Functional-Analytic and Complex methods, their Interactions, and Applications to Partial Differential Equations Proceedings of the international Graz workshop
Editors
H. Florian, N. Ortner, F. J. Schnitzer & W.Tutschke
World Scientific
Functional-Analytic and Complex methods, their Interactions, and Applications to Partial Differential Equations
Functional-Analytic and Complex methods, their Interactions, and Applications to Partial Differential Equations proceedings of the international Craz Workshop
Graz, Austria
1 2 - 1 6 February 2001
Editors H. Florian Graz University of Technology, Austria
N. Ortner University of Innsbruck, Austria
F. J. Schnitzer Mining University Leoben, Austria
W. Tutschke Graz University of Technology, Austria
V f e World Scientific ™1
New Jersey'London'Singapore'Hong
Kong
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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
FUNCTIONAL-ANALYTIC AND COMPLEX METHODS, THEIR INTERACTIONS, AND APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS Proceedings of the International Graz Workshop Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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V
PREFACE
Mathematical Analysis is a field which is used in almost all applications of Mathematics. In many cases these applications are connected with numerical methods. Electronic data processing, often in combination with computeralgebraic methods, creates tools which became absolutely essential for many applied sciences. That way Mathematical Analysis became an extremly ramified domain in Mathematics. In order to remain applicable, Mathematical Analysis has clearly to arrange its results in general methods and principles. Concerning those parts of Mathematics for which function spaces form the background, this can be done within the framework of Functional Analysis. The latter branch of Mathematical Analysis meets the needs of today's theory of Partial Differential Equations. The theory of Sobolev spaces, for instance, leads to a unified approach to analytical and numerical methods for elliptic Partial Differential Equations. Whereas classical methods are mainly restricted to special classes of Partial Differential Equations, the functional-analytic approach leads to a high generality. Another field in Mathematics originally connected with very special Partial Differential Equations is Complex Analysis. Holomorphic functions which are in the center of attention of classical Complex Analysis are connected with the Cauchy-Riemann system. The theory of Generalized Analytic Functions founded by the Georgian I. N. Vekua and the American L. Bers is aimed at investigating general linear and elliptic first order systems in the plane for two desired real-valued functions by complex methods. This goal can be reached by a functional-analytic approach based on the concept of weak (distributional) derivatives. While each special differential equation has its own fundamental solution, a general theory for linear and non-linear (elliptic) systems in the plane can be developed by using the Cauchy kernel and its square in order to generate the necessary inverse integral operators. Meanwhile not only principles of Functional Analysis but also methods of the theory of Generalized Analytic Functions became important parts of Mathematical Analysis. The development of both branches goes on continuously. Over and over again interactions of Functional Analysis and Complex Analysis turned out to be fundamental for a unified and far-reaching approach to Mathematical Analysis.
vi In order to promote further research of Mathematical Analysis, the Department of Mathematics of the Graz University of Technology organized in cooperation with the Federal Austrian Ministry of Education, Science and Culture a Workshop entitled Functional-analytic and complex methods, their interactions, and applications to Partial Differential Equations. The present book contains the Proceedings of that Workshop which took place at Graz University of Technology from February 11 to 17, 2001.
Main areas covered by the Workshop are 1. Boundary value problems and initial value problems for Partial Differential Equations. 2. Applications of functional-analytic and complex methods to Mathematical Physics. 3. Partial complex differential equations in the plane. 4. Complex methods in higher dimensions. Many contributions are aimed at the solution of initial value and boundary value problems for Partial Differential Equations. In some cases this can be done using integral representations such as those generated by fundamental solutions. That way one gets integral or integro-differential equations for the desired solution. One contribution deals with the optimization of fixed-point methods. Concerning elliptic systems, also cases are under consideration in which the ellipticity degenerates. In connexion with boundary value problems also orthogonal decompositions are investigated. Concerning hyperbolic equations the wave propagation is under consideration. Further, also a uniqueness theorem for the Dirichlet problem for hyperbolic equations is proved. The second chapter addresses applications to Mathematical Physics. While one contribution deals with a general method for the proof of convervation laws, other articles are connected with special applications: Maxwell's System, Crystal Optics, dynamical problems for cusped bars and so on. The latter leads to approximations based on I. N. Vekua's approach to shell theory.
vii
Complex methods for systems of differential equations in the plane are applied not only to first and second order systems but also to higher order ones. Generalizing the theory of poly-analytic functions, a general theory of poly-pseudoanalytic functions is developed. For the Bauer-Peschl equation a new representation formula for its solution is obtained. Complex methods in higher dimensions use not only several complex variable but also variables in Clifford Algebras. Within the framework of Clifford Analysis also higher order differential equations are solved. These Proceedings are neither as complete as a textbook is, nor they contain exclusively new results as a scientific journal does. Of course, almost all contributions contain at least partly new results. This applies to survey articles, too. On the other hand, the contributions reflect present-day investigations and, therefore, certainly not all obtained results have already reached their final version. Some of the contributions deal with questions about which so far only comparatively few investigations have been carried out. This concerns, for instance, the optimization of fixed-point methods, theory and application of difference equations in the complex domain, and the Dirichlet problem for hyperbolic equations. We hope that not only these papers but also all of the papers contained in the Proceedings will stimulate future research in Mathematical Analysis. The meeting was sponsored by • the Federal Austrian Ministry of Education, Science and Culture, • the Graz University of Technology, especially by the Faculty of Science, • the Austrian National Bank, • the State of Styria, and • the City of Graz. Without the support of those institutions, it would not have been possible to carry out the Workshop. The Organizing Committee would like to thank the mentioned institutions. The present Proceedings contain almost all contributions given at the Workshop. In case only an Abstract is available, this has been included in the Proceedings.
viii
For most competent typing and editing of this volume, we would like to thank most cordially Ms Barbara Poltl, secretary of the Mathematics Department D of Graz University of Technology. The editors would like to thank all the staff of the World Scientific Publishing Company, especially Ms Lakshmi and Ms Cai Wen, for the pleasant cooperation.
The Organizing Committee H. Florian • N. Ortner • F. J. Schnitzer • W. Tutschke.
CONTENTS Preface
v
1
Boundary value problems and initial value problems for partial differential equations
1.1
Hyperbolic equations, waves and the singularity theory
3
by M. Tsuji (Kyoto, Japan) 1.2
On the regularity of solutions of the first boundary problem for higher order hyperbolic differential equations
23
by B. Paneah (Haifa, Israel) 1.3
Zeilon's operator and lacunae by P. Wagner (Innsbruck,
1.4
33
Austria)
Representation of the Stokes potential in divergence form
47
by J. Dubinskii (Moscow, Russia) 1.5
On spectra of the operator rotor
58
by R. S. Saks (Ufa, Russia) 1.6
Algebraic properties of potential differential and pseudodifferential operators
64
by I. E. Pleshchinskaya (Kazan, Russia) 1.7
The method of weighted function spaces for solving initial value and boundary value problems
75
by W. Tutschke (Graz, Austria) 1.8
The optimization of fixed-point methods
91
by S. Graubner (Leipzig, Germany) 1.9
Principle of Telethoscope by S. Saitoh (Kiryu, Japan)
101
1.10
Isometric mappings and the problem of A. D. Aleksandrov for conservative distances
118
by T. M. Rassias (Athens, Greece) 1.11
Natural metrics of differential equations (abstract)
126
by Z. D. Usmanov (Dushanbe, Tajikistan) 1.12
Embedding theorems in anisotropic functional spaces (abstract)
129
by M. D. Ramazanov (Ufa, Russia) 1.13
Distributional analysis of boundary value problems in half-spaces-exemplifled by Melan's problem of elastostatics (abstract) by G. Kirchner (Innsbruck,
1.14
130
Austria)
Backlund transformations and nonlinear evolution equations (abstract)
132
by J. Piingel (Graz, Austria) 1.15
Nonlinear perturbations of systems of partial differential equations (abstract)
133
by C. J. Vanegas (Caracas, Venezuela) 1.16
Partial differential equations and cubature formulas (abstract)
134
by M. D. Ramazanov (Ufa, Russia) 2
Applications of functional-analytic and complex methods to mathematical physics
2.1
Conservation laws for differential equations by V. S. Vladimirov (Moscow, Russia)
137
XI
2.2
Applied quatemionic analysis. Maxwell's system and Dirac's equation
143
by V. V. Kravchenko (Mexico City, Mexico) 2.3
Integral transforms method in the conjunction problems of electromagnetic fields
161
by N. B. Pleshchinskii (Kazan, Russia) 2.4
Remarks on fundamental matrices (Green's Tensors) in elastodynamics, piezoelectricity and crystal optics by N. Ortner (Innsbruck,
2.5
178
Austria)
Dynamical problems in the (0,0) and (1,0) approximations of a mathematical model of cusped bars
188
by G. V. Jaiani (Tbilisi, Georgia) and S. S. Kharibegashvili (Tbilisi, Georgia) 2.6
Vanishing viscosity limit of the incompressible Navier-Stokes equation (abstract)
248
by K. Asano (Kyoto, Japan) 3
Partial complex differential equations in the plane
3.1
Boundary value problems for a class of nonregular elliptic equations by N. E. Tovmasyan (Yerevan,
3.2
253
Armenia)
On the solutions of the iterated Bers-Vekua equation
266
by P. Berglez (Graz, Austria) 3.3
The Bauer-Peschl equation — Derivation and solution of a partial differential equation by Laplace's method by K. W. Tomantschger (Graz, Austria)
276
3.4
Komplexe Differenzengleichungen
294
by M. Canak (Belgrad, Yugoslavia) 3.5
Stability of solutions of nonuniformly elliptic systems and boundary value problems
307
by A. Mamourian (Teheran, Iran) 3.6
An explicit solution to a class of second kind complex integral equations with singular and super-singular kernels
313
by N. Rajabov (Dushanbe, Tajikistan) 3.7
An explicit solution to a class of second kind linear systems of complex integral equations with singular and super-singular kernels (abstract)
330
by N. Rajabov (Dushanbe, Tajikistan) 3.8
Initial value problems for pseudoholomorphic functions (abstract)
333
by A. O. Celebi (Ankara, Turkey) 3.9
A boundary value problem for a special class of generalized analytic functions and its application to second order differential equations (abstract)
335
by U. Aksoy (Ankara, Turkey) 3.10
Complex methods in operator theory (abstract)
337
by H. L. Vasudeva (Chandigarh, India) 3.11
On one boundary value problem of the theory of analytic functions on a cut plane (abstract) by N. Manjavidze (Tbilisi, Georgia)
338
XIII
4
Complex methods in higher dimensions
4.1
On a class of elliptic systems in the halfplane which are singular at the boundary
341
by A. Dzhuraev (Dushanbe, Tajikistan) 4.2
Integral representations for inhomogeneous over-determined second order system of several complex equations
378
by H. Begehr (Berlin, Germany), D. Q. Dai (Guangzhou, China) and A. Dzhuraev (Dushanbe, Tajikistan) 4.3
Compactness of the canonical solution operator to B restricted to Bergman spaces by F. Haslinger (Vienna,
4.4
4.5
Austria)
The theorem of the regular continuation for partial differential equations in general form and its applications by Le Hung Son (Hanoi,
394
401
Vietnam)
Some higher order equations in Clifford analysis
414
by E. Obolashvili (Tbilisi, Georgia) 4.6
Generalizations of the complex analytic trigonometric functions to Clifford analysis by Eisenstein series
438
by R. S. Krausshar (Gent, Belgium) 4.7
Some implications of Clifford analysis (abstract)
456
by J. Ryan (Arkansas, U.S.A.) 4.8
Around zeroes of monogenic and hypermonogenic functions (abstract)
457
by T. Hempfling (Freiberg, Germany) 4.9
Identity surfaces (abstract) by W. Tutschke (Graz, Austria)
458
C H A P T E R 1:
BOUNDARY VALUE PROBLEMS AND INITIAL VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS
3
H Y P E R B O L I C EQUATIONS, WAVES AND THE SINGULARITY THEORY M. T S U J I Department
of Mathematics, Kyoto Sangyo Kita-Ku, Kyoto 603-8555, Japan E-mail:
[email protected] University
We will consider linear and nonlinear wave propagation. Equations which describe the phenomena of wave propagation are generally hyperbolic partial differential equations. In this talk "wave" means "singularity of solution". Therefore our interest is to study the singularities of solutions of hyperbolic equations. The most traditional and typical method for the singularity theory has been the "resolution of singularities", that is to say, to lift the surfaces with singularities into higher dimensional space so that the singularities would disappear. This is a survey note of our researches from the point of view of resolution of singularities.
1
Introduction
In this talk we will consider the phenomena of wave propagation which are governed by hyperbolic partial differential equations. "Wave propagation" is mathematically understood as the propagation of singularities. The most traditional and typical method for the problem of singularity has been the "resolution of singularities", that is to say, to lift the surfaces with singularities into higher dimensional space so that the singularities would disappear. In the case where the singularities do not exist, we can develop various kinds of discussion. Therefore, after studying a problem with no singularity in higher dimensional space, we project the obtained results into the base space. By this program we would like to solve the problems of singularity in analysis and to understand the meanings of the phenomena. First we will consider the singularities of elementary solutions of linear hyperbolic equations with constant coefficients. Concerning this, we will present some problem which we could not solve even now. Next we will explain the passage from linear hyperbolic equations to nonlinear first order partial differential equations and study the latter equations. Finally we will report our recent researches on nonlinear hyperbolic equations of second order. The classical method to solve first order partial differential equations is so-called the "characteristic" one. Then the characteristic strips are obtained as "solutions of canonical differential equations" defined in cotangent space. Therefore the solutions of first order partial differential equations are naturally lifted into the cotangent space. For partial differential equations of
4
second order also, we would like to develop similar discussions. Then the problem is how to lift the solution surfaces into the cotangent space. To do so, we will rewrite partial differential equations in another form defined in the cotangent space. In §4 we will consider the method of integration of second order partial differential equations from this point of view and introduce a notion of "geometric solution" defined in the cotangent space. In §6 we will consider the same problem for a certain system of conservation laws. 2
Linear hyperbolic equations with constant coefficients
Let P{DX) be a hyperbolic partial differential operator of order m with respect to fi G R n / { 0 } where x G R™ and Dx = —i{d/dx\, 8/8x2, • • •, d/dxn), i.e., its characteristic polynomial P(£) satisfies the following properties: i) P m ($) ^ 0 where P m (£) is the principal part of P(£), ii) There exists a constant 70 > 0 such that P(£ - i-yd) ^ 0 for all f G R " and 7 > 70 > 0. Let E{P)(x) be an elementary solution of P(DX) whose support is contained in {x G R n ; < x, "d >> 0}, then it is written by 1
r
E(P)(x) = (~r
i<x,t;>
/
-piTT*-
For a distribution u(x), Su is the support of u(x) and SSM is the singular support of u(x) which is the smallest closed subset such that u(x) G C°°(R n — SSw). One of the classical and unsolved problems is to "give the complete description of SSE(P) for any hyperbolic operator P(DX)". Concerning this problem, M. F. Atiyah, R. Bott and L. Garding x published the profound and elegant theory. We explain one of their results. We develop s m P ( s - 1 £ + £) in ascending power of s: smP(s-1Z
+0 =
sPPd0+O(sp+1)
where P^(C) ^ 0. The number p = m^(P) is the multiplicity of £ relative to P and the polynomial P^(C) is called the "localization of P at a point £". Then one can see that P^(C) is also hyperbolic with respect to fl. This definition means that P^(C) is the best approximation of P(() at a point £. Let us denote A = {£ G R"/{0};P m (£) = 0}. If A has no singular point, the localized polynomial P{(C) is at most of degree one. They proved the following Theorem 2.1 ([1]) lim sm-p s—»oo
x
e-™
0, equation (4.1) is hyperbolic. In this note, we will treat the equations of hyperbolic type. Let Ai and A2 be the solutions of A2 + BX + (AC + DE) — 0. Then, in the case where D ^ 0, the characteristic strip satisfies the following equations (see 3 and 7 ' ? ) :
{
dz — pdx — qdy = 0, Ddp + Cdx + \1dy = 0,
(4.4)
Ddq + \2dx + Ady = 0, or dz — pdx — qdy = 0, Ddp + Cdx + \2dy = 0, Ddq + Xxdx + Ady = 0.
(4.5)
Let us denote OJQ = dz — pdx — qdy, u)\ = Ddp + Cdx + X\dy and u>2 = Ddq + X2dx + Ady. Exchanging Ai and A2 in ui\ and LJ2, we define w\ and w2 by w\ = Ddp + Cdx + X2dy and VJ2 = Ddq + X\dx + Ady. Take an exterior product of wi and LJ2, and also of w\ and w2. Substitute into their product the relations of the contact structure U>Q = 0, dp = rdx + sdy and dq = sdx + tdy. Then we get wiAw 2 = EJiAro 2 = D{Ar + Bs + Ct + D(rt - s 2 ) - E}dx Ady .
(4.6)
10
Above we have assumed fl^O, though it is not essential for our discussion. The key point is to represent equation (4.1) as a product of one forms. For example, we will consider in §3 and §4 a certain case where D = 0. It will be shown that, though the above decomposition might be a small idea, it would effectively work to solve equation (4.1) in exact form. In a space whose dimension is greater than two, it is generally impossible to do so. The method of integration of (4.1) had been first investigated by the french school, especially by G. Monge, G. Darboux 3 and E. Goursat 7 , 8 . Their idea is to reduce the solvability of (4.1) to the integration of first order partial differential equations. For this aim, they have introduced the following definition: A function V — V(x,y,z,p,q) is called a "first integral" of {wo,wi,W2} if dV = 0 mod {W0,LJI,UJ2}If {u>o,u)i,tU2}, or {wo,n7i,ro2}, has at least two independent first integrals, the non-characteristic Cauchy problem can be reduced to the integration of certain nonlinear first order partial differential equations. Therefore the non-characteristic Cauchy problem for (4.1) can locally admit a smooth solution. This is their result. If we may assume the global existence of two independent first integrals, we can develop a global theory for (4.1) by using our results on the singularities of solutions of first order partial differential equations. Concerning this subject, see 24 . Next we will study the method of integration of (4.1) in the case where neither {U>O,OJI,U>2} nor {u>o, vo\,W2\ has two independent first integrals. We start from the point at which equation (4.1) is represented as a product of one forms as (4.6). We suppose D ^ 0 for simplicity, though it is not indispensable for our study. The essential condition for our following discussion is A ^ 0. Let us pay attention to the property that the left hand side of (4.6) is a product of one forms defined in R 5 = {(x, y, z,p, q)}. This suggests us to introduce a notion of "geometric solution" as follows: Definition 4.2 A regular geometric solution of (4.1) is a submanifold of dimension 2 defined in R 5 = {(x,y,z,p,q)} on which it holds that dz = pdx + qdy and u)\ A L02 = 0. R e m a r k In the above definition, we have added "regular" to "geometric solution". This means that we will soon introduce a "singular" geometric solution whose dimension may depend on each point. The problem to construct the geometric solution is similar to the Pfaffian problem. A difference between the classical Pfaffian problem and the above one is that we consider it in C°°-space. Therefore we need some condition which corresponds to "hyperbolicity".
11
Our problem is to find a "submanifold on which a>o = 0 and W1AW2 = 0". First we will sum up the classsical method, though it is written in J. Hadamard 9 , and also in R. Courant-D. Hilbert 2 a little. Let us consider the Cauchy problem for equation (4.1). The initial condition is given by a smooth strip Co which is defined in R 5 — {(x,y,z,p,q)} and written down as follows: C 0 : {x,y,z,p,q)
= (x0{0,yo{0,zo{0>Po{0,qo(0),
C e R1.
The idea of the classical method is to represent the solution surface by a family of characteristic strips. Then they are determined as solutions of the following system of equations (see 16 , 9 and 2 ): dz da
dx da
dy da
„ ^dp H da
„dx da
, dy da
, dx ,dy Dp-+X A22^+A^ —+ A-^ = = O, 0, da da da da da ^dp ^,dx , dy
(4.7) '
D +C
i dp+X^r^
„dq
, dx
Ady
The initial condition for system (2.9) is given by (4.8)
The local solvability of the Cauchy problem (4.7)-(4.8) is already proved first by H. Lewy 16 and afterward by J. Hadamard 9 . Let (x(a,P),y(a,P),z(a,P),p(a,f3),q(a,(3)) be a solution of (4.7)-(4.8) . Then we can prove dz/d(3 — pdx/dp — qdy/d(3 = 0. Therefore we do not need to add this equation to system (4.7). This means that (4.7) is just a "determined" system. What we must do more is to represent z = z(a, (3) as a function of (x, y). To do so, we calculate the Jacobian D(x, y)/D(a, /?). Then it holds along the initial strip Co that D{x,y) D(a,[3)
1 {Ay02 - Bx0y0 + Cx02 + D(p0x0 + q0y'o)}. Ai-A2
12
As we have assumed that the initial strip Co is not characteristic, we see by (4.3) that the Jacobian D(x, y)/D(a, 0) does not vanish in a neighbourhood of Co- Therefore we can uniquely solve the system of equations x = x(a,/3), y = y(a, j3) with respect to (a, ($) in a neighbourhood of each point of Co and denote them by a = a(x, y) and (3 =fi(x,y). Then we get the solution of the Cauchy problem for (4.1) by z(x,y) = z(a(x,y),/3(x,y)). Summing up the above discussion, we obtain the following: Theorem 4.3 ([16], [9]) Assume that the initial strip Co is not characteristic. Then the Cauchy problem for (4.1) with the initial condition Co uniquely admits a smooth solution in a neighbourhood of each point of Co. Remark If the equation and the solution are sufficiently differentiable, the solution is uniquely determined by the initial data. For example, the uniqueness of solution in C°°-space is one of the classical known results (see H. Lewy 16 and J. Hadamard 9 ) . But the uniqueness of solution in C 2 -space is a delicate problem. We will consider this subject in a forthcoming paper. 5
Nonlinear hyperbolic equations
In this section we will consider the Cauchy problem for a certain nonlinear hyperbolic equation as follows: F r
^ ^=W-dy:f(gy-)
*(0,y) = z0{y),
R2+,
(5.1)
on {a: = 0, y € R 1 }
(5.2)
= r-f'(a)t = 0
dz —(0,y) = Zi{y)
in
where R ^ = {x > 0,y G R 1 } , and f(q) is in C°°(R 1 ) and f'{q) > 0. Here z = z(x, y) is an unknown function of (x, y) G R 2 . We assume that the initial functions z»(y) (i = 0,1) are sufficiently smooth, and that z'0(y) is bounded. Equation (5.1) is of Monge-Ampere type which we have studied in §4. In fact, if we may put A=1,B = D = E = 0, and C = —f'(q) in (4.1), then we get (5.1). It is well known that the Cauchy problem (5.1)-(5.2) does not have a classical solution in the large. The first example was given by N. J. Zabusky 30 . After this paper, many people have studied the life-span of classical solutions. As there are too many papers on this subject, we do not mention it here. The above phenomenon means that singularities generally appear in finite time. Our main problem is how to extend the solutions of (5.1) beyond the singularities. We can see that the solutions take many values after the appearance of singularities. If we may consider this problem from the physical point
13
of view, we would be obliged to construct single-valued solutions of (5.1). To solve the problem of this kind, we recall what we have done for nonlinear first order partial differential equations. First we have lifted the solution surface into cotangent space so that its singularities would disappear. Then we could extend the lifted solution so that it would be defined in the whole space. Next we have projected it to the base space and gotten a multi-valued solution. Our final problem has been how to choose a single value from many values of the projected solution so that the new single-valued solution should satisfy some additional conditions attached to some physical phenomena. Now we will construct a geometric solution of (5.1)-(5.2) by the method introduced in §4. The first step is to represent equation (5.1) as a product of one forms. Let us denote u)\ = dp± \{q)dq and W2 = ±\{q)dx + dy where M
