δn(k)

Indeed, for λ ∈ Δn(k) , based on (4.11), (4.16), and (4.10), we have: Re λ = Re λn(k) + (Re λ − Re λn(k) ) ≥ Re λn(k) − |λ − λn(k) | ≥ Re λn(k) − εn(k) ≥ Re λn(k) − 1/n(k) ≥ 0 and

−n(k)−2 M (| Im λ|) < Re λ < n(k)−2 M (| Im λ|). Therefore, for λ ∈ Δn(k) , 0 ≤ Re λn(k) − 1/n(k) ≤ Re λ < n(k)−2 M (| Im λ|).

Whence, for λ ∈ Δn(k) , Re λ ≥ Re λn(k) − 1/n(k) ≥ 0 and

|λ| ≥ | Im λ| ≥ M −1 (n(k)2 [Re λn(k) − 1/n(k)]).

For any s > 0, let’s fix a sufficiently large natural N so that hN s ≥ 1, where h > 1 is the constant from condition (SBC). Using the preceding estimates, we have: e−2n(k) Re λn(k) e2M(s|λ|) e2 Re λ d(EA (λ)en(k) , en(k) ) Δn(k)

≥ e−2n(k) Re λn(k)

e2M(sM

−1

(n(k)2 [Re λn(k) −1/n(k)])) 2[Re λn(k) −1/n(k)]

e

Δn(k)

d(EA (λ)en(k) , en(k) ) by (2.3), M (sM −1 (n(k)2 [Re λn(k) −1/n(k)])) ≥ 2−N M (hN sM −1 (n(k)2 [Re λn(k) − 1/n(k)])) + [1 − 2−N ] ln(m0 l), −N −N N −1 2 e22 M(h sM (n(k) [Re λn(k) −1/n(k)])) ≥ (m0 l)2[1−2 ] e−2n(k) Re λn(k) Δn(k)

e

[Re λn(k) −1/n(k)]


recall that h s ≥ 1; N

−N

≥ (m0 l)2[1−2

] −2n(k) Re λn(k)

e2(2

e

Δn(k)


−N

n(k)2 +1)[Re λn(k) −1/n(k)]

430

M.V. Markin

for all sufficiently large natural k’s such that 2−N n(k)2 + 1 ≥ n(k); −N 1 d(EA (λ)en(k) , en(k) ) ≥ (m0 l)2[1−2 ] e−2n(k) Re λn(k) e2n(k)[Re λn(k) −1/n(k)] Δn(k)

= m0 le

−2

.

Thus, in this case, there is also a weak solution y(·) of equation (1.1) on [0, ∞) such that, for any s > 0, T 2 (s|λ|) d(EA (λ)y(1)f, y(1)f ) = ∞, 0 < s < ∞, σ(A)

i.e., by (2.1), y(1) ∈ C{mn } (A). Whence, by Proposition 3.1, y(·) ∈ C{mn } (0, ∞), H . All the possibilities for the sequence {Re λn }∞ n=1 analyzed, we arrive at the conclusion that the assumption that, for any pair of positive constants b+ and b− , the set σ(A) \ Mb− ,b+ is unbounded implies that not every of weak solution equation (1.1) on [0, ∞) belongs to the Carleman class C{mn } (0, ∞), H . Theorem 4.3. Let the sequence {mn }∞ n=0 satisfy conditions (SGR) and (BC). Then every weak solution of equation (1.1) on [0, ∞) belongs to the Carleman class C(mn ) (0, ∞), H if and only if there is such a b+ > 0 that, for an arbitrary b− > 0, the set σ(A) \ Mb− ,b+ is bounded. Proof. “If” part. Our premise is that there is a b+ > 0 such that, for an arbitrary b− > 0, the set σ(A) \ Mb− ,b+ is bounded. Consider an arbitrary weak solution of equation (1.1) on [0, ∞) (see Introduction) where f ∈

E 0≤t 0 and s > 0, let’s fix a sufficiently large natural N so that h−N 2s ≤ 1, where h > 1 is the constant from condition (BC) and set b− := 2N t−1 . Since due to condition (SGR), α > 0 in estimate (2.2) is arbitrary, assume that α := 2b− .

We have:

T 2 (s|λ|) d(EA (λ)y(t), y(t)) = σ(A)

T 2 (s|λ|) d(EA (λ)etA f, etA f ) σ(A)

The Carleman Ultradifferentiability of Weak Solutions by the properties of the s.m. and the o.c.; = T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ) = σ(A)

431


σ(A)\Mb− ,b+


+ σ(A)∩Mb− ,b+ ∩{λ|−b− M(R) 0 is the constant from estimate (2.2); < ∞. Indeed, the finiteness of the first two integrals T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ), σ(A)\Mb− ,b+

and

T 2 (s|λ|)e2t Re λ d(EA (λ)f, f ), σ(A)∩Mb− ,b+ ∩{λ|−b− M(R) 0 and s > 0, T 2 (s|λ|) d(EA (λ)y(t), y(t)) < ∞. σ(A)

By (2.1), y(t) ∈ C(mn ) (A), t > 0. By Proposition 3.1, the latter implies that an arbitrary weak solution y(·) of equation (1.1) on [0, ∞) belongs to the Carleman class C(mn ) (0, ∞), H . “Only if” part. Let’s resort to proving by contrapositive once again. Assume that for any b+ > 0, there is such a b− > 0 that the set σ(A) \ Mb− ,b+ is unbounded.

434

M.V. Markin

Let’s show that under the latter premise, its stronger version can be assumed: there is a b− > 0 such that, for any b+ > 0, the set σ(A) \ Mb− ,b+ is unbounded. Indeed, there are two possibilities: , 1. For a certain b− > 0, the set λ ∈ σ(A) −b− M (| Im λ|) < Re λ ≤ 0 is unbounded. , 2. For any b− > 0, the set λ ∈ σ(A) −b− M (| Im λ|) < Re λ ≤ 0 is bounded. In the first case, as is easily seen, the set σ(A) \ Mb− ,b+ is unbounded with a certain b− > 0 and an arbitrary b+ > 0. In the second case, the initial assumption that, for any b+ > 0, there is such a b− > 0 that the set σ(A) \ Mb− ,b+ is, unbounded immediately implies that,-for any b− > 0 and any b+ > 0, the set λ ∈ σ(A) 0 < Re λ < b+ M (| Im λ|) is unbounded and the more so is the set σ(A) \ Mb− ,b+ . Thus, the initial assumption does imply that there is a b− > 0 such that, for any b+ > 0, the set σ(A) \ Mb− ,b+ is unbounded. In particular, for any natural n, the set σ(A) \ Mb− ,n−2 is unbounded. Therefore, we can choose a sequence of points of the complex plane, {λn }∞ n=1 , in the following manner: λn ∈ σ(A),

n = 1, 2, . . . ;

− b− M (| Im λ|) < Re λn < n−2 M (| Im λ|), n = 1, 2, . . . ; λ0 := 0, |λn | > max n, |λn−1 | , n = 1, 2, . . . . The latter, in particular, implies that the points λn are distinct. Since, for any natural n, the set , λ ∈ C −b− M (| Im λ|) < Re λ < n−2 M (| Im λ|), |λ| > max n, |λn−1 | is open in C, there exists such an εn > 0 that this set contains together with the point λn the open disk centered at λn : Δn := {λ ∈ C |λ − λn | < εn , i.e., for any λ ∈ Δn , − b− M (| Im λ|) < Re λ < n−2 M (| Im λ|) |λ| > max n, |λn−1 | .

(4.19)

Moreover, since the points λn are distinct, we can regard that the radii of the disks, εn , are chosen to be small enough so that 1 0 < εn < , n = 1, 2, . . . ; n (4.20) and Δi ∩ Δj = ∅, i = j. As we observed, the subspaces EA (Δn )H are nontrivial and pairwise orthogonal.


435

Let’s choose a unit vector en ∈ EA (Δn )H thereby obtaining an orthonormal sequence: en = EA (Δn )en ,

n = 1, 2, . . . ,

(ei , ej ) = δij .

(4.21)

Concerning the sequence of the real parts, {Re λn }∞ n=1 , there are two possibilities: it is either bounded, or not. First, assume that the sequence {Re λn }∞ n=1 is bounded, i.e., there is an ω > 0 such that | Re λn | ≤ ω, n = 1, 2, . . . . , Therefore, the set σ(A) ∩ λ ∈ C −ω ≤ Re λ ≤ ω is unbounded. Whence, by [16], Theorem 5.2, we infer that there is a weak solution y(·) of equation (1.1) on [0, ∞) such that y(·) ∈ C ∞ ((0, ∞), H). Moreover, y(·) ∈ C(mn ) ((0, ∞), H). Assume now that the sequence {Re λn }∞ n=1 is unbounded. Therefore, it has a subsequence {Re λn(k) }∞ such that k=1 Re λn(k) → ∞

or

Re λn(k) → −∞

as k → ∞.

The case when Re λn(k) → ∞ as k → ∞ is considered analogously to the corresponding case in the proof of the “only if” part of Theorem 4.2, where it was shown that there is such a weak solution y(·) of equation (1.1) on [0, ∞) that y(1) ∈ C{mn } (A),

(0, ∞), H . Moreover, y(·) ∈ which, by Proposition 3.1, implies that y(·) ∈ C {m } n C(mn ) (0, ∞), H . Now, assume that there is a subsequence {Re λn(k) }∞ k=1 such that Re λn(k) → −∞ as k → ∞. Without restricting generality, we can regard that Re λn(k) ≤ −k, Let f :=

k = 1, 2, . . . .

(4.22)

∞ 1 en(k) . k k=1

By the pairwise orthogonality of the projection operators EA (Δn ), 1 EA (Δn(k) )f = en(k) , k = 1, 2, . . . , k EA (∪∞ k=1 Δn(k) )f = f.

(4.23)

In the same manner as in the proof of the “only if” part of Theorem 4.2, we can show that ; f∈ D(etA ), 0≤t 0, there is such a b− > 0 that the set σ(A)\Mb− ,b+ is unbounded implies that not every weak solution of equation (1.1) on [0, ∞) belongs to the class C(mn ) (0, ∞), H .

5. Final remarks Due to the normality of the operator A, all the above criteria are formulated exclusively in terms of its spectrum, no restrictions on the operator’s resolvent behavior required, which makes them inherently qualitative and more transparent than similar results for semigroups of linear operators (cf. [12, 24, 21, 3, 23]). As is easily seen, in Theorems 4.1–4.3, the function M (·) can be replaced by a nonnegative continuous function F (·) such that C1 F (λ) ≤ M (λ) ≤ C2 F (λ) with some positive constants C1 and C2 , for all sufficiently large nonnegative λ’s. Note that it doesn’t matter whether the function F (·) has an inverse. With the sequences mn = [n!]β , n = 0, 1, 2, . . . , 1 ≤ β < ∞, satisfying conditions (GR) and (SBC), condition (SGR) for 1 < β < ∞, and estimates

438

M.V. Markin

(6.4) (see Appendix), the corresponding function M (·) is replaceable by λ1/β , 0 ≤ λ < ∞, 1 ≤ β < ∞. Thus, all the principal theorems of paper [17] regarding the strong Gevrey ultradifferentiability (see Introduction) of the weak solutions become the special cases of Theorems 4.1–4.3 with mn = [n!]β , n = 0, 1, 2, . . . and 1 ≤ β < ∞ or 1 < β < ∞, whichever appropriate. Observe also that Theorem 4.1 immediately implies the following effect of smoothness improvement: , -∞ Corollary 5.1. Let the sequence mn n=0 satisfy conditions (GR) and (BC). Then, if all weak solution of equation (1.1) on [0, ∞) belong to the Carleman class of Roumieu type C{mn } ([0, ∞), H), they automatically belong to the Carleman class of Beurling type C(mn ) ([0, ∞), H).

6. Appendix 6.1. Instances of sequences satisfying (GR)- and (BC)-conditions Consider mn = [n!]β , n = 0, 1, 2, . . . , with 0 ≤ β < ∞. As is easily seen, for 0 ≤ β < 1, condition (GR) is not satisfied (although (WGR) is); for 1 ≤ β < ∞ condition (GR) is satisfied and, for 1 < β < ∞, so is (SGR). Based on the well-known binomial coefficients identity, for 1 ≤ β < ∞ and n = 0, 1, 2, . . . , we have: n n β n β n n n n n n ≤ 2 = ≤ (n + 1) = (n + 1) 2β ≤ 2β+1 . k k k k=0

k=0

k=0

Thus, for 1 ≤ β < ∞, condition (SBC) is met as well. Let 1 ≤ β < ∞. According to Stirling’s formula, there a C = C(β) ≥ 1 such that [n!]β ≤ nβn ≤ Ceβn [n!]β , n = 0, 1, 2, . . . .

(6.1)

Let’s consider the family of functions ρλ (x) :=

λx , 0 ≤ x < ∞, xβx

1 ≤ λ < ∞ (00 := 1).

As is easily seen, the function ρλ (·) attains its maximum value on [0, ∞) at the point xλ = e−1 λ1/β . Therefore, −1 1/β λn ≤ sup ρλ (x) = ρλ (xλ ) = eβe λ . βn n≥0 n x≥0

sup

(6.2)


439

Let, for λ ≥ eβ , N be the integer part of xλ = e−1 λ1/β . Hence, N ≥ 1 and, for all sufficiently large λ ≥ eβ , λn λN ≥ βN = exp N ln λ − βN ln N (6.3) βn N n≥0 n βe−1 1/β −1 1/β ≥ exp (xλ − 1) ln λ − βxλ ln xλ = eβe λ −ln λ ≥ e 2 λ .

sup

Taking (6.2) and (6.3) into account, for all sufficiently large positive λ’s, we have: e

βe−1 2

by (6.1);

λ1/β

∞ ∞ β n e λ λn λn ≤ sup βn ≤ T (λ) := ≤C β n [n!] nβn n≥0 n=0 n=0

β n ∞ β n ∞ β n 2e λ 2e λ 1 2e λ 1 ≤ C sup = 2C sup =C βn n βn n n 2 n 2 nβn n≥0 n≥0 n=0 n=0 ≤ 2C sup ρ2eβ λ (x) x≥0

by (6.2); −1

= 2Ceβe

(2eβ λ)

1/β

1/β

≤ e4βλ

.

Thus, for all sufficiently large positive λ’s, βe−1 1/β λ ≤ M (λ) ≤ 4βλ1/β . 2

(6.4)

2

Let’s show that the sequence mn = en , n = 0, 1, 2, . . . , satisfies conditions (SGR) and (BC). Indeed condition (SGR) is met since, for any α > 0, 2 αn n! αn nn ≤ lim = lim en ln α+n ln n−n = 0. 2 2 n n→∞ e n→∞ en n→∞ On the other hand, for n = 0, 1, 2, . . . ,

0 ≤ lim

n

n en e2k(n−k) ≥ e2(n−1) = e−2 e2 . 2 (n−k)2 = k e e k=0 k=0 2

n

Thus, condition (BC) is also satisfied. Further, for 0 ≤ λ < ∞, we have: ∞ ∞ λn λn (2λ)n 1 (2λ)n ≤ ≤ 2 sup . sup n2 ≤ T (λ) := 2 2 n2 en en 2 n n≥0 e n≥0 e n=0 n=0

For 1 ≤ λ < ∞, consider the family of functions: ϕλ (x) :=

λx , ex2

0 ≤ x < ∞.

(6.5)

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M.V. Markin

As is easily verified, for each 1 ≤ λ < ∞, the function ϕλ (·) attains its ln λ maximum value on [0, ∞) at the point xλ = . Hence, for 1 ≤ λ < ∞, 2 [ln λ]2 λn sup n2 ≤ sup ϕλ (x) = ϕλ (xλ ) = e 4 . (6.6) n≥0 e x≥0 Let N be the integer part of xλ . Then, for all sufficiently large positive λ’s, [ln λ]2 [ln λ]2 λn ln λ ln λ 2 λN N ln λ−N 2 ≥ ≥ eln λ( 2 −1)−( 2 ) = e 4 −ln λ ≥ e 8 . (6.7) 2 2 = e n N e n≥0 e

sup

By (6.5)–(6.7), for all sufficiently large positive λ’s, e

[ln λ]2 8

≤ T (λ) ≤ 2e

[ln(2λ)]2 4

≤e

[ln λ]2 2

.

Whence, for all sufficiently large positive λ’s, 1 1 [ln λ]2 ≤ M (λ) ≤ [ln λ]2 . 8 2 2

Thus, for the sequence mn = en , n = 0, 1, 2, . . . , Theorems 4.1 and 4.3 are valid, with the function M (·) being replaceable by + 0 for 0 ≤ λ < 1 def F (λ) = [ln λ]2 for λ ≥ 1. 6.2. Independence (GR)-conditions and (BC)-conditions The question concerning the independence of (GR)-conditions and (BC)-conditions naturally arises. Let’s show that they are independent. √ As is easily seen, the sequence mn := n!, n = 0, 1, 2, . . . satisfies condition (SBC), but not (GR). Thus, (SBC) ⇒ (GR). Let’s show that (SGR) ⇒ (BC). Consider the sequence:

+ mn :=

n2n 4 en

for n = n(k) otherwise,

where n(0) := 1, n(1) := 2, n(k) := n(k − 2) + n(k − 1) + 1, k = 2, 3, . . . . The sequence growing so fast, it obviously meets condition (SGR). However, it doesn’t satisfy condition (BC). Indeed, consider the subsequence of sums

n(k)

i=0

mn(k) , mi mn(k)−i

k = 0, 1, 2, . . . .


441

For 1 ≤ i ≤ n(k) − 1, k = 1, 2, . . . , there are only two possibilities. Either i or n(k) − i equals n(l) with some 0 ≤ l ≤ k − 1, e.g., i = 1 or i = n(k) − 1. Neither i nor n(k) − i equals n(l) for any 0 ≤ l ≤ k − 1. For instance, i = 3 = n(l), 0 ≤ l ≤ k − 1, and n(k) − 3 = n(l), 0 ≤ l ≤ k − 1, whenever k ≥ 4. The former being self-evident, the latter is also easily seen: n(k) − n(l) ≥ n(k) − n(k − 1) = n(k − 2) + 1 ≥ n(2) + 1 = 5,

k ≥ 4.

In the first case, for k ≥ 3 and 1 ≤ i ≤ n(k) − 1, mn(k) mn(k) ≤ , mi mn(k)−i mn(k)−n(k−1) since, if i = n(l) with some 0 ≤ l ≤ k−1, then n(k)−i = n(k)−n(l) ≥ n(k)−n(k− 1); if n(k)−i = n(l) with some 0 ≤ l ≤ k−1, then i = n(k)−n(l) ≥ n(k)−n(k−1). Observe that, when k ≥ 3, n(k)−n(k−1) = n(k−2)+1 = n(l), l = 0, 1, 2, . . . . Therefore, in the first case, for k ≥ 3 and 1 ≤ i ≤ n(k) − 1, we have: mn(k) mn(k) mn(k) n(k)2n(k) ≤ = = (n(k−2)+1)4 mi mn(k)−i mn(k)−n(k−1) mn(k−2)+1 e 4

4

= e2n(k) ln n(k)−(n(k−2)+1) ≤ e2n(k) ln n(k)−n(k−2) → 0

as k → ∞.

Since, for k ≥ 3:

2 2 n(k) ln n(k) n(k) n(k − 1) + n(k − 2) + 1 ≤ = n(k − 2)4 n(k − 2)2 n(k − 2)2 2 2 n(k − 3) + 2n(k − 2) + 2 4n(k − 2) 16 = ≤ = → 0 as k → ∞. n(k − 2)2 n(k − 2)2 n(k − 2)2

In the second case, for k ≥ 4 and 1 ≤ i ≤ n(k) − 1, mn(k) mi mn(k)−i with N (i) := max(i, n(k) − i) ≥ (n(k)/2); 4 mn(k) n(k)2n(k) 2n(k) ln n(k)− n(k) 24 ≤ = ≤ e → 0 as k → ∞. 4 mN (i) eN (i) Thus, for all sufficiently large natural k’s, mn(k) ≤ 1, mi mn(k)−i

n(k)

1 ≤ i ≤ n(k) − 1,

i.e., condition (BC) is not satisfied.

and

i=0

mn(k) ≤ n(k) + 1, mi mn(k)−i

442

M.V. Markin

Acknowledgments Eternally grateful to his mother Svetlana A. Markina, whose love, constant support and unsurpassed patience made this humble dedication possible, the author cannot but express his cordial gratitude to Mrs. Evelyn Weil and Mrs. Linda Nahin for their enduring generous kindness.

References [1] J.M. Ball, Strongly Continuous Semigroups, Weak Solutions, and the Variation of Constants Formula. Proc. Amer. Math. Soc. 63 (1977), 370–373. ´ [2] T. Carleman, Edition Compl` ete des Articles de Torsten Carleman. Institut Mathématique Mittag-Leffler, Djursholm, Suède, 1960. [3] M.G. Crandall and A. Pazy, On the Differentiability of Weak Solutions of a Differential Equation in Banach Space. J. Math. and Mech. 18 (1969), 1007–1016. [4] N. Dunford, Survey of the theory of spectral operators. Bull. Amer. Math. Soc. 64 (1958), 217–274. [5] N. Dunford and J.T. Schwartz, Linear Operators, Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Interscience Publishers, 1963. [6] , Linear Operators, Part III: Spectral Operators. Interscience Publishers, 1971. ´ [7] M. Gevrey, Sur La Nature Analytique des Solutions des Equations aux Dérivées Partielles. Ann. Ec. Norm. Sup. Paris 35 (1918), 129–196. [8] R. Goodman, Analytic and Entire Vectors for Representations of Lie Groups. Trans. Amer. Math. Soc. 143 (1969), 55–76. [9] M. L. Gorbachuk and V.I. Gorbachuk, Boundary Value Problems for Operator Differential Equations. Kluwer Academic Publishers Group, 1991. [10] V.I. Gorbachuk, Spaces of Infinitely Differentiable Vectors of a Nonnegative Selfadjoint Operator. Ukr. Math. J. 35 (1983), 531–535. [11] V.I. Gorbachuk and A.V. Knyazyuk, Boundary Values of Solutions of OperatorDifferential Equations. Russ. Math. Surveys 44 (1989), 67–111. [12] E. Hille and R.S. Phillips, Functional Analysis and Semigroups. American Mathematical Society Colloquium Publications, vol. 31, 1957. [13] H. Komatsu, Ultradistributions, I. Structure Theorems and Characterization. J. Fac. Sci. Univ. Tokyo 20 (1973), 25–105. [14] S. Mandelbrojt, Series de Fourier et Classes Quasi-Analytiques de Fonctions. Gauthier-Villars, 1935. [15] M.V. Markin, On the Ultradifferentiability of Weak Solutions of a First-Order Operator-Differential Equation in Hilbert Space. Dopov. Akad. Nauk Ukrainy (1996), no. 6., 22–26. [16] , On the Smoothness of Weak Solutions of an Abstract Evolution Equation. I. Differentiability. Appl. Anal. 73 (1999), 573–606. [17] , On the Smoothness of Weak Solutions of an Abstract Evolution Equation. II. Gevrey Ultradifferentiability. Appl. Anal. 78 (2001), 97–137.

The Carleman Ultradifferentiability of Weak Solutions [18] [19] [20] [21] [22] [23] [24] [25]

443

, On the Carleman Classes of Vectors of a Scalar Type Spectral Operator. Int. J. Math. Math. Sci. 2004 (2004), 3219–3235. , On Scalar Type Spectral Operators and Carleman Ultradifferentiable C0 Semigroups. Ukr. Math. J. (to appear). E. Nelson, Analytic Vectors. Ann. Math. 70 (1959), 572–615. A. Pazy, On the Differentiability and Compactness of Semi-Groups of Linear Operators. J. Math. and Mech. 17 (1968), 1131–1141. A.I. Plesner, Spectral Theory of Linear Operators. Nauka, 1965 (in Russian). T. Ushijima, On the Generation and Smoothness of Semi-Groups of Linear Operators. J. Fac. Sci. Univ. Tokyo 19 (1972), 65–127. K. Yosida, On the Differentiability of Semi-Groups of Linear Operators. Proc. Japan Acad. 34 (1958), 337–340. K. Yosida, Functional Analysis. 6th Edition, Springer-Verlag, 1980.

Marat V. Markin Fresno, CA, USA e-mail: [email protected]



On the Equivalence of Different Lax Pairs for the Kac–van Moerbeke Hierarchy Johanna Michor and Gerald Teschl Abstract. We give a simple algebraic proof that the two different Lax pairs for the Kac–van Moerbeke hierarchy, constructed from Jacobi respectively supersymmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations. Mathematics Subject Classification (2000). Primary 47B36, 37K15; Secondary 81U40, 39A10. Keywords. Kac–van Moerbeke hierarchy, Lax pair, Toda hierarchy.

1. Introduction There are two different Lax equations for the Kac–van Moerbeke equation: The original one, found independently by Kac and van Moerbeke [6] and Manakov [7], based on a Jacobi matrix with zero diagonal elements and its skew-symmetrized square and the second one based on super-symmetric Dirac-type matrices. Both approaches can be generalized to give corresponding hierarchies of evolution equations in the usual way and both reveal a close connection to the Toda hierarchy. In fact, the first approach shows that the Kac–van Moerbeke hierarchy (KM hierarchy) is contained in the Toda hierarchy by setting b = 0 in the odd equations. The second one relates both hierarchies via a Bäcklund transformation since the Dirac-type difference operator gives rise to two Jacobi operators by taking squares (respectively factorizing positive Jacobi operators to obtain the other direction). Both ways of introducing the KM hierarchy have their merits, however, though it is obvious that both produce the same hierarchy by looking at the first few equations, we could not find a formal proof in the literature. The purpose of this short note is to give a simple algebraic proof for this fact. As a byproduct we will also obtain some new recursions for computing the equations in the KM hierarchy. Work supported by the Austrian Science Fund (FWF) under Grants No. Y330 and J2655.

446

J. Michor and G. Teschl

In Section 2 we review the recursive construction of the Toda hierarchy via Lax pairs involving Jacobi operators and obtain the KM hierarchy by setting b = 0 in the odd equations. In Section 3 we introduce the KM hierarchy via Lax pairs involving Dirac-type difference operators. In Section 4 we show that both constructions produce the same equations. Finally, we recall how to identify Jacobi operators with b = 0 in Section 5.

2. The Toda hierarchy In this section we introduce the Toda hierarchy using the standard Lax formalism following [2] (see also [5], [10]). We will only consider bounded solutions and hence require Hypothesis H.2.1. Suppose a(t), b(t) satisfy a(t) ∈ ∞ (Z, R),

b(t) ∈ ∞ (Z, R),

a(n, t) = 0, ∞

(n, t) ∈ Z × R,

∞

and let t → (a(t), b(t)) be differentiable in (Z) ⊕ (Z). Associated with a(t), b(t) is a Jacobi operator H(t) = a(t)S + + a− (t)S − + b(t)

(2.1)

in 2 (Z), where S ± f (n) = f ± (n) = f (n ± 1) are the usual shift operators and 2 (Z) denotes the Hilbert space of square summable (complex-valued) sequences over Z. Moreover, choose constants c0 = 1, cj , 1 ≤ j ≤ r, cr+1 = 0, and set gj (n, t) =

j

cj− δn , H(t) δn ,

=0

hj (n, t) = 2a(n, t)

j

(2.2) cj− δn+1 , H(t) δn + cj+1 .

=0

The sequences gj , hj satisfy the recursion relations g0 = 1, h0 = c1 , hj+1 −

h− j+1

−

2(a2 gj+

2gj+1 − hj − h− j − 2bgj = 0,

0 ≤ j ≤ r,

(a− )2 gj− )

0 ≤ j < r.

−

− b(hj −

h− j )

= 0,

(2.3)

Introducing P2r+2 (t) = −H(t)r+1 +

r (2a(t)gj (t)S + − hj (t))H(t)r−j + gr+1 (t),

(2.4)

j=0

a straightforward computation shows that the Lax equation d H(t) − [P2r+2 (t), H(t)] = 0, dt

t ∈ R,

(2.5)

Equivalence of Lax Pairs for the Kac–van Moerbeke Hierarchy is equivalent to

⎞ + a(t) ˙ − a(t) gr+1 (t) − gr+1 (t) ⎠ = 0, TLr (a(t), b(t)) = ⎝ ˙ − hr+1 (t) − h− (t) b(t) r+1

447

⎛

(2.6)

where the dot denotes a derivative with respect to t. Varying r ∈ N0 yields the Toda hierarchy TLr (a, b) = 0. The corresponding homogeneous quantities obtained by taking all summation constants equal to zero, c ≡ 0, ∈ N, are denoted by gˆj , ˆ j , etc., resp. h R r (a, b) = TLr (a, b) . (2.7) TL c ≡0,1≤ ≤r

Next we show that we can set b ≡ 0 in the odd equations of the Toda hierarchy. Lemma 2.2. Let b ≡ 0. Then the homogeneous coefficients satisfy ˆ 2j = 0, gˆ2j+1 = h

j ∈ N0 .

Proof. We use induction on the recursion relations (2.3). The claim is true for ˆ 2j = 0 then gˆ2j+1 = 0, and h ˆ 2j = 0 follows from the last equation in j = 0. If h (2.3). In particular, if we choose c2 = 0 in TL2r+1 , then we can set b ≡ 0 to obtain a hierarchy of evolution equations for a alone. In fact, set Gj = gˆ2j ,

ˆ 2j+1 , Kj = h

(2.8)

in this case. Then they satisfy the recursion G0 = 1,

K0 = 2a2 ,

2Gj+1 − Kj − Kj− = 0, Kj+1 −

− Kj+1

− 2(a

2

G+ j

− 2

− (a )

G− j )

= 0,

0 ≤ j ≤ r, 0 ≤ j < r,

(2.9)

and TL2r+1 (a, 0) = 0 is equivalent to the KM hierarchy defined as KMr (a) = a˙ − a(G+ r+1 − Gr+1 ),

r ∈ N0 .

(2.10)

3. The Kac–van Moerbeke hierarchy as a modified Toda hierarchy In this section we review the construction of the KM hierarchy as a modified Toda hierarchy. We refer to [2], [10] for further details. Suppose ρ(t) satisfies Hypothesis H.3.1. Let ρ(t) ∈ ∞ (Z, R),

ρ(n, t) = 0, (n, t) ∈ Z × R

and let t → ρ(t) be differentiable in ∞ (Z).

(3.1)

448

J. Michor and G. Teschl Define the “even” and “odd” parts of ρ(t) by ρe (n, t) = ρ(2n, t), ρo (n, t) = ρ(2n + 1, t),

(n, t) ∈ Z × R,

(3.2)

and consider the bounded operators (in 2 (Z)) − A(t) = ρo (t)S + + ρe (t), A(t)∗ = ρ− o (t)S + ρe (t).

(3.3)

In addition, we set H1 (t) = A(t)∗ A(t),

H2 (t) = A(t)A(t)∗ ,

(3.4)

with − Hk (t) = ak (t)S + + a− k (t)S + bk (t),

k = 1, 2,

(3.5)

and a1 (t) = ρe (t)ρo (t),

2 b1 (t) = ρe (t)2 + ρ− o (t) ,

(3.6)

a2 (t) = ρ+ e (t)ρo (t),

b2 (t) = ρe (t)2 + ρo (t)2 .

(3.7)

Now we define operators D(t), Q2r+2 (t) in (Z, C ) as follows, 0 A(t)∗ , D(t) = A(t) 0 0 P1,2r+2 (t) , r ∈ N0 . Q2r+2 (t) = 0 P2,2r+2 (t) 2

2

(3.8) (3.9)

Here Pk,2r+2 (t), k = 1, 2 are defined as in (2.4), that is, Pk,2r+2 (t) = −Hk (t)r+1 +

r (2ak (t)gk,j (t)S + − hk,j (t))Hk (t)j + gk,r+1 , (3.10) j=0

{gk,j (n, t)}0≤j≤r , {hk,j (n, t)}0≤j≤r+1 are defined as in (2.2). Moreover, we choose the same integration constants in P1,2r+2 (t) and P2,2r+2 (t) (i.e., c1, = c2, ≡ c , 1 ≤ ≤ r). Analogous to equation (2.5) one obtains that d D(t) − [Q2r+2 (t), D(t)] = 0 dt

(3.11)

is equivalent to KMr (ρ) = (KMr (ρ)e , KMr (ρ)o ) ρ˙ e − ρe (g2,r+1 − g1,r+1 ) = 0. = + ρ˙ o + ρo (g2,r+1 − g1,r+1 )

(3.12)

As in the Toda context (2.6), varying r ∈ N0 yields the KM hierarchy which we denote by KMr (ρ) = 0, r ∈ N0 . (3.13) The homogeneous KM hierarchy is denoted by S r (ρ) = KMr (ρ) KM

c ≡0,1≤ ≤r

.

(3.14)

Equivalence of Lax Pairs for the Kac–van Moerbeke Hierarchy

449

One look at the transformations (3.6), (3.7) verifies that the equations for ρo , ρe are in fact one equation for ρ. More explicitly, combining gk,j , resp. hk,j , into one sequence Gj (2n) = g1,j (n) , Gj (2n + 1) = g2,j (n)

resp.

Hj (2n) = h1,j (n) , Hj (2n + 1) = h2,j (n)

(3.15)

we can rewrite (3.12) as KMr (ρ) = ρ˙ − ρ(G+ r+1 − Gr+1 ).

(3.16)

¿From (2.3) we see that Gj , Hj satisfy the recursions G0 = 1, H0 = c1 , 2Gj+1 − Hj − Hj−− − 2(ρ2 + (ρ− )2 )Gj −− − 2 −− Hj+1 − Hj+1 − 2((ρρ+ )2 G+ j − (ρ ρ) Gj ) −(ρ2 + (ρ− )2 )(Hj − Hj−− )

= 0,

0 ≤ j ≤ r,

= 0,

0 ≤ j < r.

(3.17)

ˆ j , etc., as before. ˆj , H The homogeneous quantities are denoted by G As a simple consequence of (3.11) we have d D(t)2 − [Q2r+2 (t), D(t)2 ] = 0 dt and observing

2

D(t) =

H1 (t) 0

0 H2 (t)

(3.18)

(3.19)

yields the implication KMr (ρ) = 0 ⇒ TLr (ak , bk ) = 0,

k = 1, 2,

(3.20)

that is, given a solution ρ of the KMr equation (3.13), one obtains two solutions, (a1 , b1 ) and (a2 , b2 ), of the TLr equations (2.6) related to each other by the Miuratype transformations (3.6), (3.7). For more information we refer to [4], [5], [9], [10], [11], and [12].

4. Equivalence of both constructions In this section we want to show that the constructions of the KM hierarchy outlined in the previous two sections yield in fact the same set of evolution equations. This will follow once we show that Gj defined in (2.8) is the same as Gj defined in (3.15). It will be sufficient to consider the homogeneous quantities, however, we will omit the additional hats for notational simplicity. Moreover, we will denote ˜ j to distinguish it from the one defined the sequence Gj defined in (2.8) by G ˜ j , Kj in (3.15). Since both are defined recursively via the recursions (2.9) for G respectively (3.17) for Gj , Hj our first aim is to eliminate the additional sequences ˜ j respectively Gj alone. Kj respectively Hj and to get a recursion for G

450


Lemma 4.1. The coefficients gj (n) satisfy the following linear recursion + + − gj+3 = (b + 2b+ )gj+2 − (2b + b+ )gj+2 gj+3 + + − (2b + b+ )b+ gj+1 + b(2b+ + b)gj+1 + kj+1 + kj+1

+

bkj+

− b kj ,

kj = a2 gj+ − (a− )2 gj− ,

j ∈ N.

b(b+ )2 gj+

− b b gj − + 2

(4.1)

+

where (4.2)

Proof. It suffices to consider the homogeneous case gj (n) = δn , H j δn . Then (compare [10, Sect 6.1]) g(z, n) = δn , (H − z)−1 δn = −

∞ gj (n) j=0

z j+1

satisfies [10, (1.109)] (a+ )2 g ++ − a2 g a2 g + − (a− )2 g − = (z − b+ )g + − (z − b)g, + z − b+ z−b and the claim follows after comparing coefficients.

˜ j , defined by (2.8) and corresponding Corollary 4.2. For j ∈ N0 , the sequences G to the TL hierarchy with b ≡ 0, satisfy ˜+ − G ˜ j+1 = (a+ )2 G ˜+ − G ˜ j ) − (a− )2 G ˜ ++ + a2 (G ˜−. G j+1 j j j

(4.3)

The corresponding sequences Gj for the KM hierarchy defined in (3.15) satisfy + 2 − 2 2 2 (a ) + (a++ )2 Gj Gj+3 − G++ j+3 = (a ) + a 2 + 2 + (a−− )2 (a− )2 G−− j+1 + a (a ) Gj+1 + 2 + (a ) + (a++ )2 2(a− )2 + 2a2 + (a+ )2 + (a++ )2 G++ j+1 + 2(a− )2 + 2a2 + (a+ )2 + (a++ )2 Gj+2 2 − (a− )2 + a2 (a+ )2 + (a++ )2 G++ j (4.4) −− 2 − 2 −− + 2 ++ 2 − (a ) + (a ) (a ) (a ) Gj − a2 (a+ )2 G++ j − (a− )2 + a2 a2 (a+ )2 Gj − (a++ )2 (a+++ )2 G++++ j − (a− )2 + a2 (a− )2 + a2 + 2(a+ )2 + 2(a++ )2 Gj+1 ++ 2 +++ 2 ++++ − a2 (a+ )2 G++ ) (a ) Gj+1 j+1 − (a − 2 − (a ) + a2 + 2(a+ )2 + 2(a++ )2 G++ j+2 .

Proof. Use (4.1) with b ≡ 0 for (4.3) resp. (3.6), (3.7) with a = ρ for (4.4).

Lemma 4.3. For all n ∈ Z, ˜ j (n) = Gj (n), G

j ∈ N0 .

(4.5)


451

˜ j satisfy the linear recursion relation (4.4) for Proof. Our aim is to show that G ˆ j . We start with (4.3), G ˜+ + G ˜+ − G ˜ ++ = −(a+ )2 G ˜ j+2 − G ˜ + ) + (a− )2 G ˜ ++ + a2 (G ˜− ˜ j+3 − G G j+3 j+3 j+3 j+2 j+2 j+2 ++ ++ 2 ˜ +++ + 2 ˜+ 2˜ ˜ − (a ) G + (a ) (G − G ) + a Gj+2 , j+2

j+2

j+2

(4.6) and observe that the right hand side of (4.4) only involves even shifts of Gj . Hence ˜ j by (4.3), we systematically replace in (4.6) odd shifts of G + ˜ + − (a+ )2 G ˜ j−1 − G ˜ + ) + (a− )2 G ˜ ++ + a2 (G ˜− G1,j := G j j−1 j−1 j−1 ˜j = G − − 2 ˜+ − 2 ˜ −− 2 ˜ −− , ˜ ˜ G2,j := Gj + a Gj−1 + (a ) (Gj−1 − Gj−1 ) − (a ) Gj−1 as follows: ˜ +++ → G+++ , G j+2 2,j+2

+ ˜ + → xG+ G j+2 1,j+2 + (1 − x)G2,j+2 ,

˜ − → G− , G j+2 1,j+2

with x=

(a− )2 + a2 + (a++ )2 . a2 − (a+ )2

In the resulting equation we replace ˜ +++ → G+++ , G j+1 2,j+1

+ ˜ + → yG+ G j+1 1,j+1 + (1 − y)G2,j+1 ,

˜ − → G− , G j+1 1,j+1

where y=

(a− )2 (a++ )2 + a2 (a++ )2 . a2 (a++ )2 − (a− )2 (a+ )2

˜j . This gives (4.4) for G

Hence both constructions for the KM hierarchy are equivalent and we have Theorem 4.4. Let r ∈ N0 . Then TL2r+1 (a, 0) = KMr (a).

(4.7)

KM provided cTL and cTL 2j+1 = cj 2j = 0 for j = 0, . . . , r.

Remark 4.5. As pointed out by M. Gekhtman to us, an alternate way of proving equivalence is by showing that (in the semi-infinite case, n ∈ N) both constructions give rise to the same set of evolutions for the moments of the underlying spectral measure (compare [1]). Our purely algebraic approach has the advantage that it does neither require the semi-infinite case nor self-adjointness.

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5. Appendix: Jacobi operators with b ≡ 0 In order to get solutions for the Kac–van Moerbeke hierarchy out of solutions of the Toda hierarchy one clearly needs to identify those cases which lead to Jacobi operators with b ≡ 0. For the sake of completeness we recall some folklore results here. Let H be a Jacobi operator associated with the sequences a, b as in (2.1). Recall that under the unitary operator U f (n) = (−1)n f (n) our Jacobi operator transforms according to U −1 H(a, b)U = H(−a, b), where we write H(a, b) in order to display the dependence of H on the sequences a and b. Hence, in the special case b ≡ 0 we infer that H and −H are unitarily equivalent, U −1 HU = −H. In particular, the spectrum is symmetric with respect to the reflection z → −z and it is not surprising, that this symmetry plays an important role. Denote the diagonal and first off-diagonal of the Green’s function of a Jacobi operator H by g(z, n) = δn , (H − z)−1 δn , h(z, n) = 2a(n)δn+1 , (H − z)−1 δn − 1.

(5.1)

Then we have Theorem 5.1. For a given Jacobi operator, b ≡ 0 is equivalent to g(z, n) = −g(−z, n) and h(z, n) = h(−z, n). ˜ = −U −1 HU , then the corresponding diagonal and first off-diagonal Proof. Set H ˜ n) = h(−z, n). Hence the elements are related via g˜(z, n) = −g(−z, n) and h(z, claim follows since g(z, n) and h(z, n) uniquely determine H (see [10, Sect. 2.7] respectively [8] for the unbounded case). Note that one could alternatively use recursions: Since gj (n) and hj (n) are just the coefficients in the asymptotic expansions of g(z, n) respectively h(z, n) around z = ∞ (see [10, Chap. 6]), our claim is equivalent to g2j+1 (n) = 0 and h2j (n) = 0. Similarly, b ≡ 0 is equivalent to m± (z, n) = −m± (−z, n), where m± (z, n) = δn±1 , (H±,n − z)−1 δn±1

(5.2)

are the Weyl m-functions. Here H±,n are the two half-line operators obtained from H by imposing an additional Dirichlet boundary condition at n. The corresponding spectral measures are of course symmetric in this case. For a quasi-periodic algebro-geometric solution (see, e.g., [10, Chap. 9]), this implies b ≡ 0 if and only if both the spectrum and the Dirichlet divisor are symmetric with respect to the reflection z → −z (cf. [3, Chap. 3]). For an N soliton solution this implies b ≡ 0 if and only if the eigenvalues come in pairs, E and −E, and the norming constants associated with each eigenvalue pair are equal.


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Acknowledgments We thank Michael Gekhtman and Fritz Gesztesy for valuable discussions on this topic and hints with respect to the literature. G.T. would like to thank all organizers of the international conference on Modern Analysis and Applications in honor of Mark Krein, Odessa, April 2008, for their kind invitation and the stimulating atmosphere during the meeting.

References [1] Y. Berezansky and M. Shmoish, Nonisospectral flows on semi-infinite Jacobi matrices, J. Nonlinear Math. Phys. 1, no. 2, 116–146 (1994). [2] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc. 135–641, (1998). [3] B.A. Dubrovin and V.B. Matveev and S.P. Novikov, Non-linear equations of the Korteweg–de Vries type, finite-zone linear operators and Abelian varieties, Russ. Math. Surv. 31, 59–146, (1976). [4] F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, On the Toda and Kac-van Moerbeke systems, Trans. Amer. Math. Soc. 339, 849–868 (1993). [5] F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1+1)-Dimensional Discrete Models, Cambridge Studies in Advanced Mathematics 114, Cambridge University Press, Cambridge, 2008. [6] M. Kac and P. van Moerbeke, On an explicitly soluble system of nonlinear differential equations, related to certain Toda lattices, Adv. Math. 16, 160–169 (1975). [7] S.V. Manakov, Complete integrability and stochastization of discrete dynamical systems, Soviet Physics JETP 40, 269–274 (1975). [8] G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators, Comm. Math. Phys. 196, 175–202 (1998). [9] G. Teschl, On the Toda and Kac–van Moerbeke hierarchies, Math. Z. 231, 325–344 (1999). [10] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000. [11] M. Toda, Theory of Nonlinear Lattices, 2nd enl. edition, Springer, Berlin, 1989. [12] M. Toda and M. Wadati, A canonical transformation for the exponential lattice, J. Phys. Soc. Jpn. 39, 1204–1211 (1975).

Johanna Michor Imperial College 180 Queen’s Gate London SW7 2BZ and

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International Erwin Schr¨ odinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien, Austria e-mail: [email protected] URL: http://www.mat.univie.ac.at/∼jmichor/ Gerald Teschl Faculty of Mathematics Nordbergstrasse 15 1090 Wien, Austria and International Erwin Schr¨ odinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien, Austria e-mail: [email protected] URL: http://www.mat.univie.ac.at/∼gerald/


Elliptic Problems and Hörmander Spaces Vladimir A. Mikhailets and Aleksandr A. Murach Abstract. The paper gives a survey of the modern results on elliptic problems on the H¨ ormander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic H¨ ormander spaces parametrized by a real number and a function slowly varying at +∞ in the Karamata sense. This refined scale is finer than the Sobolev scale and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this scale. A local refined smoothness of the elliptic problem solution is studied. An abstract construction of classes of function spaces in which the elliptic problem is a Fredholm one is found. In particular, some generalizations of the Lions-Magenes theorems are given. Mathematics Subject Classification (2000). Primary: 35J30, 35J40; Secondary: 46E35. Keywords. H¨ ormander spaces, generalized smoothness, interpolation with a function parameter, elliptic operator, elliptic boundary-value problem, the Fredholm property, local regularity of solutions, the Lions-Magenes theorems.

0. Introduction The paper gives a survey of the modern results [32–49] devoted to elliptic problems on the Hilbert scale of the isotropic H¨ ormander spaces 1/2 ·s ϕ(·) s,ϕ H := H2 , ξ := 1 + |ξ|2 . (0.1) Here s ∈ R and ϕ is a functional parameter slowly varying at +∞ in the Karamata sense. In particular, every standard function ϕ(t) = (log t)r1 (log log t)r2 . . . (log . . . log t)rk , {r1 , r2 , . . . , rk } ⊂ R,

t ! 1,

k ∈ Z+ ,

is admissible. This scale contains the Sobolev scale {H s } ≡ {H s,1 }, is attached to it by the number parameter s, and much finer than {H s }.

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V.A. Mikhailets and A.A. Murach

Spaces of form (1) arise naturally in different spectral problems: convergence of spectral expansions of self-adjoint elliptic operators almost everywhere, in the norm of the spaces Lp with p > 2 or C (see survey [6]); spectral asymptotics of general self-adjoint elliptic operators in a bounded domain, the Weyl formula, a sharp estimate of the remainder in it (see [30, 31]) and others. They may be expected to be useful in other “fine” questions. Due to their interpolation properties, the spaces H s,ϕ occupy a special position among the spaces of a generalized smoothness, which are actively investigated and used today (see survey [23], recent articles [19, 14] and the bibliography given therein). The paper consists of six sections. In Section 1 the refined scale of the Hörmander spaces (0.1) is introduced and studied. In particular, important interpolation properties of this scale are under investigation. In Section 2 an elliptic pseudodifferential operator on the refined scale on a closed compact smooth manifold is considered. We show that this operator is a Fredholm one and establishes a collection of isomorphisms on the two-sided refined scale. The local refined smoothness of a solution to the elliptic equation is studied. We also give an equivalent definition of the refined scale on the closed manifold by means of certain functions of a positive elliptic operator. Next we study a regular elliptic boundary problem on a bounded Euclidean domain with the smooth boundary. In Section 3 we show that the operator of this problem is a Fredholm one on the upper part of the refined scale. A local refined smoothness up to the boundary of a solution to the problem is studied. As an important application, we give a sufficient condition for the solution to be classical. Section 4 is devoted to semihomogeneous elliptic boundary problems. We show that these problems are Fredholm on the two-sided refined scales. Since the operator of the general nonhomogeneous boundary problem cannot be defined correctly on the lower part of the refined scale, we consider in Section 5 a special modified refined scale on which the operator is well defined, bounded, and Fredholm everywhere. This modification depends solely on the order of the problem, so that the theorem on the Fredholm property is generic for the class of elliptic problems having the same order. The last Section 6 is devoted to some individual theorems on the Fredholm property. We give an abstract construction of classes of function spaces on which the elliptic problem operator is a Fredholm one. A characteristic feature of this construction is that the domain of the operator depends on coefficients of the elliptic expression. So, we have the individual theorems on the Fredholm property. As an important application, we give some generalizations of the known LionsMagenes theorems.

1. A refined scale of H¨ ormander spaces Let us denote by M the set of all functions ϕ : [1, +∞) → (0, +∞) such that: a) ϕ is a Borel measurable function;

Elliptic Problems and H¨ ormander Spaces

457

b) the functions ϕ and 1/ϕ are bounded on every closed interval [1, b], where 1 < b < +∞; c) ϕ is a slowly varying function at +∞ in the Karamata sense (see [61, Sec. 1.1]), i.e., lim ϕ(λ t)/ϕ(t) = 1 for each λ > 0. t→ +∞

Let s ∈ R and ϕ distributions w on the the distribution w is a the condition

∈ M. We denote by H s,ϕ (Rn ) the space of all tempered Euclidean space Rn such that the Fourier transform w 2 of locally Lebesgue integrable on Rn function which satisfies 2 ξ2s ϕ2 (ξ) |w(ξ)| 2 dξ < ∞. Rn

+ · · · + ξn2 )1/2 is the smoothed modulus of a vector ξ = Here ξ = (1 + n (ξ1 , . . . , ξn ) ∈ R . An inner product in the space Hs,ϕ (Rn ) is defined by the formula ξ2s ϕ2 (ξ) w R1 (ξ) w R2 (ξ) dξ. (w1 , w2 )Hs,ϕ (Rn ) := ξ12

Rn

The inner product induces the norm in Hs,ϕ (Rn ) in the usual way. Note that we consider distributions which are antilinear functionals on the space of test functions. The space H s,ϕ (Rn ) is a special isotropic Hilbert case of the spaces introduced and investigated by L. H¨ ormander [20, Sec. 2.2], [21, Sec. 10.1] and the different spaces studied by L.R. Volevich and B.P. Paneah [65, Sec. 2], [53, Sec. 1.4.2]. In the simplest case where ϕ(·) ≡ 1, the space H s,ϕ (Rn ) coincides with the Sobolev space H s (Rn ). The inclusions ; < H s+ε (Rn ) =: H s+ (Rn ) ⊂ H s,ϕ (Rn ) ⊂ H s− (Rn ) := H s−ε (Rn ) ε>0

imply that in the set of separable Hilbert spaces , s,ϕ n H (R ) : s ∈ R, ϕ ∈ M ,

ε>0

(1.1)

the functional parameter ϕ defines an additional (subpower) smoothness with respect to the basic (power) s-smoothness. Otherwise speaking, ϕ refines the power smoothness s. Therefore, the collection of spaces (1.1) is naturally called the refined scale over Rn (with respect to the Sobolev scale). We are going to study an application of the refined scale to elliptic boundary problems in a bounded domain Ω ⊂ Rn . Therefore, we need to have the refined scales over the domain Ω and over its boundary ∂Ω. The refined scale over the closed domain Ω := Ω ∪ ∂Ω is also of use. We construct these scales from (1.1) in the standard way. Let us denote , H s,ϕ (Ω) := u = w Ω : w ∈ H s,ϕ (Rn ) , , u H s,ϕ (Ω) := inf w H s,ϕ (Rn ) : w ∈ H s,ϕ (Rn ), w = u ∈ Ω .

458


The norm in the space H s,ϕ (Ω) is induced by the inner product u1 , u2 H s,ϕ (Ω) := w1 − Πw1 , w2 − Πw2 H s,ϕ (Rn ) . Here wj ∈ H s,ϕ (Rn ), wj = uj ∈ Ω for j = 1, 2, and Π is the orthogonal projector of the space H s,ϕ (Rn ) onto the subspace {w ∈ H s,ϕ (Rn ) : supp w ⊆ Rn \ Ω}. The space H s,ϕ (Ω) is a separable Hilbert one. We also denote , HΩs,ϕ (Rn ) := w ∈ H s,ϕ (Rn ) : supp w ⊆ Ω . This space is a separable Hilbert one with respect to the inner product in the space H s,ϕ (Rn ). Thus the space H s,ϕ (Ω) consists of the distributions given in the open domain Ω, whereas the space HΩs,ϕ (Rn ) consists of the distributions supported on the closed domain Ω. The collections of Hilbert spaces , s,ϕ , s,ϕ n H (Ω) : s ∈ R, ϕ ∈ M and HΩ (R ) : s ∈ R, ϕ ∈ M (1.2) are called the refined scales over Ω and over Ω respectively. The boundary ∂Ω is assumed to possess an infinitely smooth field of unit vectors of normals. So, ∂Ω is a particular case of a compact closed infinitely smooth manifold. Let us define the refined scale over a closed infinitely smooth manifold Γ of an arbitrary dimension n. We choose a finite atlas from the C ∞ -structure on the manifold Γ consisting of the local charts αj : Rn ↔ Uj , j = 1, . . . , r. Here the open sets Uj form the finite covering of the manifold Γ. Let functions χj ∈ C ∞ (Γ), j = 1, . . . , r, form a partition of unity on Γ satisfying the condition supp χj ⊂ Uj . We set H s,ϕ (Γ) := {h ∈ D (Γ) : (χj h) ◦ αj ∈ H s,ϕ (Rn ),

j = 1, . . . , r} .

Here, as usual, D (Γ) is the topological space of all distributions on Γ, and (χj h)◦αj is the representation of the distribution χj h in the local chart αj . The inner product in the space H s,ϕ (Γ) is defined by the formula (h1 , h2 )H s,ϕ (Γ) :=

r

((χj h1 ) ◦ αj , (χj h2 ) ◦ αj )H s,ϕ (Rn )

j=1

and induces the norm in the usual way. The Hilbert space H s,ϕ (Γ) is separable and does not depend (up to equivalence of norms) on the choice of the atlas and the partition of unity. The collection of function spaces {H s,ϕ (Γ) : s ∈ R, ϕ ∈ M} (1.3) is called the refined scale over the manifold Γ. Specifically, we need the refined scale of spaces H s,ϕ (∂Ω).


459

We note the following properties of the refined scales: Theorem 1.1. Let s ∈ R and ϕ, ϕ1 ∈ M. The following assertions are true: (i) The set C ∞ ( Ω ) is dense in the space H s,ϕ (Ω). (ii) The set C0∞ (Ω) := {w ∈ C ∞ (Rn ) : supp w ⊂ Ω} is dense in the space HΩs,ϕ (Rn ). (iii) If |s| < 1/2, then the mapping w → w Ω establishes a topological isomorphism from HΩs,ϕ (Rn ) onto H s,ϕ (Ω). (iv) For each ε > 0 the compact and dense embeddings hold: H s+ε, ϕ1 (Ω) → H s,ϕ (Ω),

HΩs+ε, ϕ1 (Rn ) → HΩs,ϕ (Rn ).

(1.4)

(v) Suppose that the function ϕ/ϕ1 is bounded in a neighborhood of +∞. Then continuous dense embeddings (1.4) are valid for ε = 0. They are compact if ϕ(t)/ϕ1 (t) → 0 as t → +∞. (vi) For every fixed integer k ≥ 0 the inequality +∞ dt 1/2. Assertions (iv)–(vi) show that the refined scale is much finer than the classical Sobolev scale (the case of ϕ ≡ ϕ1 ≡ 1). Note also that ϕ ∈ M ⇔ 1/ϕ ∈ M, so the −s,1/ϕ space HΩ (Rn ) in assertion (vii) is defined as an element of the refined scale. Theorem 1.2. Let s ∈ R and ϕ, ϕ1 ∈ M. Then: (i) Assertions (i) and (iv)–(vi) of Theorem 1.1 hold true if we replace both notations (Ω) and ( Ω ) with (Γ). (ii) The spaces H s,ϕ (Γ) and H −s,1/ϕ (Γ) are mutually dual (up to equivalence of norms) with respect to the inner product in the space L2 (Γ, dx), where dx is a C ∞ -smooth density on Γ. The refined scale of spaces (1.1), (1.2), and (1.3) were introduced and investigated by authors in [32, 34, 39]. Theorems 1.1, 1.2 were proved in [34, Theorem 3.6] and [39, Theorem 4.2]. All assertions of these theorems, except (iii), follow from the properties of H¨ ormander spaces [20, Sec. 2.2], [21, Sec. 10.1] (see also [65, Sec. 2], [53, Sec. 1.4.2]). The refined scale possesses the interpolation property which selects the scale from among the spaces of generalized smoothness. Namely, every space of this scale is obtained by the interpolation, with an appropriate function parameter, of

460


a couple of the Sobolev spaces. We recall the definition of such an interpolation in the case of general separable Hilbert spaces. Let an ordered couple X := [X0 , X1 ] of complex Hilbert spaces X0 and X1 be such that these spaces are separable and the continuous dense embedding X1 → X0 holds true. We call this couple admissible. For the couple X there exists an isometric isomorphism J : X1 ↔ X 0 such that J is a self-adjoint positive operator in the space X 0 with the domain X1 . This operator is uniquely determined by the couple X. Let a Borel measurable function ψ : (0, +∞) → (0, +∞) be given. We denote by [X0 , X1 ]ψ or simply by Xψ the domain of the operator ψ(J) endowed with the graphics inner product and the corresponding norm: (u, v)Xψ := (u, v)X0 + (ψ(J)u, ψ(J)v)X0 ,

1/2

u Xψ = (u, u)Xψ .

The space Xψ is a separable Hilbert one. The function ψ is called an interpolation parameter if the following condition is fulfilled for all admissible couples X = [X0 , X1 ], Y = [Y0 , Y1 ] of Hilbert spaces and an arbitrary linear mapping T given on X0 : if the restriction of the mapping T to the space Xj is a bounded operator T : Xj → Yj for each j = 0, 1, then the restriction of the mapping T to the space Xψ is also a bounded operator T : Xψ → Yψ . Theorem 1.3. Let a function ϕ ∈ M and positive numbers ε, δ be given. We set ψ(t) := t ε/(ε+δ) ϕ(t1/(ε+δ) ) for t ≥ 1

and

ψ(t) := ϕ(1) for 0 < t < 1.

Then the function ψ is an interpolation parameter and, for each s ∈ R, the following equalities of spaces with equivalence of norms in them are true: s−ε,1 H (G), H s+δ,1 (G) ψ = H s,ϕ (G) for G ∈ {Rn , Ω, Γ}, s−ε,1 n HΩ (R ), HΩs+δ,1 (Rn ) ψ = HΩs,ϕ (Rn ). The refined scale is closed with respect to the interpolation with a function parameter ψ(t) := tθ χ(t) where 0 < θ < 1, whereas χ(t) is a Borel measurable positive function slowly varying at +∞. Theorem 1.4. Let s0 , s1 ∈ R, s0 ≤ s1 , and ϕ0 , ϕ1 ∈ M. In the case where s0 = s1 we suppose that the function ϕ0 /ϕ1 is bounded in a neighborhood of +∞. Let a Borel measurable function ψ : (0, +∞) → (0, +∞) is of the form ψ(t) := tθ χ(t), where 0 < θ < 1 and χ(t) is a function slowly varying at +∞. Then ψ is an interpolation parameter, and the following equalities of spaces with equivalence of norms in them are true: s0 ,ϕ0 H (G), H s1 ,ϕ1 (G) ψ = H s,ϕ (G) for G ∈ {Rn , Ω, Γ}, s0 ,ϕ0 n HΩ (R ), HΩs1 ,ϕ1 (Rn ) ψ = HΩs,ϕ (Rn ). Here s := (1 − θ)s0 + θs1 , and the function ϕ ∈ M is given by the formula ϕ(t) := ϕ1−θ (t) ϕθ1 (t) χ ts1 −s0 ϕ1 (t)/ϕ0 (t) for t ≥ 1. 0


461

The interpolation of general Hilbert spaces with a function parameter was studied in [15, 12, 54, 40]. The class of all interpolation parameters was described in [54] (see also [40, Theorem 2.7]). Theorem 1.3 was proved in [34, Theorems 3.1, 3.5] and [39, Theorem 4.1]. Theorem 1.4 was proved in [40, Theorem 3.7] for the refined scale over Γ (the proof for the scales (1.1) and (1.2) is analogous). Various normed spaces of generalized smoothness over Rn were studied by means of the interpolation with a function parameter in [29, 11].

2. An elliptic operator over a closed manifold We recall that Γ is a closed (compact and without a boundary) infinitely smooth manifold of an arbitrary dimension n ≥ 1 and a certain C ∞ -density dx is defined on Γ. We interpret D (Γ) as a space antidual to C ∞ (Γ) with respect to the extension of the inner product in L2 (Γ, dx) by continuity. This extension is denoted by (f, w)Γ for f ∈ D (Γ), w ∈ C ∞ (Γ). Let A be a classical (polyhomogeneous) pseudodifferential operator on Γ of an arbitrary order r ∈ R. The complete symbol of A is an infinitely smooth complexvalued function on the cotangent bundle T ∗ Γ. We assume that pseudodifferential operator A is elliptic on Γ. The mapping u → Au is a linear continuous operator on the space D (Γ). We will investigate the restriction of this operator to spaces of the refined scale over Γ. Let us denote by A+ a pseudodifferential operator formally adjoint to A with respect to the sesquilinear form (·, ·)Γ . Since both A and A+ are elliptic on Γ, both spaces , N := { u ∈ C ∞ (Γ) : Au = 0 on Γ } , N + := v ∈ C ∞ (Γ) : A+ v = 0 on Γ are finite-dimensional. Let us recall the following: a linear bounded operator T : X → Y is called a Fredholm one if its kernel is finite-dimensional and its range T (X) is closed in the space Y and has the finite codimension therein. Here X and Y are Hilbert spaces. The Fredholm operator T has the finite index ind T := dim ker T − dim(Y / T (X)). Theorem 2.1. A restriction of the mapping u → Au, u ∈ D (Γ), establishes the linear bounded operator A : H s,ϕ (Γ) → H s−r, ϕ (Γ)

for each s ∈ R, ϕ ∈ M.

(2.1)

This operator is a Fredholm one, has the kernel N and the range , f ∈ H s−r, ϕ (Γ) : (f, v)Γ = 0 ∀ v ∈ N + . The index of the operator (2.1) is equal to dim N − dim N + and does not depend on s and ϕ.

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Theorem 2.2. For arbitrarily chosen parameters s ∈ R, ϕ ∈ M, and σ < s, the following a priori estimate holds true: u H s,ϕ (Γ) ≤ c Au H s−r, ϕ (Γ) + u H σ,ϕ (Γ) , u ∈ H s,ϕ (Γ). Here the number c > 0 does not depend on u. If the spaces N and N + are trivial, then the operator (2.1) is a topological isomorphism. Generally, it is convenient to construct the isomorphism with the help of two projectors. Let us decompose the spaces from (2.1) into the following direct sums of (closed) subspaces: , H s,ϕ (Γ) = N u ∈ H s,ϕ (Γ) : (u, w)Γ = 0 ∀ w ∈ N , , H s−r, ϕ (Γ) = N + f ∈ H s−r, ϕ (Γ) : (f, v)Γ = 0 ∀ v ∈ N + . We denote by P and P + respectively the projectors of these spaces on the second terms in the sums in parallel to the first terms. The projectors do not depend on s, ϕ. Theorem 2.3. Let s ∈ R and ϕ ∈ M. The restriction of the operator (2.1) to the subspace P (H s,ϕ (Γ)) establishes the topological isomorphism A : P (H s,ϕ (Γ)) ↔ P + (H s−r, ϕ (Γ)). Theorems 2.1–2.3 were proved in [48, Sec. 4]. They specify, with regard to the refined scale, the known theorems on properties of an elliptic pseudodifferential operator on the Sobolev scale (see [22, Theorem 19.2.1] or [4, Theorems 2.3.3, 2.3.12]). Note that the boundedness of the operator (2.1) holds true without the assumption about ellipticity of A. If dim Γ ≥ 2, then the index of operator (2.1) is equal to zero [7], [4, Sec. 2.3 f]. In the case where dim Γ = 1, the index can be nonzero. There is a class of elliptic operators depending on a complex parameter (so-called parameter elliptic operators) such that N = N + = {0} for all values of the parameter sufficiently large in modulus [4, Sec. 4.1]. Moreover for a solution to a parameter elliptic equation, a certain two-sided a priory estimate holds with constants independent of the parameter. Such an estimate was obtained for the refined scale in [48, Theorem 6.1]. The analogs of Theorems 2.1–2.3 for different types of elliptic matrix operators were proved in [46, 49, 42]. Let us study a local smoothness of an elliptic equation solution in the refined scale. Let Γ0 be an nonempty open set on the manifold Γ. We denote , s,ϕ (Γ0 ) := f ∈ D (Γ) : χ f ∈ H s,ϕ (Γ) ∀ χ ∈ C ∞ (Γ), supp χ ⊆ Γ0 . Hloc Theorem 2.4. Let u ∈ D (Γ) be a solution to the equation Au = f on Γ0 with s,ϕ s+r, ϕ (Γ0 ) for some s ∈ R and ϕ ∈ M. Then u ∈ Hloc (Γ0 ). f ∈ Hloc This theorem and the analog of Theorem 1.1 (vi) for the refined scale over Γ imply the following sufficient condition for a solution u to have continuous derivatives of a prescribed order.


463

Theorem 2.5. Let u ∈ D (Γ) be a solution to the equation Au = f on Γ0 , where k−r+n/2, ϕ f ∈ Hloc (Γ0 ) for a certain integer k ≥ 0 and a function parameter ϕ satisfying inequality (1.5). Then u ∈ C k (Γ0 ). Theorems 2.4 and 2.5 were proved in [48, Sec. 5]. Theorem 2.5 shows an advantage of the refined scale over the Sobolev scale when a classical smoothness of a solution is under investigation. Indeed, if we restrict ourselves to the case of k−r+n/2, ϕ (Γ0 ) with the condition f ∈ ϕ ≡ 1, we have to replace the condition f ∈ Hloc k−r+ε+n/2, 1 Hloc (Γ0 ) for some ε > 0. The last condition is far stronger than previous one. The analogs of Theorems 2.4 and 2.5 for elliptic matrix operators were proved in [46, 49, 42]. A local regularity of an elliptic system solution in the Sobolev scale was investigated in [20, Sec. 10.6]. We also note that, in the H¨ ormander spaces, regularity properties of solutions to hypoelliptic partial differential equations with constant coefficients were studied in [20, Ch. IV], [21, Ch. 11] At the end of this section we give, with the help of A, an alternative and equivalent definition of the refined scale over the closed manifold Γ. Let us assume that ord A = r > 0 and that the operator A : C ∞ (Γ) → C ∞ (Γ) is positive in the space L2 (Γ, dx). We denote by A0 the closure of this operator in L2 (Γ, dx). Let s ∈ R, ϕ ∈ M, and ϕs,r (t) := ts/r ϕ(t1/r ) for t ≥ 1

and ϕs,r (t) := ϕ(1) for 0 < t < 1.

The operator ϕs,r (A0 ) is regarded in L2 (Γ, dx) as the Borel function ϕs,r of the self-adjoint operator A0 . Theorem 2.6. For arbitrary s ∈ R and ϕ ∈ M, the space H s,ϕ (Γ) coincides with the completion of the set of all functions u ∈ C ∞ (Γ) with respect to the norm ϕs,r (A0 ) u L2 (Γ) , which is equivalent to the norm u H s,ϕ (Γ) . An important example of the operator A mentioned above is the operator 1 − /Γ , where /Γ is the Beltrami-Laplace operator on the Riemannian manifold Γ (then r = 2). Theorem 2.6 was proved in [40, Sec 3.8]. For equivalent definition of the Sobolev scale over Γ, the powers of A0 is used instead of the regular varying function ϕs (see [4, Sec 5.3]).

3. An elliptic boundary problem on the one-sided scale Let us recall that Ω is a bounded domain in Rn , were n ≥ 2, and that its boundary ∂Ω is a closed infinitely smooth manifold of the dimension n − 1. We consider the nonhomogeneous boundary problem in the domain Ω: Lu ≡ lμ Dμ u = f in Ω, (3.1) Bj u ≡

|μ|≤mj

|μ|≤2q

bj,μ Dμ u = gj on ∂Ω, j = 1, . . . , q.

(3.2)

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Here L and Bj are linear partial differential expressions with complex-valued coefficients lμ ∈ C ∞ ( Ω ) and bj,μ ∈ C ∞ (∂Ω). We suppose that ord L = 2q is an even positive number and ord Bj = mj ≤ 2q − 1 for all j = 1, . . . , q. Let m := max {m1 , . . . , mq }. In what follows the boundary problem (3.1), (3.2) is assumed to be regular elliptic. It means that the expression L is proper elliptic in Ω, and the system B := (B1 , . . . , Bq ) of the boundary expressions is normal and satisfies the complementing condition with respect to L on ∂Ω (see [27], [63, Sec. 5.2.1]). It follows from the condition of normality that all numbers mj , j = 1, . . . , q, are distinct. We will investigate the mapping u → (Lu, Bu) in appropriate spaces of the refined scales. To describe the range of this mapping, we consider the boundary problem L+ v = ω in Ω, (3.3) Bj+ v = hj on ∂Ω, j = 1, . . . , q,

(3.4)

formally adjoint to the problem (3.1), (3.2) with respect to the Green formula (Lu, v)Ω +

q

(Bj u, Cj+ v)∂Ω = (u, L+ v)Ω +

j=1

q

(Cj u, Bj+ v)∂Ω , u, v ∈ C ∞ ( Ω ).

j=1

(3.5) Here L+ is the linear differential expression formally adjoint to L, and {Bj+ }, {Cj }, {Cj+ } are some normal systems of linear differential boundary expressions. Their coefficients are infinitely smooth, and their orders satisfy the equalities ord L+ = 2q,

ord Bj + ord Cj+ = ord Cj + ord Bj+ = 2q − 1.

+ We denote m+ j := ord Bj . In (3.5) and bellow, the notations (·, ·)Ω and (·, ·)∂Ω stand for the inner products in the spaces L2 (Ω) and L2 (∂Ω) respectively, and also denote the extensions by continuity of these products. We set

N := {u ∈ C ∞ ( Ω ) : Lu = 0 in Ω, Bj u = 0 on ∂Ω, N + := {v ∈ C ∞ ( Ω ) : L+ v = 0 in Ω, Bj+ v = 0 on ∂Ω,

j = 1, . . . , q}, j = 1, . . . , q}.

Since both problems (3.1), (3.2) and (3.3), (3.4) are regular elliptic, both spaces N and N + are finite dimensional. Theorem 3.1. Let s > m + 1/2 and ϕ ∈ M. The mapping (L, B) : u → (Lu, B1 u, . . . , Bq u),

u ∈ C ∞ ( Ω ),

(3.6)

is extended by a continuity to the bounded linear operator (L, B) : H s,ϕ (Ω) → H s−2q, ϕ (Ω) ⊕

q Q j=1

H s−mj −1/2, ϕ (∂Ω) =: Hs,ϕ (Ω, ∂Ω). (3.7)


465

This operator is a Fredholm one. Its kernel coincides with N , and its range is equal to the set q % & (f, g1 , . . . , gq ) ∈ Hs,ϕ (Ω, ∂Ω) : (f, v)Ω + (gj , Cj+ v)∂Ω = 0 ∀ v ∈ N + . (3.8) j=1

The index of the operator (3.7) is equal to dim N − dim N + and does not depend on s, ϕ. In this theorem and in the next theorems of the section, the condition s > m + 1/2 is essential. Indeed, if s < mj + 1/2 for some j = 1, . . . , q, then the mapping u → Bj u, u ∈ C ∞ ( Ω ), cannot be extended to the continuous linear operator Bj : H s,ϕ (Ω) → D (∂Ω). Thus the operator (3.6) is correctly defined on the upper refined one-sided scale {H s,ϕ (Ω) : s > m + 1/2, ϕ ∈ M . Hence the left-hand sides of equations (3.1), (3.2) is defined for each u ∈ H s,ϕ (Ω) with s > m + 1/2, whereas these equations are understood in the theory of distributions. Theorem 3.2. For arbitrarily chosen parameters s > m + 1/2, ϕ ∈ M, and σ < s, the following a priori estimate holds true: u H s,ϕ (Ω) ≤ c (L, B)u Hs,ϕ (Ω,∂Ω) + u H σ,ϕ (Ω) , u ∈ H s,ϕ (Ω). Here the number c > 0 does not depend on u. If the spaces N and N + are trivial, then the operator (3.7) is a topological isomorphism. In general, we can get the isomorphism with the help of two projectors. Let the spaces in which the operator (3.7) acts be decomposed into the following direct sums of subspaces: , H s,ϕ (Ω) = N u ∈ H s,ϕ (Ω) : (u, w)Ω = 0 ∀ w ∈ N , , Hs,ϕ (Ω, ∂Ω) = (v, 0, . . . , 0) : v ∈ N + (3.8). We denote by P and Q+ respectively the projectors of these spaces on the second terms in the sums in parallel to the first terms. The projectors are independent of s and ϕ. Theorem 3.3. Let s > m + 1/2 and ϕ ∈ M. The restriction of the operator (3.7) to the subspace P(H s,ϕ (Ω)) establishes the topological isomorphism (L, B) : P(H s,ϕ (Ω)) ↔ Q+ (Hs,ϕ (Ω, ∂Ω)). Theorems 3.1–3.3 were proved in [34, Sec. 4]. The boundedness of the operator (3.7) holds true without the assumption that the boundary problem (3.1), (3.2) is elliptic. In the paper [62] this problem was studied in a different scale of the Hörmander spaces (also called a refined one). Theorems 3.1–3.3 specify, with regard to the refined scale, the known theorems on properties of an elliptic boundary problem in the Sobolev one-sided scale (see [1, Ch. V], [27, Ch. 2, Sec. 5.4], [22, Ch.

466


20], [5, Sec 2, 4]). The analogs of Theorems 3.1–3.3 are valid for nonregular elliptic boundary problems [34] and for elliptic problems for systems of partial differential equations [47]. The case where the boundary operators have distinct orders on different connected components of the domain Ω was considered especially in [45]. There is a class of elliptic boundary problems depending on a parameter λ ∈ C such that N = N + = {0} for |λ| ! 1, and hence the index of the corresponding operator is equal to 0 for all λ (see [2, 3], [5, Sec. 3]). For a solution to such a parameter elliptic problem, a certain two-sided a priory estimate holds with constants independent of the parameter λ ∈ C with |λ| ! 1. Such an estimate was obtained for the refined scale in [35, Theorem 7.2]. Regular elliptic boundary problems in positive one-sided scales of different normed spaces were studied in [1, 63, 64]. Now we study an increase in a local smoothness of an elliptic boundary problem solution. Let U be an open subset in Rn . We set Ω0 := U ∩ Ω = ∅ and Γ0 := U ∩ ∂Ω (the case were Γ0 = ∅ is possible). Let us introduce the following local analogs of spaces of the refined scales: , σ,ϕ Hloc (Ω0 , Γ0 ) := u ∈ D (Ω) : χ u ∈ H σ,ϕ (Ω) ∀ χ ∈ C ∞ (Ω), supp χ ⊆ Ω0 ∪ Γ0 , , σ,ϕ Hloc (Γ0 ) := h ∈ D (∂Ω) : χ h ∈ H σ,ϕ (∂Ω) ∀ χ ∈ C ∞ (∂Ω), supp χ ⊆ Γ0 . Here σ ∈ R, ϕ ∈ M and, as usual, D (Ω) denotes the topological space of all distributions in Ω. Theorem 3.4. Let s > m + 1/2 and η ∈ M. Suppose that the distribution u ∈ H s,η (Ω) is a solution to the problem (3.1), (3.2), where s−2q+ε, ϕ (Ω0 , Γ0 ) f ∈ Hloc

and

s−mj −1/2+ε, ϕ

gj ∈ Hloc

for some ε ≥ 0 and ϕ ∈ M. Then u ∈

(Γ0 ),

j = 1, . . . , q,

s+ε, ϕ Hloc (Ω0 , Γ0 ).

Note that in the case where Ω0 = Ω and Γ0 = ∂Ω we have the global smoothness increase (i.e., the increase in the whole closed domain Ω). If Γ0 = ∅, then we get an interior smoothness increase (in an open subset Ω0 ⊆ Ω). Theorems 3.4 and 1.1 (vi) imply the following sufficient condition for the solution u to be classical. Theorem 3.5. Let s > m + 1/2 and χ ∈ M. Suppose that the distribution u ∈ H s,χ (Ω) is a solution to the problem (3.1), (3.2) in which n/2, ϕ

f ∈ Hloc

(Ω, ∅) ∩ H m−2q+n/2, ϕ (Ω),

gj ∈ H m−mj +(n−1)/2, ϕ (∂Ω), j = 1, . . . , q, and the function parameter ϕ ∈ M satisfies condition (1.5). Then the solution u is classical, that is u ∈ C 2q (Ω) ∩ C m ( Ω ). Theorems 3.4, 3.5 were proved in [35, Sec. 5, 6] (generally, for a non regular elliptic problem). The analog of Theorem 3.4 is valid for elliptic boundary problems for systems of partial differential equations [47]. In the Sobolev positive one-sided scale (s ≥ 0, ϕ ≡ 1), a smoothness of solutions to elliptic boundary problems was investigated in [52, 10, 59], [9, Ch. 3, Sec. 4] (see also [5, Sec. 2.4]).


467

4. Semihomogeneous elliptic problems 4.1. As we have mentioned, the results of Section 3 are not valid for s < m + 1/2 because the mapping (3.6) cannot be extended to the bounded linear operator (3.7). But if the boundary problem (3.1), (3.2) is semihomogeneous (i.e., f ≡ 0 or all gj ≡ 0), it establishes a bounded and Fredholm operator in the two-sided refined scale (for all real s). We will consider separately the case of the homogeneous elliptic equation (3.1) and the case of the homogeneous boundary conditions (3.2). 4.2. A boundary problem for a homogeneous elliptic equation Let us consider the regular elliptic boundary problem (3.1), (3.2), provided that f ≡ 0: Lu = 0 on Ω, Bj u = gj on ∂Ω, j = 1, . . . , q. (4.1) We will connect the following spaces with this problem: , KL∞ (Ω) := u ∈ C ∞ ( Ω ) : L u = 0 in Ω , , KLs,ϕ (Ω) := u ∈ H s,ϕ (Ω) : L u = 0 in Ω for s ∈ R, ϕ ∈ M. It follows from a continuity of the embedding H s,ϕ (Ω) → D (Ω) that KLs,ϕ (Ω) is a closed subspace in H s,ϕ (Ω). We can consider KLs,ϕ (Ω) as a Hilbert space with respect to the inner product in H s,ϕ (Ω). Theorem 4.1. Let s ∈ R and ϕ ∈ M. The set KL∞ (Ω) is dense in the space KLs,ϕ (Ω). The mapping u → Bu = (B1 u, . . . , Bq u),

u ∈ KL∞ (Ω),

is extended by a continuity to the bounded linear operator B : KLs,ϕ (Ω) →

q Q

H s−mj −1/2, ϕ (∂Ω) =: Hs,ϕ (∂Ω).

(4.2)

j=1

This operator is a Fredholm one. Its kernel coincides with N , and its range is equal to the set q & % (g1 , . . . , gq ) ∈ Hs,ϕ (∂Ω) : (gj , Cj+ v)∂Ω = 0 ∀ v ∈ N + . j=1

The index of the operator (4.2) is equal to dim N − dim G + where , G + := C1+ v, . . . , Cq+ v : v ∈ N + , and does not depend on s, ϕ. Theorem 4.1 was proved in [38, Sec. 6]. In contrast to Theorem 3.1, the ellipticity condition is essential for the boundedness of the operator (4.2) in the case where s ≤ m + 1/2. Note that dim G + ≤ dim N + where the strict inequality is possible that results from [21, Theorem 13.6.15]. In the case where ϕ ≡ 1 and s ∈ R\{−1/2, −3/2, −5/2, . . .} Theorem 4.1 is a consequence of the Lions–Magenes Theorems [27, Ch. 2, Sec. 6.6, 7.3] (see also [25, 26] and [28, Sec. 6.10, 6.12]).

468


4.3. An elliptic problem with homogeneous boundary conditions Now we will consider the regular elliptic boundary problem (3.1), (3.2), provided that all gj ≡ 0: Lu = f in Ω,

Bj u = 0 on ∂Ω,

j = 1, . . . , q.

(4.3)

Let us introduce the function spaces in which the operator of the problem (4.3) acts. For the sake of brevity, we denote by (b.c.) the homogeneous boundary conditions in (4.3). In addition, we denote by (b.c.)+ the homogeneous boundary conditions Bj+ v = 0 on ∂Ω, j = 1, . . . , q. They correspond to the formally adjoint boundary problem (3.3), (3.4). We set , C ∞ (b.c.) := u ∈ C ∞ ( Ω ) : Bj u = 0 on ∂Ω, j = 1, . . . , q , , C ∞ (b.c.)+ := v ∈ C ∞ ( Ω ) : Bj+ v = 0 on ∂Ω, j = 1, . . . , q . Let s ∈ R and ϕ ∈ M. We define the Hilbert space H s,ϕ,(0) (Ω) in the following way:

+ H

s,ϕ,(0)

(Ω) :=

H s,ϕ (Ω) HΩs,ϕ (Rn )

for s ≥ 0, for s < 0.

According to Theorem 1.1 (iii), (vii), the spaces H s,ϕ,(0) (Ω) and H −s,1/ϕ,(0) (Ω) are mutually dual for every s ∈ R with respect to the inner product in L2 (Ω). It also follows from Theorem 1.1 (i), (ii) that the set C ∞ ( Ω ) is dense in the space H s,ϕ,(0) (Ω) for each s ∈ R. Here we identify each function f ∈ C ∞ ( Ω ) with its extension by zero + Of (x) :=

f (x) 0

for x ∈ Ω, for x ∈ Rn \ Ω

(4.4)

which is a regular distribution in HΩs,ϕ (Rn ) for s < 0. Now one may conclude that Theorem 1.1 (iii), (iv) implies the continuous dense embedding H s1 ,ϕ1 ,(0) (Ω) → H s,ϕ,(0) (Ω) for − ∞ < s < s1 < ∞, and ϕ, ϕ1 ∈ M. Finally, let us define the Hilbert spaces H s,ϕ (b.c.) and H s,ϕ (b.c.)+ of distributions satisfying the homogeneous boundary conditions. In the case where s∈ / {mj + 1/2 : j = 1, . . . , q} we denote by H s,ϕ (b.c.) the closure of C ∞ (b.c.) in the space H s,ϕ,(0) (Ω). In the case where s ∈ {mj + 1/2 : j = 1, . . . , q} we define the space H s,ϕ (b.c.) by means of the interpolation with the parameter ψ(t) = t1/2 : (4.5) H s,ϕ (b.c.) := H s−1/2, ϕ (b.c.), H s+1/2, ϕ (b.c.) t1/2 . If we change (b.c.) for (b.c.)+ , and mj for m+ j in the last two sentences, we give the s,ϕ + definition of the space H (b.c.) . Note that in the case where s ∈ {mj +1/2 : j = 1, . . . , q} the norms in the spaces H s,ϕ (b.c.) and H s,ϕ,(0) (Ω) are not equivalent. The analogous fact is true for H s,ϕ (b.c.)+ .


469

Proposition 4.1. Let s > 0, s = mj + 1/2 for all j = 1, . . . , q, and ϕ ∈ M. Then , H s,ϕ (b.c.) = u ∈ H s,ϕ (Ω) : Bj u = 0 on ∂Ω for all j = 1, . . . , q such that s > mj + 1/2 . If s < 1/2, then H s,ϕ (b.c.) = H s,ϕ,(0) (Ω). This proposition remains true if we + + change mj for m+ j , (b.c.) for (b.c.) , and Bj for Bj . Theorem 4.2. Let s ∈ R and ϕ ∈ M. The mapping u → Lu, u ∈ C ∞ (b.c.), is extended by a continuity to the bounded linear operator L : H s,ϕ (b.c.) → (H 2q−s, 1/ϕ (b.c.)+ ) .

(4.6)

Here the function Lu is interpreted as the functional (Lu, · )Ω , whereas (H 2q−s, 1/ϕ (b.c.)+ ) denotes the antidual space to H 2q−s, 1/ϕ (b.c.)+ with respect to the inner product in L2 (Ω). The operator (4.6) is a Fredholm one. Its kernel coincides with N , and its range is equal to the set , f ∈ (H 2q−s, 1/ϕ (b.c.)+ ) : (f, v)Ω = 0 ∀ v ∈ N + . The index of the operator (4.6) is equal to dim N − dim N + and does not depend on s, ϕ. Theorem 4.2 was proved in [39, Sec. 5], provided that s = j − 1/2 for each j = 1, . . . , 2q. For the rest values of s, the theorem is deduced by means of the interpolation formula (4.5). The analogs of Theorems 3.2–3.4 was obtained for the operator (4.6) as well (see also [32]). Theorem 4.2 specifies, with regard to the refined scale, the theorem of Berezansky, Krein and Roitberg on homeomorphisms realized by the elliptic operator L on the two-sided Sobolev scale [8], [9, Ch. 3, Sec. 6], [57, Sec. 5.5]. In the case of s ≤ m + 1/2 the ellipticity condition is essential for the boundedness of the operator (4.6). The interpolation space (4.5) was studied in the Sobolev case of ϕ ≡ 1 in [16, 60] (see also [63, Sec. 4.3.3]). 4.4. We note that the general nonhomogeneous boundary problem (3.1), (3.2) cannot be reduced to the semihomogeneous boundary problems in the lower part of the refined scale, namely for s < m + 1/2. Indeed, if s < −1/2, then solutions to these problems belong to the spaces of distributions of the different nature; solutions to the problem (4.1) belong to KLs,ϕ (Ω) ⊂ H s,ϕ (Ω) being distributions defined in the open domain Ω, whereas solutions to the problem (4.3) belong to H s,ϕ (b.c.) ⊂ HΩs,ϕ (Rn ) being distributions supported on the closed domain Ω. If −1/2 < s < m + 1/2, then solutions to the semihomogeneous problems are distributions defined in Ω (see Theorem 1.1 (iii) in the case −1/2 < s < 0), but the operator (L, B) cannot be correctly defined on KLs,ϕ (Ω) ∪ H s,ϕ (b.c.) because of the inequality (KLs,ϕ (Ω) ∩ H s,ϕ (b.c.)) \ N = ∅. (4.7) Note also that in the case where s > m + 1/2 we have the equality of sets in (4.7). Hence the nonhomogeneous problem (3.1), (3.2) is reduced to the semiho-

470


mogeneous problems (4.1) and (4.3); i.e., Theorem 3.1 is equivalent to Theorems 4.1 and 4.2 taken together.

5. Generic theorems for elliptic problems in two-sided scales In [55, 57, 58] Ya.A. Roitberg introduced a special modification of the Sobolev two-sided scale in which the operator of an elliptic boundary problem is bounded and a Fredholm one for every parameter s ∈ R (see also [9, Ch. 3, Sec. 6], [5, Sec. 7.9]). This modification does not depend on coefficients of the elliptic differential expression but depends solely on the order of the expression. Therefore, the theorems on properties of elliptic problems in such modified scale is naturally to call generic (for the class of the problems having the same order). We will consider these theorems with regard to the refined scale. Let s ∈ R, ϕ ∈ M, and integer r > 0. We set Er := {k − 1/2 : k = 1, . . . , r}. In the case where s ∈ R \ Er we denote by H s,ϕ,(r) (Ω) the completion of C ∞ ( Ω ) with respect to the Hilbert norm r 1/2 > k−1 > >(Dν u) ∂Ω >2 s−k+1/2,ϕ u H s,ϕ,(r) (Ω) := u 2H s,ϕ,(0) (Ω) + . H (∂Ω) k=1

Here Dν := i ∂/∂ν, with ν being the unit vector of the inner normal to ∂Ω. In the case where s ∈ Er we set H s,ϕ,(r) (Ω) := H s−1/2,ϕ,(r) (Ω), H s+1/2,ϕ,(r) (Ω) t1/2 . The collection of separable Hilbert spaces {H s,ϕ,(r) (Ω) : s ∈ R, ϕ ∈ M }

(5.1)

is called the refined scale modified in the Roitberg sense. The number r is called the index of this modification. The scale (5.1) admits the following description. Let us denote by Υs,ϕ,(r) (Ω, ∂Ω) the space of all vector-functions (u0 , u1 , . . . , ur ) ∈ H s,ϕ,(0) (Ω) ⊕

r Q

H s−k+1/2, ϕ (∂Ω)

(5.2)

k=1

such that uk = (Dνk−1 u0 ) ∂Ω for every integer k = 1, . . . r satisfying the inequality s > k − 1/2. In view of Theorem 1.1 (viii), Υs,ϕ,(r) (Ω, ∂Ω) is a Hilbert space with respect to the inner product in the space (5.2). Proposition 5.1. The mapping Tr : u → u, u ∂Ω, . . . , (Dνr−1 u) ∂Ω ,

u ∈ C ∞ ( Ω ),

is extended by a continuity to the bounded linear injective operator Tr : H s,ϕ,(r) (Ω) → Υs,ϕ,(r)(Ω, ∂Ω)

(5.3)

far all s ∈ R and ϕ ∈ M. If s ∈ / Er , then the operator (5.3) is an isometric isomorphism.


471

Thus, we can interpret an element u ∈ H s,ϕ,(r) (Ω) as the vector-valued function (u0 , u1 , . . . , ur ) := Tr u ∈ Υs,ϕ,(r)(Ω, ∂Ω). Note that in view of Theorem 1.1 (viii)

(5.4)

u H s,ϕ,(r) (Ω) 0 u0 H s,ϕ,(0) (Ω) = u0 H s,ϕ (Ω) if s > r − 1/2. Therefore H s,ϕ,(r) (Ω) = H s,ϕ (Ω) with equvivalence of norms if s > r − 1/2.

(5.5)

Theorem 5.1. Let s ∈ R and ϕ ∈ M. The mapping (3.6) is extended by a continuity to the bounded linear operator (L, B) : H s,ϕ,(2q) (Ω) → H s−2q,ϕ,(0) (Ω)⊕

q Q

H s−mj −1/2, ϕ (∂Ω) =: Hs,ϕ,(0) (Ω, ∂Ω).

j=1

(5.6) This operator is a Fredholm one. Its kernel coincides with N , and its range is equal to the set q & % (gj , Cj+ v)∂Ω = 0 ∀ v ∈ N + . (f, g1 , . . . , gq ) ∈ Hs,ϕ,(0) (Ω, ∂Ω) : (f, v)Ω + j=1

The index of the operator (5.6) is equal to dim N − dim N + and does not depend on s, ϕ. This theorem is generic because the spaces in which the operator (5.6) acts are the same for all boundary problems of the common order (2q, m1 , . . . , mq ). It follows from (5.5) that Theorem 5.1 coincides with Theorem 3.1 for s > 2q − 1/2. Using Proposition 5.1 we give the following interpretation of a solution u ∈ H s,ϕ,(2q) (Ω) to the boundary problem (3.1), (3.2) in the sense of the distribution theory. Let us write down the differential expressions L and Bj in a neighborhood of ∂Ω in the form mj 2q L= Lk Dνk , Bj = Bj,k Dνk . (5.7) k=0

k=0

Here Lk and Bj,k are certain tangent differential expression. Integrating by parts we arrive at the (special) Green formula (Lu, v)Ω = (u, L v)Ω − i +

(k)

02q

2q

(Dνk−1 u, L(k) v)∂Ω ,

k=1

r−k + Lr , with L+ r being r=k Dν to Lr . By passing to the limit s,ϕ,(2q)

:= Here L formally adjoint the next equality for u ∈ H

u, v ∈ C ∞ ( Ω ).

the tangent differential expression and using the notation (5.4) we get

(Ω):

(Lu, v)Ω = (u0 , L+ v)Ω − i

2q k=1

(uk , L(k) v)∂Ω ,

v ∈ C ∞ ( Ω ).

(5.8)

472


Now it follows from (5.7), (5.8) that the element u ∈ H s,ϕ,(2q) (Ω) is a solution to the boundary problem (3.1), (3.2) with f ∈ H s−2q,ϕ,(0) (Ω), gi ∈ H s−mj −1/2, ϕ (∂Ω) if and only if the following equalities hold true: (u0 , L+ v)Ω − i

2q

(uk , L(k) v)∂Ω = (f, v)Ω ∀ v ∈ C ∞ ( Ω ),

k=1

mj

Bj,k uk+1 = gj on ∂Ω,

j = 1, . . . , q.

k=0

Theorem 5.1 was proved in [41, Sec. 5]. The analogs of Theorems 3.2–3.4 were obtained for the operator (5.6) as well. Theorem 5.1 specifies, with regard to the refined scale, the theorem of Ya.A. Roitberg on the Fredholm property of a regular elliptic boundary problem in the modified Sobolev scale (so-called theorem on a complete collections of homeomorphisms) [55], [57, Sec. 4.1, 5.3] (see also [9, Ch. 3, Sec. 6], [5, Sec. 7.9]). The analogs of Theorem 5.1 are also valid for nonregular elliptic boundary problems both for one and for system of partial differential equations. Note that the boundedness of the operator (5.6) holds true without the ellipticity assumption. Elliptic boundary problems in the modified twosided scales of different normed spaces were studied in [57] (the Sobolev Lp -spaces) and in [50, 51] (non-Sobolev spaces). A certain classes of non-elliptic problems were investigated in the two-sided modified scales as well (see [58], [13] and the references therein).

6. Individual theorems for elliptic problems In the individual theorems, the domain of the operator (L, B) depends on coefficients of the elliptic expression L. Namely, we consider the operator s,ϕ (L, B) : DL,X (Ω) → X(Ω) ⊕

q Q

H s−mj −1/2, ϕ (∂Ω) =: Xs,ϕ (Ω, ∂Ω).

(6.1)

j=1

Here s ∈ R, ϕ ∈ M, and X(Ω) is a certain Hilbert space consisting of distributions in Ω, and satisfying the continuous embedding X(Ω) → D (Ω). The domain of the operator (6.1) is the Hilbert space , s,ϕ DL,X (Ω) := u ∈ H s,ϕ (Ω) : Lu ∈ X(Ω) endowed with the graphics inner product s,ϕ (u, v)DL,X (Ω) := (u, v)H s,ϕ (Ω) + (Lu, Lv)X(Ω) .

In the case where s > m + 1/2 we can set X(Ω) := H s−2q, ϕ (Ω) that leads us to Theorem 3.1. But in the case where s ≤ m + 1/2 we cannot do so if we want to define the operator (L, B) on the non-modified refined scale. The space X(Ω) must be narrower than H s−2q, ϕ (Ω).


473

Let us formulate the conditions on X(Ω) under which the operator (5.1) is bounded and has the Fredholm property for some s and ϕ. Condition 1. The set X ∞ (Ω) := X(Ω) ∩ C ∞ ( Ω ) is dense in the space X(Ω). Condition 2. There exists a number c > 0 such that Of H s−2q,ϕ (Rn ) ≤ c f X(Ω) ,

f ∈ X ∞ (Ω).

We recall that the function Of is given by formula (4.4). It follows from the Conditions 1 and 2 that the mapping f → Of , f ∈ X ∞ (Ω), is extended by continuity to the linear bounded operator O : X(Ω) → HΩs−2q, ϕ (Rn ). It satisfies the condition Of = f in Ω; i.e., O is an operator extending a distribution from Ω onto Rn . This implies the continuous embedding X(Ω) → H s−2q, ϕ (Ω). Theorem 6.1. Let s < 2q − 1/2, s + 1/2 ∈ / Z, and ϕ ∈ M. We assume that a Hilbert space X(Ω) is continuously embedded into D (Ω) and satisfies Conditions 1, 2. Then the following assertions hold true: ∞ (i) The set DL,X (Ω) := { u ∈ C ∞ ( Ω ) : Lu ∈ X(Ω) } is dense in the space s,ϕ DL,X (Ω). ∞ (Ω), is extended by a continuity to the (ii) The mapping (3.6), where u ∈ DL,X linear bounded operator (6.1). (iii) The operator (6.1) is a Fredholm one. Its kernel coincides with N , and its range is equal to the set q % & (f, g1 , . . . , gq ) ∈ Xs,ϕ (Ω, ∂Ω) : (f, v)Ω + (gj , Cj+ v)∂Ω = 0 ∀ v ∈ N + . j=1

(iv) If the set O(X ∞ (Ω)) is dense in the space HΩs−2q, ϕ (Rn ), then the index of the operator (6.1) is equal to dim N − dim N + . Conditions 1 and 2 allow us to vary the space X(Ω) in a broad fashion. We especially note two possible options of X(Ω). The first of them is the choice X(Ω) := H σ,η (Ω) for arbitrary fixed parameters σ > −1/2 and η ∈ M. Theorem 6.2. Let s < 2q−1/2, s+1/2 ∈ / Z, σ > −1/2, and ϕ, η ∈ M. The mapping (3.6) is extended by a continuity to the bounded and the Fredholm operator q Q , (L,B) : u ∈ H s,ϕ (Ω) : Lu ∈ H σ,η (Ω) → H σ,η (Ω) ⊕ H s−mj −1/2,ϕ (∂Ω),

(6.2)

j=1

provided that its domain is endowed with the graphics norm 1/2 . u 2H s,ϕ (Ω) + Lu 2H σ,η (Ω) The index of the operator (6.2) is equal to dim N − dim N + and does not depend on parameters s, σ, ϕ, and η.

474


The case where σ = 0 and η ≡ 1, i.e., X(Ω) := H 0,1 (Ω) = L2 (Ω), is of great importance in the spectral theory of elliptic operators [17, 18, 30, 31]. The condition σ > −1/2 is essential in Theorem 2, that does not allow us to consider the boundary problem (3.1), (3.2) for an arbitrary distribution f ∈ D (Ω) supported on a compact subset in Ω. Here the important example is f (x) := δ(x − x0 ), where x0 ∈ Ω. The following construction of the space X(Ω) has not this demerit. We consider the set of weight functions , Wk∞ ( Ω ) := ρ ∈ C ∞ ( Ω ) : ρ > 0 in Ω, Dνj ρ = 0 on ∂Ω, j = 0, . . . , k , where integer k ≥ 0. ∞ Let s < 2q − 1/2, ϕ ∈ M, and ρ ∈ W[2q−s−1/2] ( Ω ). (As usual, [t] denotes the integral part of t.) We consider the space , ρH s−2q, ϕ (Ω) := f = ρv : v ∈ H s−2q, ϕ (Ω) endowed with the inner product f1 , f2 ρH s−2q, ϕ (Ω) := ρ−1 f1 , ρ−1 f2 H s−2q, ϕ (Ω) . The space X(Ω) = ρH s−2q, ϕ (Ω) is Hilbert separable and satisfies Conditions 1, 2. ∞ ( Ω ). The Theorem 6.3. Let s < 2q −1/2, s+1/2 ∈ / Z, ϕ ∈ M, and ρ ∈ W[2q−s−1/2] ∞ s−2q, ϕ (Ω), is extended by a continuity mapping (3.6), where u ∈ C ( Ω ), Lu ∈ ρH to the bounded and the Fredholm operator , (L, B) : u ∈ H s,ϕ (Ω) : Lu ∈ ρH s−2q, ϕ (Ω)

→ ρH s−2q, ϕ (Ω) ⊕

q Q

H s−mj −1/2, ϕ (∂Ω),

(6.3)

j=1

provided that its domain is endowed with the graphics norm 1/2 . u 2H s,ϕ (Ω) + Lu 2ρH s−2q,ϕ (Ω) The index of the operator (6.3) is equal to dim N − dim N + and does not depend on s, ϕ, and ρ. ∞ As an example of ρ ∈ W[2q−s−1/2] ( Ω ), we can chose every function ρ ∈ C ∞ ( Ω ) such that ρ is positive in Ω and

ρ(·) = (dist(·, ∂Ω))δ in a neighborhood of ∂Ω for δ = [2q − s + 1/2].

(6.4)

Theorems 6.1–6.3 were proved in [43, 44]. They are closely connected with the theorems of J.-L. Lions and E. Magenes on a solvability of elliptic boundary problems in the two-sided Sobolev scale [25, 26, 27, 28]. A theorem similar to Theorem 6.1 were proved in [28, Sec. 6.10] in the case of s ≤ 0, ϕ ≡ 1 and the Dirichlet boundary conditions. In this paper, certain different conditions depending on the problem under consideration were imposed on X(Ω) (see also [27, Ch. 2, Sec. 6.2]). Theorem 6.2 was proved in [25, 26] in the important case ϕ ≡ χ ≡ 1


475

and σ = 0. Theorem 6.3 was proved in [27, Ch. 2, Sec. 6,7] in the case where ϕ ≡ 1 and the weight function ρ satisfies the condition (6.4) with δ = 2q − s. The similar questions were considered in [56, 24], [58, Sec. 1.3] for the modified Sobolev scale. We note that Theorems 6.2 and 6.3 are also true for half-integer values of s if we define the spaces with the help of the interpolation.

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[17] G. Grubb, A characterization of non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa 22 (1968), no. 3, 425–513. [18] G. Grubb, Functional Calculas of Pseudo-Differential Boundary Problems. 2nd ed., Birkh¨ auser, Boston, 1996. [19] D.D. Haroske, S.D. Moura, Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers. J. Approximation Theory 128 (2004), no. 2, 151–174. [20] L. H¨ ormander, Linear Partial Differential Operators. Springer-Verlag, Berlin, 1963. [21] L. H¨ ormander, The Analysis of Linear Partial Differential Operators. vol. 2, Springer-Verlag, Berlin, 1983. [22] L. H¨ ormander, The Analysis of Linear Partial Differential Operators. vol. 3, Springer-Verlag, Berlin, 1985. [23] G.A. Kalyabin, P.I. Lizorkin, Spaces of functions of generalized smoothness. Math. Nachr. 133 (1987), 7–32. [24] Yu.V. Kostarchuk, Ya.A. Roitberg, Isomorphism theorems for elliptic boundary value problems with boundary conditions that are not normal. Ukrainian Math. J. 25 (1973), no. 2, 222–226. [25] J.-L. Lions, E. Magenes, Problèmes aux limites non homog` enes, V. Ann. Sci. Norm. Sup. Pisa 16 (1962), 1–44. [26] J.-L. Lions, E. Magenes, Problèmes aux limites non homog` enes, VI. J. d’Analyse Math. 11 (1963), 165–188. [27] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications. vol. 1, Springer-Verlag, New York, 1972. [28] E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali. Atti VII Congr. Un. Mat. Italiana (Genova, 1963), Edizioni Cremonese, Rome, 1965, 134–197. [29] C. Merucci, Application of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. Proc. Lund Conf. 1983, Lecture Notes in Math. 1070, Springer-Verlag, Berlin, 1984, 183–201. [30] V.A. Mikhailets, Asymptotics of the spectrum of elliptic operators and boundary conditions. Soviet. Math. Dokl. 26 (1982), no. 5, 464–468. [31] V.A. Mikhailets, A precise estimate of the remainder in the spectral asymptotics of general elliptic boundary problems. Funct. Anal. Appl. 23 (1989), no. 2, 137–139. [32] V.A. Mikhailets, A.A. Murach, Elliptic operators in a refined scale of functional spaces. Ukrainian. Math. J. 57 (2005), no. 5, 817–825. [33] V.A. Mikhailets, A.A. Murach, Improved scales of spaces and elliptic boundary-value problems. I. Ukrainian. Math. J. 58 (2006), no. 2, 244–262. [34] V.A. Mikhailets, A.A. Murach, Improved scales of spaces and elliptic boundary-value problems. II. Ukrainian. Math. J. 58 (2006), no. 3, 398–417. [35] V.A. Mikhailets, A.A. Murach, Refined scales of spaces and elliptic boundary-value problems. III. Ukrainian Math. J. 59 (2007), no. 5, 744–765. [36] V.A. Mikhailets, A.A. Murach, The interpolation of spaces with functional parameter and the spaces of differential functions. Dopov. Nats. Acad. Nauk. Ukr. Mat. Pryr. Tehn. Nauky (2006), no. 6, 13–18. (Russian)


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Connection of Solutions of Abstract Paired Equations in Rings with Factorization Pairs Gennadiy Poletaev Abstract. The connection between solutions of the abstract paired equations with respect to an unknown x ∈ (R1 ∩ R2 ): (a1 x)− = c− , (a2 x)+ = b+ , is considered. It is assumed that the coefficients aj ∈ Rj , j = 1, 2, where R1 , R2 are the associative rings with factorization pairs (Rj+ , Rj− ) and unity e ∈ R1 ∩ R2 . Mathematics Subject Classification (2000). 45N05. Keywords. Paired equations, ring, factorization, convolution, integral.

A connection between solutions of abstract paired equations with an arbitrary right-hand side and the same equations with the multiplicative unity of a ring in right-hand side is analyzed. The cases when the coefficients belong to the same or different rings with factorization pairs are considered. The possibility of application to paired integral and paired matrix equations is pointed out. The appearance of the notion of rings with factorization pairs [1–6] and the paired equations in the forms (a1 x)− = c− (1) (a2 x)+ = b+ , is connected with the penetration of Banach algebra ideas into the theory of convolution type integral equations, which was initiated by M.G. Krein [7]. The equations of type (1) are found in investigations of integral equations with kernels depending on the difference of the arguments, including the paired equations of the convolution type: ⎧ 3∞ ⎪ ⎪ k1 (t − s)ϕ(s) ds = f (t), −∞ < t < 0, ⎨ ϕ(t) − −∞ (2) 3∞ ⎪ ⎪ k2 (t − s)ϕ(s) ds = f (t), 0 < t < ∞; ⎩ ϕ(t) − −∞

480

G. Poletaev

where kj (t) exp{cj t} ∈ L1 (−∞, ∞), cj ∈ R, j = 1, 2, and also in special applied tasks [1, 3, 7 – 11]. The equations (1) are subtypes of the general form of paired equations: − [a11 x a12 ] = c− (3) + [a21 x a22 ] = b+ , with respect to an unknown element x [2, 3, 11]. The abstract paired equations (1), (3) with factorizable coefficients in rings with factorization pairs were, in particular, studied in [3, 11]. The results concerning the solvability of these equations through the factorization of elements which may be built with the help of coefficients were presented during the international conferences [5, 12] and others. This article is devoted to the investigation of connection between solutions of the abstract paired equations (1) with respect to the unknown element x.

1. Notations, definitions, and general provisions 1.1. Following [3, 4, 6, 13], by R we shall denote any, in general, non-commutative, and, probably, non-associative ring with unity e. Let p+ , p− be commutative projectors, i. e. additive and idempotent mappings from R into R. Let us assume: p0 := p+ p− (= p− p+ ); p∓ := p∓ − p0 . For any subset B ⊆ R we shall denote: B ∓,0 := p∓,0 B; B∓ := p∓ B; B ∗ := B + + B − ; B∗ := B+ + B− . For any x ∈ R we note x∓,0 := p∓,0 x; x∓ := p∓ x. The inverse in R of an element x ∈ R invertible in R will be denoted by the symbol x , if necessary, additionally supplied. For any subsets A, B ⊆ R we shall define the set inv(A, B) := {x ∈ A, x exists and belongs to B}. Let us denote inv(A, A) := invA. The element u+ , [the element v 0 , the element w− ] will be called correct [6], if u+ ∈ invR+ , [v 0 ∈ invR0 , w− ∈ invR− ]. 1.2. Supplementing [2,3,13], where, in particular, the concept of factorization of structures [6] is developed, and [8], we shall introduce the following definitions. Definition 1. A pair of subrings (R+ , R− ) [≡ (R− , R+ )] in ring R with unity e will be called the left factorization pair (LFP) of the ring R, if there are commutative projectors p+ , p− generated which act on R∓ := p∓ R and satisfy the following axioms: e ∈ R0 ; (4) p0 is a ring homomorphism from R+ and R− into R0 ; −

(5)

∗

(6) R R ⊆R . The right factorization pair (RFP) [2] is defined similarly. It should be noted that factorizations of structures in R [6] correspond here to LFP of R. If R is commutative and whenever the pair (R+ , R− ) is simultaneously LFP and RFP of R, this pair will be called a factorization pair (FP) of the ring R. +

Definition 2. Any ring R with unity e, considered together with its fixed FP (R+ , R− ) [≡ (R− , R+ )] will be called a ring with factorization pair.

Connection of Solution of Abstract Paired Equations

481

Nontrivial examples of rings with FP can be constructed by starting, for example, from rings of matrices, rings of absolute integrable functions, their appropriate transformations, and others [3, 4, 6, 7, 10, 13]. 1.3. We shall say ([3], compare with [6]) that an element a ∈ R allows in R a left factorization, l.f., (right factorization, r.f.) by a pair (R+ , R− ) if there are elements r+ ∈ R+ , s0 ∈ R0 , t− ∈ R− such that a = r + s 0 t− ,

(a = t− s0 r+ ).

The multipliers r+ ∈ R+ , s0 ∈ R0 , t− ∈ R− are referred to as plus-, diagonaland minus-factors, respectively. An l.f. (r.f.) is referred to as correct left factorization, c.l.f., (correct right factorization, c.r.f.) if r+ ∈ R+ , s0 ∈ R0 , t− ∈ R− are correct elements; – as normalized left factorization, n.l.f., (normalized right factorization, n.r.f.) if t0 = r0 = e, and as normalized correct left factorization, n.c.l.f., (normalized correct right factorization, n.c.r.f.) if it is both c.l.f. (c.r.f.) and t0 = r0 = e. 1.4. When the problem of solvability of the abstract paired equations (1), (3) is posed in a ring R with the factorization pair (R+ , R− ), an element x ∈ R∗ will be considered as unknown, and the rest of the elements will be assumed to be given. We also assume that c− ∈ R− , b+ ∈ R+ and the coefficients a1 , a2 , a11 ,a12 ,a21 ,a22 are invertible in R. It is assumed that a solution to the paired equation (1) is an element x ∈ R such that the corresponding right-hand and left-hand sides of equations (1) coincide after x is substituted into them. The problem of solvability of paired equation (1) in R1∩2 := R1 ∩ R2 in the case where the coefficients ai ∈ Ri ; i = 1, 2, R1 and R2 are the rings with factorization pairs, with unity e ∈ R1∩2 and common multiplication [3], may be posed as well. In this case, we define R1∪2 := (R1 + R2 ). Below, we shall assume that the following conditions hold: R1− ⊆ R2− ,

R1+ ⊇ R2+ .

(7)

2. Main result ∗ 2.1. Solution xe ∈ R1∩2 of abstract paired equation (1) with coefficients ai ; i = 1, 2 invertible in their rings Ri for the right-hand side c− = b+ = e plays a special role in the theory of solvability of these equations. Under some conditions, the ∗ solution x ∈ R1∩2 of (1) with arbitrary right-hand part c− ∈ R1− , b+ ∈ R2+ can be expressed through it.

Theorem 1. (Connection of solutions). Let R1 , R2 be associative and, in general, non-commutative rings with common multiplication, common unity e ∈ R1∩2 , and FP (Rj+ , Rj− ) generated by commutating projectors p+ , p− : R1∪2 → R1∪2 , so that conditions (7) hold true, and the coefficients of equations (1) aj ∈ invRj∗ ; j = 1, 2.

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G. Poletaev

Let the abstract paired equation:

(a1 xe )− (a2 xe )+

= e, = e,

(8)

∗ in R1∩2 which has the inverse have a solution xe ∈ R1∩2

[xe ]R∗1 := [xe ]R1 ∈ R1∗ , [xe ]R∗2 := [xe ]R2 ∈ R2∗ . Then for any right-hand part (c− , b+ ), c− ∈ R1− , b+ ∈ R2+ , by the compatibility condition [(a1 xe )R∗1 c− ]0 = [(a2 xe )R∗2 b+ ]0 , (9) ∗ the solution x ∈ R1∩2 of (1) in R1∩2 can be represented as

x = xe {[(a2 xe )R∗2 b+ ]+ + [(a1 xe )R∗1 c− ]− }.

(10)

[Here, (aj xe )R∗ are the inverses of aj xe in Rj , which exist and belong to Rj∗ ; j j = 1, 2 under the assumptions of Theorem 1.] Proof. Under the assumptions of Theorem 1 and for any c− ∈ R1− , b+ ∈ R2+ , ∗ the right-hand side of formula (10) is meaningful and is an element x ∈ R1∩2 . Substituting this element x into the left-hand side of the paired equation (1) and transforming it with the help of ring operations acting in Rj ; j = 1, 2, and projectors p+ , p− , one can be convinced that it satisfies the equation (1) if the compatibility condition (9) holds. Indeed, (a1 x)− = (a1 xe {[(a2 xe )R∗2 b+ ]+ + [(a1 xe )R∗1 c− ]− })− = {(a1 xe )+ [(a2 xe )R2 b+ ]+ }0 + (a1 xe {(a1 xe )R1 c− − [(a1 xe )R1 c− ]+ })− = (a1 xe )0 [(a2 xe )R2 b+ ]0 + [a1 xe (a1 xe )R1 c− ]− − {(a1 xe )+ [(a1 xe )R1 c− ]+ }0 = [(a2 xe )R2 b+ ]0 + c− − [(a1 xe )R1 c− ]0 = c− . Similarly, it can be proved that (a2 x)+ = (a2 xe {[(a2 xe )R∗2 b+ ]+ + [(a1 xe )R∗1 c− ]− })+ = b+ .

Theorem is proved. 2.2. If R1 = R2 = R, then from Theorem 1 it follows

Theorem 2. Let R be an associative and, in general, non-commutative ring with unity e and FP (R+ , R− ) generated by commutating projectors p+ , p− : R → R. Let aj ∈ invR∗ be the coefficients and there exists a solution xe ∈ invR∗ of the paired equation (8) invertible in R. Then for any right-hand part satisfying the compatibility condition [(a1 xe )R∗ c− ]0 = [(a2 xe )R∗ b+ ]0 ,

Connection of Solution of Abstract Paired Equations

483

one of the solutions x ∈ R∗ of (1) has the representation x = xe {[(a2 xe )R∗ b+ ]+ + [(a1 xe )R∗ c− ]− }.

(11)

In this formula, (aj xe )R∗ , j = 1, 2, denotes the corresponding inverse element in R. Let us point out that the solutions xe , having the required inverses, exist, in particular, under corresponding correct factorizations by the factorization pairs of the ring of the elements, which defined by the coefficients and also in some more general situations. 1 R2 = L 1 ; c 0 [2, 3, 10], then theorems 2.3. For example, if R = R1 = L, 1, 2 can be applied to paired integral equations of the convolution type (2), in particular, with kernel functions k1,2 (t) ∈ L1 (−∞, ∞) or k1 (t), k2 (t) exp(ct) ∈ L1 (−∞, ∞), c 0 [8–10]. When R = Rn×n ; n 2 [6, 14], from theorem 2 it follows the formula of connection of solutions of the corresponding paired matrix equations with unknown matrix X ∈ Rn×n and projectors p+ , (p− ) : Rn×n → Rn×n , which map each matrix from Rn×n onto the corresponding lower (upper) triangular: [A1 X]− = C − (12) + [A2 X] = B+; − + where A1 , A2 ∈ Rn×n , C − ∈ Rn×n , b+ ∈ Rn×n are given matrices. Thus, C − and + B are right- and left-triangular matrices from Rn×n respectively. 2.4. Note that for the abstract paired equation (1) in rings with factorization pairs the connection between the solutions corresponding to c− := a1− and b+ := a2+ is established as well.

References [1] G.S. Poletaev, On some integral equations in mechanics and their abstract analogs. Book of abstracts of the VIIIth Winter Mathematical School, Voronezh (1974), 87– 89. [2] G.S. Poletaev, About equations and systems of one type in rings with factorization pair. Preprint Math. Institute, Acad. Sci. Kiev 88. 31 (1988), 20 p. [3] G.S. Poletaev, The abstract analogue paired equations of convolution type in a ring with factorization pair. Ukraine Math. J. 43 (1991), no. 9, 1201–1213. [4] G.S. Poletaev, About one-projector of second order equations with correct factorized coefficients in a ring with factorization pair. Bull. Kherson Tech. Univ. 2 (8) (2000), 191–195. [5] G.S. Poletaev, On paired equations in different rings with factorization pairs. Abstracts of Int. Conf. on Analytic methods of analysis and differential equations (AMADE-2001). Minsk, Feb. 15–19, (2001), 127–128. [6] A. McNabb, A. Schumitzky, Factorization of operators – I: Algebraic Theory and Examples. J. Funct. Anal. 9 (1972), 262–295.

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[7] M.G. Krein, Integral equations on a half-line with kernels dependent on the difference of the arguments. Uspechi Math. Nauk 13 (1958), no. 5 (83), 3–120 (in Russian); Transl. Amer. Math. Soc. Ser. 2, 22 (1962), 163–288. [8] I.Z. Gohberg, M.G. Krein, On a paired integral equation and its transposed I. Theoret. Appl. Math. 1 (1958), 58–81 (in Russian). [9] F.D. Gakhov, Y.I. Cherskiy, The equations of convolution type. Nauka, Moscow, 1978, 296 p. [10] G.S. Poletaev, Paired equations of convolution type with kernels from different Banach algebras. Ukraine Math. J. 43 (1991), no. 6, 803–813. [11] G.S. Poletaev, Some results about pairs of equations in rings with factorization pairs. Progress in Analysis. Vol. II. Proc. of the 3rd Int. ISAAC Congr., Berlin, Germany, 20–25 Aug. 2001. Editors: H.G.W. Begehr, R.P. Gilbert, M. W. Wong. World Scientific, New Jersey, London, Singapore, Hong Kong, 2003, 851–855. [12] G.S. Poletaev, The paired equations with correct factorized coefficients. Ukraine Math. Congress. Int. Conf. Funct. anal., Kiev, 2001, 79. [13] G.S. Poletaev, To the abstract analogue theory of some equations of convolution type. Math. Phys. 24 (1978), 104–106. [14] G.S. Poletaev, About the formulation and matrix models of some return problems of beam mechanics and the influence of factorized representation matrices. The Math. Models in Education, Science and Industry. St. Petersburg, 2000, 146–148. Gennadiy Poletaev Department of High Mathematics Odessa State Academy of Buildings and Architecture 4 Didrihsona St. 65029 Odessa, Ukraine e-mail: poletayev [email protected]


The Dynamic Problems About the Definition of Stress State Near Thin Elastic Inclusions Under the Conditions of Perfect Coupling V.G. Popov, O.V. Litvin and A.P. Moysyeyenok Abstract. The isotropic unbounded elastic body (matrix), which is in the condition of plane strain and which contains a thin elastic inclusion in the form of a strip is considered. It occupies the area: |x| ≤ a, |y| ≤ h2 in the plane Oxy. It is necessary to determine the stress state in the matrix caused by the non-stationary or harmonic plane waves interacting with the inclusion. The problem is reduced to the construction of the solution of the Lame equations for plane strain, which satisfies the given boundary conditions on the inclusion. It is considered that under these conditions the inclusion is so thin that the displacements of any point of it coincide with the displacements of the appropriate point of a middle plane. The method of the solution is based on the presentation of the displacements and stresses caused by scattered waves in the form of the discontinuous solution of the Lame equations (in the nonstationary case in the space of the Laplace images). Stress intensity factors (SIF) are taken as amounts characterizing the stress state near the inclusion, as in a number of publications, where analogous problems were considered in the static formulation. The transition from the Laplace images to the originals is implemented numerically for non-stationary problems calculations. The numerical research of the dependence of SIF on time or frequency and the ratio of elastic constants of the matrix and the inclusion has been done. The possibility of the consideration of inclusions of large rigidity as absolutely rigid ones is analyzed. Mathematics Subject Classification (2000). 74J20;74K20. Keywords. Elastic waves, thin elastic inclusions, discontinuous solution, stress intensity factor, numerical Laplace transformation, singular integral equations.

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V.G. Popov, O.V. Litvin and A.P. Moysyeyenok

Introduction At present the dynamic stress concentration in the unbounded bodies containing thin absolutely rigid inclusions is researched fully enough. The solutions of similar harmonic and non-stationary problems in 2D and 3D formulations can be found in the articles [1], [2], [3], [4], [5], [6]. The works where the elastic properties of the inclusion are taken into account are much fewer. For example, in [7], [8], [9] the approach based on the use of the method of asymptotic decompositions is developed. Two small parameters – the ratio of the thickness of the inclusion to the typical geometrical size and the ratio of elastic properties of matrix and inclusion – are used. Basically, in these articles harmonic oscillations are considered and inclusions are considered as soft or weakly contrasting. Therefore the peculiarities of stress state near thin inclusions of large rigidity aren’t practically researched. In the present work the problems about the determination of the stress state in the unbounded body near thin elastic inclusions in the form of a strip with the interaction of elastic non-stationary and steady harmonic waves are considered.

1. Problem formulation Let the unbounded elastic body (matrix) be under the conditions of plane strain and contain an inclusion in the form of a plate with the thickness h 0 such that for any {ck }∞ k=1 and {nk }k=1 , where nk+1 > nk and nk ∈ N, we have 1/2 π ∞ ∞ |cν |2 ink x dx ≥ c c e k ν −π s−1 s s=1 k=1

2

≤ν :> > ψ(k) > > > inf ≤K 0 > φ(k) >M(Z) 0 (the greatest lower bound is actually used when choosing the values of fractions of the form 00 ). Proofs of sharp inequalities are essentially based on positive definiteness of certain functions, i.e., those representable as the Fourier transform of a positive measure (B + (Rd )). Because of this, let us give for d = 1 a criterion of positive definiteness. Using it, the author succeeded to construct compactly supported polynomial radial splines of a given degree and maximal smoothness (basis radial functions, see [3]). Theorem 1.8. Let f be a function continuous at zero. It is in B + (R) if and only if the following conditions are fulfilled: α) f ∈ C(R) and bounded; 3 →+∞ f (t)−f (−t) β) the improper integral →0 dt converges; t 3N 1 γ) limN →+∞ 2N −N f (t)dt ≥ 0; δ) there exists k0 ∈ N such that for all k ≥ k0 and x = 0 ∞ f (t)dt k+1 (sign x) ≥ 0. k+1 −∞ (x + it) Let us give two examples of applications of multipliers. Example 1. Comparison 0r of differential operators. 0r Let Q(x) = s=0 qs (x) and P (x) = s=0 ps (x), where ps and qs − are homogeneous polynomials in x1 , . . . , xd of degree s. If for some constant γ for all functions from W r (Rd ), where || · ||∞ is the sup-norm in Rd ∂ |α| f d α ∈ Z ||Q(D)f ||∞ ≤ γ||P (D)f ||∞ , Dα f = , |α| = αj , αd + 1 ∂xα 1 · · · ∂xd then qr = c · pr .

Fourier Multipliers and Comparison of Linear Operators

503

Indeed, replacing x with λx (λ > 0), dividing by λr and passing to the limit as λ → +∞, we obtain ||qr (D)f ||∞ ≤ γ||pr (D)f ||∞ . By 1.7 and 1.6 we may consider the fraction qr /pr to be continuous everywhere (in particular, at zero). But this homogeneous function of order zero is constant on the rays going from the origin and thus constant. Example 2. The Rogosinski-Bernstein method of summability of double Fourier series. In the multiple case (we restrict ourselves to dimension d = 2) we define the Rogosinski-Bernstein type means by γu Rn (f ; x) = Rn (f ; W, γ, μ, x) = )dμ(u), Sn (f ; W ; x − n 2 R where n ∈ N, Sn (f ; W ; x) = fˆ(k)ei(k,x) k∈nW

is the nth partial sum of the Fourier series (of the function f ∈ L1 (T2 )) generated 2 by the bounded 3 set W ⊂ R , γ > 0, and μ is a finite complex-valued Borel measure 2 on R with dμ = 1. The case of the circle is studied by Chandrasekharan and Minakshisundaran (1947). Clearly, k fˆ(k)ei(k,x) Rn (f ; x) = φ n k

with (χW is the indicator of W )

φ(x) = χW (x)

e−iγ(x,u) dμ(u). R2

In order that limn→∞ Rn (f ) = f on C(T2 ) it is necessary, and if in addition the measure μ have a compact support, W is a bounded connected set containing the neighborhood of zero and satisfying two additional conditions: the plane Lebesgue measure of ∂W is zero, and in an arbitrary neighborhood of any boundary point there are interior points from both W and R2 \ W, it is also sufficient that for all x ∈ ∂W e−iγ(x,u) dμ(u) = 0.

R2

Necessity follows from 1.6, while the sufficiency was proved by V.P. Zastavnyi. Theorem 1.1 is announced in lecture notes [4]. One can find the proof of Theorem 1.1 in [19]. For 1.4–1.7, see [5], [6]. The items 1.2, 1.3, and 1.8 can be found in the book [3]. Besides, in [3]–[6] necessary conditions for belonging to A(Rd ) are given, as well as the asymptotics of the Fourier transform of a function satisfying condition of convexity type including the multiple radial case, sufficient conditions for multipliers of power series of functions from the Hardy space Hp (D), D− being a disk in the complex plane C, p ∈ (0, 1], and expansions in eigenfunctions of the Sturm-Liouville operator (regular case).

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2. The comparison of linear differential operators Let G be a bounded domain in Rd , C0 (G) be the set of functions from C ∞ (Rd ) which vanish off a compact subset of G. Let P and Q be polynomials of x1 , . . . , xd . As Hörmander proved (see [7], Ch. 2), in order that a constant γ exist such that for all f ∈ Co∞ (G) (the norms are those in L2 (Rd )) ||Q(D)f ||2 ≤ γ||P (D)f ||2 , it is necessary and sufficient one of the following two conditions to fulfil: 0 |Q(α) (ix)| α∈Zd |Q(ix)| + < ∞, sup 0 < ∞. sup 0 (α) (α) (ix)| (ix)| x∈Rd x∈Rd α∈Zd |P α∈Zd |P +

+

We note that the necessity of the given condition is valid in a more general situation. Theorem 2.1. If for some p ≥ 1 and q > 0 for all f ∈ C0∞ (G) ||Q(D)f ||q ≤ γ||P (D)f ||p , then

0 sup 0

α∈Zd +

|Q(α) (ix)|

x∈Rd

α∈Zd +

|P (α) (ix)|

< ∞.

In what follows we consider the simplest case d = 1 when the condition in question reduces to deg Q ≤ deg P . Theorem 2.2. Let G = (−1, 1) ⊂ R. Inequality ||Q(D)f ||q ≤ γ||P (D)f ||p holds true for all functions f ∈ C0∞ (G) for q > 0 and p ≥ 1 provided deg Q < deg P, and only for q > 0 and p ≥ max{q, 1} provided deg Q = deg P . Let us study a different problem. Let Q and Pj (1 ≤ j ≤ m) be polynomials (algebraic polynomials with comd . When plex coefficients), and D = dx ||Q(D)f ||Lq ≤ γ

m

||Pj (D)f ||Lpj

(∗)

j=1

with a constant γ independent of the function f ∈ Wpr ? First, established is a criterion of validity of such inequalities for m = 1, separately in the cases of functions on the circle T, real axis R and half-axis R+ = [0, ∞) : ||Q(D)f ||q ≤ γ||P (D)f ||p (∗∗) provided s = deg Q ≤ r = deg P and {p, q} ⊂ [1, ∞]. Theorem 2.3. Let p and q ∈ [1, ∞]. For (∗∗) to take place when for all periodic functions from Wpr (T) when s < r, it is necessary and sufficient that P(ik) = 0 (k ∈ Z) implies Q(ik) = 0. When s = r assumption q ≤ p to be added.


505

Theorem 2.4. Let (p, q) = (1, ∞). For (∗∗) to take place for all functions from Q(ix) Wpr (R) when s < r, it is necessary and sufficient that supx∈R P(ix) < ∞ and q ≥ p. When s = r assumption q = p to be added. If simultaneously p = 1 and q = |Q(ix)| ∞, for (∗∗) to hold it is necessary and sufficient that supx∈R |P(ix)|+|P (ix)| < ∞ and s < r. Theorem 2.5. Let (p, q) = (1, ∞). For (∗∗) to take place for all functions from Wpr (R+ ) when s < r, it is necessary and sufficient that supz:Re z≤0 Q(z) P(z) < ∞ and q ≥ p. When s = r assumption q = p to be added. If p = 1 and q = ∞, for (∗∗) to hold it is necessary and sufficient that s < r and, with δ being an arbitrary negative number, Q(z) |Q(ix)| < ∞, sup sup < ∞. P (z) |P (ix)| + |P (ix)| z:Re z≤δ 0 and αs ∈ N (1 ≤ s ≤ m). Then for each p ∈ [1, ∞] and any function f ∈ Wpr (R+ )

||Q(D)f ||∞ ≤ γ0 ||P (D)f ||p , where the least constant γ0 = ||g||p , with 1p + p1 = 1, and for R(z) = T 1 g(x) = R(ζ)e−xζ dζ. 2πi Γ

Q(z) P (z)

Here λ1 , . . . , λm are within the closed contour Γ, while the conjugates to other zeros of P are outside of Γ. We then pass to the case m ≥ 2 (see (∗)). Let degree deg Pj = rj (1 ≤ j ≤ m), max rj = r1 and s = deg Q ≤ r1 . We consider in the sequel that s < r1 . When s = r1 one has to apply the result obtained for s < r1 to the polynomial Q1 = Q − μP1 of a lower degree. Theorem 2.7. Let I be the greatest common divisor of polynomials Pj , 1 ≤ j ≤ m, all with zeros on the imaginary axis iR, while the polynomial Q is divisible by I. Then for s < r1 and any q ∈ [1, ∞] and 1 ≤ p1 ≤ q m ||Q(D)f ||Lq (R) ≤ γ ||Pj (D)f ||Lpj (R) j=1

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where pj ∈ [1, p1 ] for j ≥ 2 if Pj has zeros off iR and pj = p1 otherwise. The rj constant γ does not depend of f ∈ ∩m j=1 Wpj (R). If s = r1 one has to add condition p1 = q. And if Q is divisible only by the greatest common divisor of I and I , the indicated inequality holds true only for q = ∞ and pj ≡ 1. Proof. We may assume that Pj = I · Ij · P˜j , where I · Ij = 0 off iR, and P˜j = 0 on iR (1 ≤ j ≤ m). The polynomials {Ij }m 1 have no common zeros. If I1 ≡ 1, that is, P1 = 0 on iR only at zeros of I, then, by Theorem 2.4 for s < r1 and p1 ∈ [1, q] Q(D)f Lq (R) ≤ γ P1 (D)f Lp1 (R) , while p1 = q in this inequality when s = r1 . It is possible to add now any nonnegative summands on the right-hand side. We further suppose that I1 (ix1 ) = 0 for some x1 ∈ R. If I1 (z) = aΠhν=1 (z − ixν ), where xν ∈ R (1 ≤ ν ≤ h), then I1 (ix) ∈ R when a = ih for all x ∈ R. Thus one may assume that I(ix) and Ij (ix), 1 ≤ j ≤ m, are real for all x ∈ R. Assume that for n = deg P˜1 (deg P I ≤ deg P1 = r1 ) P (z) = iH(z)I1 (z) +

m

λj Ij (z),

H(z) = (iz + x1 )n ,

j=2

0m where real numbers {λj }m 2 are specified so that j=2 λj Ij (ix) = 0 at zeros of I1 (ix) (this will be justified below). Then P (ix) = 0 for all x ∈ R, since the sum does not vanish at the points where I1 (ix) = 0, while the imaginary part ImP (ix) = 0 when I1 (ix) = 0. Replacing i with −i if needed, we may assume that degP = degHI, in other words the degree does not become smaller when adding the sum. Applying Theorem 2.4 to the operators Q(D) and P I(D), we obtain, with p1 ∈ [1, q] when s < r1 and p1 = q when s = r1 , Q(D)f Lq (R) ≤ γ P I(D)f Lp1 (R) . Using then the triangle inequality for the norms, we get Q(D)f Lq (R) ≤ γ( H · I · I1 (D)f Lp1 (R) +

m

I · Ij (D)f Lp1 (R) ).

j=2

It remains to apply Theorem 2.4 to each summand, m times all together. Let now Q is divisible only by the greatest common divisor of I and I . By Theorem 2.4 Q(D)f L∞ (R) ≤ γ P I(D)f L1 (R) , and the above proven inequality works with q = pj ≡ 1. It remains to make sure of that if {Ij }r1 are real polynomials with no common real zeros for all r ≥ 2 and E is a finite subset of R, then the numbers λj ∈ (0, 1],


507

1 ≤ j ≤ r, exist such that r

λj Ij (x) = 0

(x ∈ E).

j=1

When r = 2 one can choose λ1 = 1, while any nonzero real number satisfying λ2 = − II12 (x) (x) when I2 (x) = 0 and x ∈ E can be taken as λ2 . For r ≥ 2 induction argument is applicable. Let us denote by I2,r the greatest common divisor of the polynomials {Ij }r2 with real zeros. By inductive hypothesis, there exist real numbers λj ∈ (0, 1], 2 ≤ j ≤ r, such that r Ij (x) = 0 (x ∈ E). λj I2,r (x) j=2 Consider the sum, in which λ1 is yet to be specified, r r Ij (x) . λj Ij (x) = λ1 I1 (x) + I2,r (x) λj I2,r (x) j=1 j=2 Since I1 and I2,r has no common real zeros, the sum does not vanish for any λ1 when x ∈ E and I1 (x) = 0. If x ∈ E and I1 (x) = 0, it suffices to choose any real λ1 satisfying r −1 λ1 = λj Ij (x) (x ∈ E). I1 (x) j=2

The proof is complete. A similar result is valid for periodic functions.

Theorem 2.8. If the condition Pj (ik) = 0 for k ∈ Z and all j ∈ [1, m] implies Q(ik) = 0 (which is also necessary), then for s < r1 and any q, p1 ∈ [1, +∞] m Q(D)f Lq (T) ≤ γ Pj (D)f Lpj (T) . j=1

When j ≥ 2 here, pj ∈ [1, +∞] if Pj has zeros off iZ, while pj ∈ [1, p1 ] if Pj has rj no zeros off iZ. The constant γ is independent of function f, f ∈ ∩m j=1 Wpj (T). When s = r1 one has to add the condition p1 ∈ [1, q]. The case of the semi-axis is more difficult. Theorem 2.9. Let deg Q < deg P and polynomials P1 and P2 , as well as Q, are 2 (z) = 0 when divisible by I, the latter with no zeros when Re z > 0. Let also PI(z) Re z ≤ 0. Then for each q ∈ [1, +∞] and both p1 , p2 ∈ [1, q] ||Q(D)f ||Lq (R+ ) ≤ γ(||P1 (D)f ||Lp1 (R+ ) + ||P2 (D)f ||Lp2 (R+ ) ), with constant γ independent of f ∈ Wpr11 (R+ ) ∩ Wpr22 (R+ ). If under the same assumptions on I and P2 we have I = I1 · I0 , where I0 (z) = 0 for Re z < 0, I1 (z) = 0 when Re z = 0, and polynomials P1 and Q are

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divisible by I1 and by the greatest common divisor of I0 and I0 , then for 1 ≤ p1 ≤ ∞ and all f ∈ Wpr11 (R+ ) ∩ W1r2 (R+ ) ||Q(D)f ||L∞ (R+ ) ≤ γ(||P1 (D)f ||Lp1 (R+ ) + ||P2 (D)f ||L1 (R+ )||). It was suggested in [8] to give direct proofs of two sharp inequalities for differential operators of the second order of functions on the half-axis known in the theory of operators. This is fulfilled by the author in [9]. We now give an example when sharp inequalities are different for periodic and non-periodic functions (even for p = ∞). Here ε > 0. Theorem 2.10. a) For each p ∈ [1, +∞] and any function f ∈ Wp2 (T) f ∞ ≤ γ0 f − ε2 f p , where the least constant γ0 =

1 21/p sinh πε

p−1 πε p−1 p p 1 p (sinh t) p−1 dt . ε 0

b) For any function f ∈ Wp2 (R) and p ∈ [1, +∞] p−1 p−1 1 p p−1 p ||f − ε2 f ||p . ||f ||∞ ≤ p ε One can find the proofs of Theorems 2.3–2.6 in [9].

3. Methods of summability of Fourier series and K-functionals We go on to problems of approximation of periodic functions by linear polynomial operators. Modulus of smoothness of a 2π-periodic function f ∈ C(T), T = [−π, π], of order r ∈ N and step h > 0 is defined as ωr (f ; h) = sup{|Δrδ f (x)| : x ∈ T, δ ∈ (0, h]},

Δδ f (x) = f (x) − f (x + δ).

It is known, for example, that ωr (f ; h) ≤ hr if and only if f (r−1) is absolutely continuous on T and almost everywhere |f (r) (x)| ≤ 1 (D.A. Raikov). For each r ∈ R there exists a sequence of trigonometric polynomials of degree not higher than n satisfying the inequality 1 max |f (x) − Tr,n (f ; x)| ≤ γ(r)ωr f ; x n (r = 1 – Jackson, r = 2 – Akhiezer, r ≥ 3 – Stechkin). One can replace the norm in C(T) in this inequality by that in Lp (T) with 1 ≤ p < ∞. It was the author who had built long ago polynomials τr,n (of degree not higher than n) which satisfy the two-sided inequality 1 1 ≤ ||f − τr,n (f )|| ≤ γ2 (r)ωr f ; . γ1 (r)ωr f ; n n


509

These were the means of the Fourier series of the function f of Rogosinski-Bernstein type first. To date exact rates of approximation of individual functions by all classical methods of summability of Fourier series are already found (see [6], and also [10] and [3]). We set for each integer r ≥ 2 π π 1 r Jr,n (f ) = Jr,n (f, x) = f (x + t)Dn (t) dt, αr,n = Dnr (t) dt, αr,n −π −π where Dn is the nth Dirichlet kernel. These are the Fejér-Jackson type polynomials od degree rn. Theorem 3.1. For each r ≥ 3 there exist positive constants γ1 (r) and γ2 (r) such that for any function f ∈ C(T) 1 1 ≤ ||f − Jr,n (f )|| ≤ γ2 (r)ω2 f ; . γ1 (r)ω2 f ; n n The proof is fulfilled by comparison with polynomials τ2,n (see above). It turned out that this result was already proved in [11] by a different method for only even r ≥ 4 (the case of positive operators). However, in author’s opinion the most interesting case is that of minimal r, r = 3. The problem of comparison in norm of various methods of summability of Fourier series arose after obtaining the two-sided inequalities (publications of 1968 by the author and also by H. Shapiro). It turned out that all (C, α) methods as well as that of Abel-Poisson (for definitions, see, e.g., [12]) are equivalent in this sense provided their parameters are connected in a specific way. 1 Theorem 3.2. For any α > 0, p ∈ [1, +∞], f ∈ Lp (T), and r ∈ (0, 1) , n = [ 1−r ] (integral part)

γ1 (α) f (·) − Ur (f ; (·)) p ≤ f − σnα (f ) p ≤ γ2 (α) f (·) − Ur (f ; (·)) p . Further, J.L. Lions and J. Peetre introduced K-functionals for finding interpolation spaces between two Banach spaces (see, e.g., [13], real method of interpolation). Let us consider a differential operator Dr defined by a matrix {μk,r } (k ∈ Zd \ 0, μk,r = 0, lim|k|→∞ μk,r = ∞): μk,r fˆ(k)ek ; Dr f ∼ k=0

if μk,r = −|k|

2r

this is the poly-harmonic operator Δr . Let W (Dr )p =: {f ∈ Lp (Td ) : Dr f ∈ Lp (Td )},

K(t, f ; Lp , W (Dr )p ) :=

inf

g∈W (Dr )p

{||f − g||p + t||Dr g||p }

(t > 0). 1

It is well known that if d = 1 and Dr f = f (r) then K(t, f ) 0 ωr (f ; t r ) (see the same reference).

510

R.M. Trigub

Usual moduli are not suitable for d ≥ 2 (in C(Td ) and L1 (Td )). To clarify the situation, we will consider here only the case of even r. We introduce the linearized moduli (r ∈ N) > > 2r > > (2r)! ν > ω ˜ 2r (f, μ; h) := > (−1) f (· + (ν − r)hu) dμ> >. Rd ν=0 ν!(2r − ν)! The operation of taking the upper bound with respect to h is replaced by the integral averaging (μ is a finite Borel measure).0 If dμ = χE du and E is the unit ball 0 in Rd we will write ω ˜ 2r (f, h); while for dμ = χEj duj and Ej = [−1, 1] ⊂ Oxj , + for all 1 ≤ j ≤ d, we will write ω ˜ 2r (f ; h). Let 2r d (2r)! Δ+ (−1)ν f (x + (ν − r)δe0j ), f (x) = r,δ ν!(2r − ν)! j=1 ν=0 here e0j is the unit vector of the axis Oxj . We define + (f, h)p = sup ||Δ+ ω2r r,δ f (·)||p , 0

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