Complex Analysis and Potential Theory
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Potential Theory Proceedings of the Conference Satellite to ICM 2006 Gebze Institute of Technology, Turkey
8-14 September 2006
Editors
Tahir Aliyev Azeroglu Gebze Institute cf Technology, Turkey
Promarz M. Tamrazov National Academy of Sciences, Ukraine
Gebze Institute of Technology
World Scientific NEW JERSEY • L O N D O N . SINGAPORE • B E I J I N G • SHANGHAI • HONGKONG • TAIPEI • C H E N N A I
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COMPLEX ANALYSIS AND POTENTIAL THEORY Proceedings of the Conference Satellite to ICM 2006 Copyright 0 2007 by World Scientific Publishing Co. Pte. Ltd All rights reserved. This book, orparts thereoJ niay 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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ISBN-13 978-981-270-598-3 ISBN-10 981-270-598-8
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V
PREFACE The satellite to ICM (2006) International Conference on Complex Analysis and Potential Theory was held at the Gebze Institute of Technology in Gebze, Turkey during the period 8-14 September, 2006. The Proceedings involves the most of presentations delivered at the Conference by the participants, among of which many of the top-notch mathematicians were present. Topics discussed include Grunsky inequalities and Moser's conjecture, speed of approximation to degenerate quasiconformal mappings, combinatorial Theorems of complex analysis, geometry of the general Beltrami equations, polyharmonic Dirichlet problem, isoperimetric inequalities for sums of reciprocal eigenvalues of the Laplacian, contour-solid theorems for finely meromorphic functions, residues on a Klein surface, functional analytic approach to the analysis of nonlinear boundary value problems, generalized quasiconformal mappings, properties of separately quasi-nearly subharmonic functions, implicit function theorem for Sobolev Mappings, The Martin boundary and the restricted mean value property for harmonic functions, approximate properties of the Bieberbach polynomials on the complex domains, asymptotic expansions of solutions of the heat equation with generalized functions initial data, Hausdorff operators, harmonic transfinite diameter and Chebyshev constants, analogues of the Mittag-Leffler and the Weierstrass theorems for harmonic differential forms on noncompact Riemannian manifolds, harmonic commutative Banach algebras and spatial potentials fields, parameter space of error functions and others. Besides, Prof. C. C. Yang proposed to include in the Proceedings a set of the articles devoted to so-called "open problems," i.e., the problems of great importance, but unsolved yet. This suggestion was approved by the scientific peers and accepted. In this connection, the Proceeding is composed of two Parts. The first one, Part A, involves the talks, which were presented and discussed at the Conference. The second one is devoted to the open problems completely. However, one can find some open problems in the first part as well, where they are given in passing with the main contents of the talks. The articles published in this Proceedings are oriented on the active researchers, who works in these areas directly and in the adjacent fields of mathematics and would like to update recent developments in the field, This book will be useful for the Ph.D. and M.S. students as well as researchers who just start only or continue their activity in this area of mathematics and its applications in engineering. We would like to thank all participants for their invaluable contributions. We acknowledge great efforts of our colleagues, Dr. Faik Mikailov, Tugba Akyel and
others who made major contribution t o the organization of the meeting. Special thanks t o Prof. Alinur Buyukaksoy, Rector of Gebze Institute of Technology, who supported every stages of preparation and holding of the meeting, and t o the Scientific and the Technical Council of Turkey (TUBITAK) for the support of this conference. Tahir Aliyev Azerojjlu Department of Mathematics, Gebze Institute of Technology, Gebze, 41410 Kocaeli, Turkey
Promarz M. Tamrazov Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 01601, Ukraine
vii
PARTICIPANTS Olli Martio, University of Helsinki, Finland Vladimir Mazya, Ohio State University, USA Samuel Krushkal, Bar-Ilan University, Ramat Gan, Israel Hakan Hedenmalm, Royal Institute of Technology, Stockholm, Sweden Promarz M. Tamrazov, Institute of Mathematics of NAS, Ukraine Cliung Chun Yang, HKUST, Hong Kong Vladimir Miklyukov, Volgograd State University, Russia Reiner Kuhnau, Martin-Luther Universitat Halle-Wittenberg, Germany Bodo Dittmar, Martin-Luther-Universitat Halle-Wittenberg, Germany Bogdan Bojarski, Polish Academy of Sciences, Warsaw, Poland Heinrich Begehr, I. Math. Inst., F U Berlin, Germany Tahir Aliyev Azeroglu, Gebze Institute of Technology, Turkey Sergei Favorov, Kharkov, Ukraine Tatyana Shaposhnikova, Linkoping, Sweden Lev Aizenberg, Bar-Ilan University, Ramat-Gan, Israel Akif Gadjiev, Institute of Mathematics and Mechanics, Baku, Azerbaijan Massimo Lanza De Cristoforis, Universita Degli Studi di Padova, Italy Arturo Fernandez Arias, UNED, Madrid, Spain Yurii Zelinskii, Institute of Mathematics of NAS, Ukraine Anatoly Golberg, Bar-Ilan University, Ramat-Gan, Israel A. V.Pokrovskii, Institute of Mathematics of NAS, Ukraine Juhani Riihentaus, University of Joensuu, Joensuu, Finland Sergiy Plaksa, Institute of Mathematics of NAS, Ukraine Matti Vuorinen, University of Turku, Finland Oleg F. Gerus, Zhytomyr, Ukraine Shunsuke Morosawa, Kochi University, Japan Victor V. Starkov, Petrozavodsk State University, Russian Allami Benyaiche, Universiti! Ibn Tofail, Kenitra, Morocco Igor V. Zhuravlev, Volgograd, Russia Elijah Liflyand, Bar-Ilan University, Ramat-Gan, Israel Mubariz T. Karayev, Suleyman Demirel University, Isparta, Turkey Kunio Yoshino, Sophia University, Tokyo, Japan Eugenia Malinnikova, Trondheim, Norway Daniyal M.Israfilov, Balikesir University, Turkey Aydin Aytuna, Sabanci University, Turkey
...
Vlll
Aydin Aytuna, Sabanci University, Turkey Ana,toliy Pogoruy, Zhytomyr, Ukraine Andrey L. Targonskii, Institute of Mathematics of NAS, Ukraine Yasuyuki O h , Sophia University, Tokyo, Japan Bulent N. Ornek, Gebze Institute of Technology, Turkey Shahram Rezapour, Azarbaidjan University of Tarbiat Moallem, Iran Hiroshige Shiga, Tokyo, Japan Vyacheslav Zakharyuta, Istanbul, Turkey H. Turgay Kaptanoglu, Ankara, Turkey Olena Karupu, Institute of Mathematics of NAS, Ukraine Mehmet Acikgoz, University of Gaziantep, Turkey Yu.V. Vasil’eva, Institute of Mathematics of NAS, Ukraine Coskun Yakar, Gebze Institute of Technology, Kocaeli, Turkey Allaberen Ashyralyev, Fatih University, Istanbul, Turkey Ali Sirma, Gebze Institute of Technology, Gebze, Kocaeli, Turkey Mehmet Kucukaslan, Mersin University, Turkey Peter Tien-Yu Chern, Kaohsiung, Taiwan
1x
CONTENTS
Preface
V
vii
Participants
Part A
TALKS
Strengthened Moser’s Conjecture and Finsler Geometry of Grunsky Coefficients S. Krushkal Decompositions of Meromorphic Functions Over Small Functions Fields C.-C. Yang and P. La Speed of Approximation to Degenerate Quasiconformal Mappings and Stability Problems V. M. Miklyukov
3
17
33
Grunsky Inequalities, Fredholm Eigenvalues, Reflection Coefficients R. Kuhnau
46
Sums of Reciprocal Eigenvalues B. Dittmar
54
Geometry of the General Beltrami Equations B. Bojarski
66
A Particular Polyharmonic Dirichlet Problem H. Begehr
84
Finely Meromorphic Functions in Contour-Solid Problems T. Aliyeu Azeroilu and P. M. Tamrazou
116
A Generalized Schwartz Lemma at the Boundary T. Aliyev Azeroilu and B. N . Ornek
125
X
Singular Perturbation Problems in Potential Theory and Applications M. Lanza de Gristoforis
131
Residues on a Klein Surface A . Ferna’ndez Arias and J . Pirez Alvarez
140
Combinatorial Theorems of Complex Analysis Yu. B.Zelinskii
145
Geometric Approach in the Theory of Generalized Quasiconformal Mappings 148
A . Golberg Separately Quasi-Nearly Subharmoriic Functions J. Riihentaus
156
Harmonic Commutative Banach Algebras and Spatial Potential Fields S. A . Plaksa
166
The Parameter Space of Error Functions of the Form a e-w2dw S. Morosawa
s;
174
On Potential Theory Associated to a Coupled PDE A. Benyaiche
178
An Implicit Function Theorem for Sobolev Mappings I. V. Zhuravlev
187
A Relation Among Ramanujan’s Integral Formula, Shannon’s Sampling Theorem and Plana’s Summation Formula K. Yoshino
191
Asymptotic Expansions of the Solutions to the Heat Equations with Generalized Functions Initial Value K. Yoshino and Y . Oka
198
On the Existence of Harmonic Differential Forms with Prescribed Singularities E. Malinnikova
207
Approximate Properties of the Bieberbach Polynomials on the Complex Domains
D.M. Israfilov
214
xi
Harmonic Transfinite Diameter and Chebyshev Constants N . Skiba and V. Zakharyuta
222
On Properties of Moduli of Smoothness of Conformal Mappings 0. W. Karupu
231
Strict Stability Criteria of Perturbed Systems with Respect to Unperturbed Systems in Terms of Initial Time Difference C. Yakar
239
Piecewise Continuous Riemann Boundary Value Problem on a Closed Jordan Rectifiable Curve Yu. V. Vasil’eva
249
A Note on the Modified Crank-Nicholson Difference Schemes for Schrodinger Equation A. Ashyralyev and A. Sirma
Part B
256
OPEN PROBLEMS
Some Old (Unsolved) and New Problems and Conjectures on Functional Equations of Entire and Meromorphic Functions C.-C. Yang
275
An Open Problem on the Bohr Radius L. Aizenberg
279
Open Problems on Hausdorff Operators E. Laflyand
280
Author Index
287
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PART A
TALKS
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3
STRENGTHENED MOSER’S CONJECTURE AND FINSLER GEOMETRY OF GRUNSKY COEFFICIENTS SAMUEL KRUSHKAL Department of Mathematics, Bar-Ilan University,52900 Ramat-GanJsrael and Department of Mathematical Sciences, University of Memphis, Memphis, T N 38152, USA T h e Grunsky and Teichmiiller norms ~ ( f and ) k(f) of a univalent2unction f in a finitely connected domain D 3 00 with quasiconformal extension t o @. are related by ~ ( f 5) k(f). In 1985, Jurgen Moser conjectured that any univalent function in the disk A* = ( 2 : 121 > 1) can be approximated locally uniformly by functions with ~ ( f 1) if and only if
w
m,n=l
5 1,
,,h% a,,z,x,I
are generated by
x = ( 2 , ) runs over the unit sphere S(12)of the Hilbert space 1’ with
w 11~1= 1 ~C lxnl 2 , 1
and the principal branch of logarithmic function is chosen (cf.Gr).The quantity N(f)
:= sup
{I
c 00
Jmn mnz,,z,l
: x = ( 2 , ) E S(Z’))
m,n=l
is called the Grunsky norm o f f . We denote by C the class of univalent holomorphic functions f ( z ) = z bo b1z-l .. . mapping A* into \ {0}, and by C ( k ) its subclass of f with kquasiconformal extensions to the unit disk = 1.( < l} so that f(0) = 0. Let CO = C(k). These functions are intrinsically connencted with the universal Teichmiiller space ‘IT modelled as a bounded domain in the Banach space IB of holomorphic functions in A* with norm llyll~= S U P ~ . ( [ Z-~ ~1)’1(z)1. All E IB can be regarded as the Schwarzian derivatives
+ +
+
Uk
Sf = (f”/f’)‘- (f”/f’)”2 of locally univalent holomorphic functions in A*. The points of whose minimal dilatation
T represent f
E Co
k ( f ) := inf{k(wfi) = 11p1: wfil8A* = f }
determines the Teichmuller metric on ‘IT. Here Ilpllw = esssupcIp(z)I. Grunsky’s theorem has been essentially strengthened for the functions with quasiconformal extensions, for which the Grunsky and Teichmuller norms of f € C are related as follows
4’) 5 k(f)
(3)
(seeKU1);on the other hand, by theorem of Pommerenke and Zhuravlev, any f E C with x(f) 5 k < I, has kl-quasiconformal extensions t o with Icl = k l ( k ) 2 k (seep0 ,Zh ;KK1 , pp. 82-84). An explicit not sharp bound k l ( k ) is given inKu4. We will discuss this problem in the last section. A point is that for a generic function f E Co, we have in (3) the strict inequality x(f) < k ( f ) (see e.g.KK2).On the other hand, the functions with ~ ( f =) k ( f ) are crucial in many applications of the Grunsky inequality technique. In 1985, Jurgen Moser conjectured that the set of functions with x(f) = k ( f ) is sparse in Co so that any function f E C is approximated by functions satisfying (1.8) uniformly on compact sets in A*. This conjecture was recently proved inKK2.
5
A related conjecture, posed inKK2and which remains open, states that f E Co satisfying (1.7) cannot be the limit functions of locally uniformly convergent sequences { f n } C Co with ~ ( f , ) = k ( f n ) . It is proved only for the sequences of maps f n which are asymptotically conformal on the unit circle S1= aA*. In applications of Schwarzian derivatives, especially to Teichmuller spaces, one has to use a much stronger topology than locally uniform convergence. A question is how sparse is the set of derivatives p = Sfin T representing the maps with equal Grunsky and Teichmuller norms. The answer is given by the following key theorem. Theorem 1. The set of points p = Sf,which represent the maps f E Co with x(f)< I c ( f ) , is open and dense in the space T. 1.2.
The first application of this theorem is to Fredholm eigenvalues theory. We consider the eigenvalues ,on of oriented quasiconformal Jordan curves L c (quasicircles) which in the case of smooth curves coincide with eigenvalues of double-layer potential over L . These values are intrinsically connected with Grunsky coefficients of the corresponding conformal maps, which is qualitatively expressed by Kuhnau-Schiffer’s theorem on reciprocity of x(f) to the least positive F’redholm eigenvalue p ~ This . value is defined for any quasicircle L c by
e
where G and G* are, respectively, the interior and exterior of L; D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on @. and harmonic on G u G* (cf.K”2 ,”). A basic ingredient for estimating p~ is the well-known Ahlfors inequality 1 - 5 qL (4) h
PL
where q L is the reflection coefficient of L , i.e., the minimal dilatation of quasiconformal reflections across L (cf.’ ,Kr4 , K u 2 ) . It suffices to take the images L = f”(S’) of the unit circle under quasiconformal self-maps of with Beltrami coefficients p = a,f/a,f supported in the unit disk . Then q~ equals the minimal dilatation k ( f ” ) = IIplI of such maps, and the inequality (4) is reduced inequality (3). Applying Theorem 1, one obtains Theorem 2 The set of quasiconformal curves L , for which Ahlfors’inequality (4) is fulfilled in the strict form l / p ~> q L , is open and dense in the strongest topology determined b y the norm of the space B. 2. Sketch of the proof of Theorem 1
The openness follows from continuity of both quantities ~ ( f and ) k ( f ) as functions of the Schwarzian derivatives Sf on T (cf.Sh) , and the main counterpart of the
6
proof concerns the density. The proof involves the density of Strebel points in T and relies on curvature properties of certain Finsler metrics on this space which are briefly presented below. 2.1.
Recall that if f o := f P o is an extremal representative of its class [ f o ] with dilatation k ( f o ) = Ilpolloo= inf(k(fb) : fPIS1 = folS'} = k , and if there exists in this class a quasiconformal map f l whose Beltrami coefficient pfl satisfies the strong inequality A, Ipfl (.)I < k in some annulus A, := ( 2 : r < IzI < l},then f l is called the frame map for the class [ f o ] and the corresponding point of the space T is called the Strebel point. We use the following important properties of Strebel's points adopted to our case. Proposition 1. (i) If a class [ f ]has a frame map, then the extremal map f o in this class is unique and either conformal or a Teichmuller map with Beltrami coeficient po = kI$oI/$o o n , defined b y an integrable holomorphic quadratic differential $ on A and a constant k E ( 0 ,1 ) 1 2 . (ii) The set of Strebel points is open and dense in T.GL We shall use the following construction exploited inGL . Suppose f o with Beltrami coefficient po is extremal in its class. Fix a number t between 0 and l , and take an increasing sequence ( r n } l with 0 < T, < 1 approaching 1. Put
and let f n be a quasiconformal map with Beltrami coefficient p,. Then, for sufficiently large n, f , is a frame map for its class, and the dilatation k , of the extremal map in the class of f, approaches ko = k ( f 0 ) . 2.2.
As well-known, the universal Teichmuller space T is the space of quasisymmetric homeomorphisms of the unit circle S1 = dA factorized by Mobius maps. The canonical complex Banach structure on T is defined by factorization of the ball of coefficients Belt(A)1 = { P E Loo() : plA* = 0, 11p11 < 11,
(6)
letting p, v E Belt(A)l be equivalent if the corresponding maps wp,w" E C" coincide on S 1 (hence, on F) and passing to Schwarzian derivatives S f , . The defining projection g 5 :~ p + S,, is a holomorphic map from &(A) t o B . The equivalence class of a map w k will be denoted by [wp].
7
It is generated by the Finsler structure on the tangent bundle T(T) = T x T defined by
lo
B of
B e W ) 1 ; v,v* E G o ( q . (8) The space T as a complex Banach manifold has also the contractible invariant metrics. The Kobayashi metric d~ on T is the largest pseudometric d on 11‘ contracted by holomorphic maps h : A + T so that for any two points $1, $2 E T,we have FT($T(P),dMPb)= inf{ llv*(l-lPlz)-l
: 4!Irb)v* = &(P)v; P E
5 i n f ( d ~ ( 0 , t :) h(0) =$I,
d~($1,+2)
h(t)=$2),
where dA is the hyperbolic Poincare metric on A of Gaussian curvature -4, with the differential form ds = X(z)Jdzl := Idzl/(l - 1
~1~).
(9) The Caratheodory metric c~ is the least pseudometric on T , which does not increase the holomorphic maps A + T. We shall use also the infinitesimal Kobayashi metric I c ~ ( + , v defined ) on T ( T ) . The following result is a strengthened version of the fundamental GardinerRoyden theorem for universal Teichmiiller space. Proposition 2.Kr4The differential metric Ic,(cp,v) on the tangent bundle T(T) of the universal Teichmuller space T is logarithmically plurisubharmonic in cp E T, equals the canonical Finsler structure F~ ( cpw) , on T(T) generating the Teichmuller metric of T and has constant holomorphic sectional curvature ~ ~ ( c p , w=) -4 on
T(T). The generalized Gaussian curvature nx of a upper semicontinuos Finsler metric d s = X(t)IdtI in a domain R c C is defined by A log X(t) nx(t) = X(t)2
’
where A is the generalized Laplacian
(provided that -co 5 X(t) < co), and the sectional holomorphic curvature of a Finsler metric on T the supremum of curvatures (10) over appropriate collections of holomorphic maps from the disk into T. Similar to C2 functions, for which A coincides with the usual Laplacian, one obtains that X is subharmonic on R if and only if A X ( t ) 2 0; hence, at the points to of local maximuma of X with X(t0) > -03, we have AX(t0) 5 0. For details see e.g.Di
,EKK >GL KO > 15,
Rol
>
.
8
2.3. Letting Al(6) = {$ E &(A)
:
+ }, A;
= {+ E Al(A) : $ = u 2 ,w }, and
we have the following key result. Proposition 3.14 ,Kr3 The equality (1.6) holds if and only if the function f is the restriction to ’ of a quasiconformal seu-map wfio of with Beltrami coeficient PO satisfying the condition sup I(PO,cp)A/= IlPOllm,
(11)
where the supremum is taken over holomorphic functions cp E AS(A) with IIPIIA~(A) = 1. If, an addition, the class [f]contains a frame map (is a Strebel point), then is of the form PO(Z)
= 11Po11m1~0(z)1/+0(~) with
$0
E
4 in A.
(12)
Geometrically the condition (11) means that the Caratheodory metric on the holomorphic extremal disk { $ q ( t ~ 0 / ~ ~ :p to E ~ ~A} ) in T coincides with the Teichmuller metric of this space. For analytic curves f(S1) the equality (12) was obtained by a different method inKu3
2.4.
The proof of Theorem 1 involves generic holomorphic disks in T and a new Finsler structure on determined by generalized Grunsky coefficients. The method of Grunsky inequalities is extended to bordered Riemann surfaces X with a finite number of the boundary components, in particular, to multiply connected domains on the complex plane though to a somewhat less extent (cf.G‘ ,Mi ,”). In the general case, the generating function must be replaced by the bilinear differential 03
- log f ( z ) - f ( C )
-Rx(z,C)=
z-I
P m n c p m ( z ) p n ( ) :X x X + C ,
(13)
m,n=l
where the surface kernel R,y(z,C) relates to conformal map j e ( z , I ) of X onto the sphere C slit along arcs of logarithmic spirals inclined at the angle 0 E [0,T ) to a ray issuing from the origin so that j o ( C , = 0 and j o ( z ) = z - z ~ + c o n s t + O ( l / ( z - z o ) ) as z = jkl(co). Here {pn}yis a canonical system of holomorphic functions on X such that (in a local parameter)
c)
cpn(.)
= a n J X n + Un+l,nZ
-n-I
+ ...
with
> 0, n = 1 , 2,...,
9
and cp; form a complete orthonormal system in H 2 ( X ) . We shall deal only with simply connected domains X 3 00 with quasiconformal boundaries. For any such domain, the kernel Rx vanishes identically on X x X , and the expansion (13) assumes the form 00
where f denotes a conformal map of X onto the disk * so that f(00) = 00, f ' ( 0 0 ) > 0, and a,, = P m n / f i are the normalized generalized Grunsky coefficients. These coefficients also depend holomorphically on Schwarzian derivatives Sf. A theorem of Milin extending the Grunsky univalence criterion to multiply connected domains X states that a holomorphic function f ( z ) = z const O(2-l) in a neighborhood of the infinite point z = 00 is continued to a univalent function in the whole domain X if and only if the coefficients Pmn in (2.10) satisfy the
+
03
inequality
C Pmn I m,n=l
IC,Z,~
5 1 1 1 ~ 1 1 for ~ any point
IC
=
(IC,)
+
E S(Z2) (seeMi).Ac00
cordingly, we have the generalized constant xx ( f ) = sup
I m,n=l ,Bmn zmxnI over
x = (2,) E S(12),which coincides with (2) for X = A*. 2.5.
Now the proof of Theorem 1 continues as follows. In view of continuity of both functions 2(Sf):= ~ ( f and ) z ( S f ) := I c ( f ) on T mentioned above, we need to establish only that each point cp* = Sp representing a function f * E Co is the limit point of a sequence { c p , = S,} c T with x ( f n )= k ( f n ) . We may assume that cp* is a Strebel point and IIcp*Ilo0 < 2. Its class [ f * ] contains a Teichmuller extremal map with Beltrami coefficient p* ( z ) = Ic* I$* (z)l/$*( z ) in A and p * ( z ) = 0 in A* determined by a holomorphic quadratic differential $* which has in the unit disk A only zeros of even order. We fix T E ( 0 , l ) and define a family of Beltrami coefficients pt = p ( . ,t ) depending on a complex parameter t (complexify (5)), letting
and p(z, t ) = 0 for IzI > 1. The admissible values o f t are those for which Ip(z,t)I 1. This inequality holds, provided t ranges over the disk
L: = {t' E:
It'fal
R ( a ) } with a = a ( k * ) = l/[l-(Ic*)27 > 1, R ( a ) = a(a-1).
(15) For t = 00 and t = 0, we have, respectively, , L L ~ = p* a.nd p o ( z ) = p * ( z ) if 1zI < T , pLg(z) = 0 otherwise. We must establish that if the disk (15) contains a
10
sequence { t n }going to infinity and such that lim lip(., tn) - p ( . , 0 0 ) l l ~= 0;
~ ( f f i ( ’ > ~ -= ) ) /c(fp(’,t”)),
n+m
n = I, 2 , . . .
,
then
x ( f f i t= ) k ( f f i t ) for all t E A*,.
(16)
Here ,ut is the extremal Beltrami coefficients in the classes [ f f i ( , , t ) ] , t €2,and the points cpt = q5T(,ut) run over a holomorhic disk in T,which we denote by R,. To establish (16), we construct on R, a Finsler metric of generalized Gaussian curvatures at most -4 and compare it with the Kobayashi metric. The underlying fact is that coefficients S,,(S,) generate for each z = (z,) E S(12)the holomorphic map 00
h x ( ~ )=
C
6h
Consider in the tangent bundle T(T) K:(v) = T x covering the disk q5,(A:) in T.Their points are pairs (cp, v), where v = &[cp]p E B is a tangent vector to T at the point cp, and /I runs over the ball
-
Belt(D,)l = { P E
L ( c ):
PID; = 0,
I I P I I ~< 11.
Here D and D sdenote the images of and A* under f = fp E Co with Sf = cp. To get the maps A -+ T preserving the origins, we transform the functions (17) by the chain rule for Beltrami coefficients w” = w‘(”) o ( f ” O ) - ’ , where
.(v)
0 f”0
=v
-
& f ”O
vol - vov-,
W
”
.
0
denote the composed maps by gz Using these maps, we pull-back the hyperbolic metric (9) onto the disks in 7 ( T ) covering R, and get on these disks the conformal subharmonic metrics ds = Agz[u,(”)](~)Idtl on Go, with
with curvature -4 at nonsingular points. Consider the upper envelope of these metrics A
h
A d t ) = SUP with supremum over all z E S(12)and all regularization
[u,]
oipE
( t ),
Belt(A)l.Its upper semicontinuous
h
A,@)
= lim sup A,@’) t’+t
descends to an upper semicontinuous metric on 0,. Similar tolKr4one derives the following basic properties of this metric.
B th
11
Lemma 1. (a) The metric A, is a logarithmically subharmonic Finsler metric o n 0,; (b) For small r = (t - a ( k ) ) / R ( k ) ,we have
+
= x(rcp*) o ( r )
A,(T)
as r
-+
0,
where .t(rcp*)denotes the Grunsky constant of the map f E Co with Sfla%= r p * ; ( c ) The generalized Gaussian curvature of A, satisfies k x , 5 -4. Equivalently, log A, 2 4A; or u, 2 4e2ux, where u , = log .A, 2.6. The next step consists of comparison of A, with restriction of the infinitesimal Kobayashi metric A, of T onto 0,. This restriction provides a logarithmically subharmonic metric of generalized Gaussian curvature -4. To this end, we define on T(T) a new Finsler structure FN((P0,W)
= sup{Id&(cpo; cP0)vl : z E S ( 1 2 ) ) ,
(18)
using the form
It is dominated by the canonical Finsler structure (8). The structure (18) allows us t o construct in a standard way on embedded holomorphic disks (A) the Finsler metrics A, ( t )= F,(g(t), g ' ( t ) ) and, accordingly, the corresponding distance d(cpl,cp2) = inf
1
A, ( t ) d s t ,
P
taking the infimum over C1 smooth curves ,8 : [0,1]+ T joining the points
cp1
and
P2.
Lemma 2. O n any extremal disk A ( p 0 ) = (qbT(tp0) : t E A}, we have the equality
Taking into account that the disk 0, touches the Teichmuller disks A ( p * ) and A ( p n ) at points cp* and cpn = q b ~ ( p ( . , t ~and ) ) that A does not depend on tangent unit vectors with one end at points of R,, one derives that A, relates t o Kobayashi metric A K l R , as follows A,(O) = *A, ( y * )= A,(()), A,(t,) =,,A, ( p n )= A,(t,) and A,(t) 5 A,(t) for t E R, \ ( 0 , tn}. Hence, by the maximum principle of MindaMi" , the metrics A and must coincide on the entire disk R,, which is equivalent t o the desired equality (16). In the same way, one obtains that x()= k(cp) in a ball B(cp*,)C T centered at p*. Moving this ball from the point p* along the segment [ - c p * , * ] , one derives that this equality must hold for all points of a ball centered at the origin 'p = 0. But the latter is impossible, since it contradicts the existence of points cp = Sf in a neighborhood of 0 a t which x(f)< k ( f ) . This contradiction proves the theorem.
12
2.7.
A similar approach allows to establish that the set of the Schwarzian derivatives p = Sf E representing the functions f E Co with ~ ( = f )k ( f ) is convex (cf.K‘6). 3. Inversion of Ahlfors and Grunsky inequalities 3.1.
The important problem on the sharp estimation of the dilatation k ( f ) by Grunsky norm of f , or equivalently, by Fredholm eigenvalue of f(S1)was first stated by Kuhnau in 1981 and remains open. There was an explicit bound kl ( k ) for dilatations of quasiconformal extensions of f E C with ~ ( f5 )k found in.Ku4It is given by
where X ( K ) = m a x w ( l ) , and the maximum is taken over K-quasiconformal selfmaps w of C with the fixed points -1,0, ca.The distortion function A. For small K there is a somewhat better estimate which is also not sharp. 3.2.
The following theorem solves the problem and has many other applications. Theorem 3.Kr5For f E Co we have the estimate
(and accordingly for Fredholm eigenvalues pf(si)), which is asymptotically sharp as
-+0. The equality holds for the map
with t = const E ( 0 , l ) . Note that the Beltrami coefficient of this map in the disk is p 3 ( z ) = tlzl/z. The proof of this theorem consists of several independent stages. It suffices to establish its assertion for f E Co having Teichmuller extremal quasiconformal extensions onto , i.e., with Beltrami coefficient of the form p f ( z ) = kIcp(z)l/cp(z), where k = const E ( 0 , l ) and ‘p is an integrable holomorphic function in , which again means that f is represented in by a Strebel point [f].Put p * ( z ) = p ( z ) / ~ ~ p ~ ~ m to have IIp*llm = 1. 3.3.
We first prove
13
Theorem 4. For every function f we have the sharp bound
Co with unique extremal extension
fPo
to A,
with
a ( f P o= )
SUP
I(&,P)Al
V'EA?,ll'fllA,=l
The proof of Theorem 4 is geometric and relies on properties of conformal met0 of negative integral curvature rics ds = X(z)ldzI on the disk A with X(z) bounded from above. The curvature is understanding in the supporting sense of Ahlfors or, more generally, in the potential sense of Royden (see e.gRo2).For such metrics, we have the following result which underlies getting the desired lower estimate. Lemma 3.Ro2If a circularly symmetric conformal metric X(lzl)ldzl in A has curvature at most -4 in the potential sense, then X(r) a ( 1 - a'?), where a = X(0). On the extremal disk A(&) = {q!q(tp*.c;): t E A} c 'IT the infinitesimal Kobayashi-Teichmuller metric AK of mathbbT is isometrically equivalent to hyperbolic metric (9) on A of curvature -4. The Grunsky coefficients o f f E Co allows us to construct the holomorphic maps
>
>
-
c Jmn 03
hx(t) := hx(Pt) =
amn(vt) xmxn :
A + a,
m,n=l
where pt = S f t p ; and x = (xn) E S(12).Then sup {Ihx(t)l : x E S(12)}= Pull-backing the hyperbolic metric to A(&) by applying these maps, we get conformal metrics
N(f t b ; ) .
A-,*(t) := h:(X) = lL;(t)1/(1
-
Ihx(t)12)
of Gaussian curvature -4 at noncritical points. Take their upper envelope(t) = sup{-,x(t) : x E S(L2)}and pass to upper semicontinuous regularization X,(t) = limsup X,(t'). t'+t
This yields a logarithmically subharmonic metric on A whose curvature in the supporting and in the potential sense both are less than or equal -4. Its circular mean
lzT
M[Ll(ltl)= ( 2 ~ ) ~X,(reis)dO ~ is a circularly symmetric metric with curvature also at most -4 in the potential sense. To calculate the value of M[X,](O),one can use the standard variational method t o the maps f p E Co and to their Grunsky coefficients, which yields
M[X](O)= X,(O)
= a(fPo)).
14
Thereafter, applying Lemma 3, we get
M[zI(r)2 a ( f o ) / [ l- 4 f o ) 2 r 2 1 and, integrating both sides of this inequality over a radial segment [0, Q] with
e=
IIPoll,
ie
M[,](r)dr 2 tanhY1[a(fbo))e] = tanh-'[a(fbo)k(feb~)]= tanh-'[a(fbo)k(fbo)].
On the other hand, by (19),
1"
X,(t)dt = tanh-l x,
N
= x(fb0).
Using these relations, one obtains the desired estimate ( 2 2 ) . To get ( 2 l ) , we have to estimate the quantity (6) from below. Applying Proposition 3 and Theorem 4, we can restrict ourselves by finding the minimal value of the functionals l b ( $ ) = l(p*,y)i on the set {'p E A:, ll'plll = l} for p* = l$l/$ defined by integrable holomorphic functions in A of the form
$ ( z ) = z m ( c O + c l z + ...), This minimum equals
m = l , 3 , 5 ,... .
and is attained on the map (21).
3.4. Geometric applications Theorem 3 has interesting geometric consequnces. The inequalities ( 3 ) and (20) result in
4 f )5 k ( f ) 5 3 4 f ) / ( 2 J Z ) ,
(23)
and similarly for reciprocals of Fredholm eigenvalues of quasicircles f ( 5 " ) . Since x ( f ) 5 c ~ ( 0 , S fand ) the universal Teichmuller space m a t h b b T is a homogeneous domain, one obtains from (26) the following inequalities estimating the behavior of invariant metrics on this space. Theorem 5 . For a n y t w o p o i n t s (PI, cp2 E T,t h e i r Caratheodory and Kobayashi distances are related by
3.5. Note that the equality in ( 2 1 ) is attained by map (21) only asymptotically as N while for small N , we have k ( f 3 , t )=
It1 = N* - O ( N * ~ n. Let P ( f ) , Q ( g )E P with coeficients being small functions of f and g , respectively, and P (f ) = a, f + a,-l f n p l + ... + alf + ao, a, Z0.
If
and if f and g satisfy the following functional equation:
then P ( f ) assumes one of the form described in Theorem A . Corollary 2. Let ao, a l , ..., a, be given meromorphic functions. Suppose that f is a transcendental meromorphic functions with ao, a l , ...,a, being its small functions, and P ( f ) E P is as defined in Corollary 1. If there exist a positive integer k > 2n and a nonconstant meromorphic function g satisfying the functional equation:
then
Remark. The condition k > 2n in Corollary 2 is necessary. For example, let h be the elliptic function which satisfy (h')2= 4h3 + 6h2 4h + 1, and let g = 1 l / h , f = h'/h2,then g4 = f 2 - 1. Also, Corollary 2 shows that the functional equation (6) has no admissible transcendental meromorphic solution when k > 2n 2 4, a,-l E 0 and aoan # 0. Here and in the sequel, we call a meromorphic function f an admissible solution of a differential or functional equation if the coefficients of the equation are small functions off. Corollary 3. Let f , g , P ( f ) , Q ( g )be as defined in Corollary 1. Suppose that k and n are two relatively prime positive integers and k > 4, k # 6 , 2 5 n < k . Then the
+
+
20
functional equation (5) has no admissible transcendental meromorphic solutions f and g provided that an-1 2 0 and U O U , 2 0. Remark. The assumption that k > 4 and k # 6 are necessary. In fact, let hl and h2 are solutions of the equations (h:)’ = hl(hT 3) and (hh)2 = h$ 5, respectively. Let f1 = hf 2, g1 = h‘, and f 2 = h$ + 3, g2 = h2hh. Then we have gf = (fi - 2)(fl + 1)’ and 92 = ( f 2 - 3)’(fz + 2)3 Suppose that f is a transcendental meromorphic function with al,..., a, being distinct small functions of f. Then by the second fundamental theorem related to n small functions (see, [ 6 ] ) the , inequality
+
+
holds for any positive number
E.
+
Let
Theorem 2 shows that (9) is still true if P ( f ) is a polynomial in f of degree 3, and P ( f ) has no factor of the form (f where (Y is a small function of
f. It is natural to ask: whether or not the inequality (9) remains to be valid for arbitrary polynomials which can not be decomposed into form of (8)? We just realized that the question can be resolved, according to a very recent paper of Yamanoi (Corollary 6.1 in [7]). Also we would like to point out that both the methods used in the present paper and [7] are pertinent to discuss the existence or non-existence of admissible solutions f , g for functional equations of the form: P(f)= Q ( g ) ,with P(f),Q ( g ) E P of higher degree, see [3]. Furthermore, it appears that both methods may not be useful to deal the similar question with P and Q of lower degree. For further investigations, we propose such a simple looking conjecture as follows: Conjecture. Suppose that a E C and b ( z ) a nonconstant meromorphic funcb ( z )f a has no admissible merotion. Then the functional equation g2 = f morphic solutions f and g.
+
+
2. Some Lemmas
Lemma 1 (Clunie’s lemma, see [l]).Suppose that f ( z ) is meromorphic and transcendental in the complex plane C,and
21
where P ( f ) and Q ( f ) are differential polynomials in f with the coefficients being small functions of f, and the degree of Q ( f ) is at most n. Then
m ( r ,P ( f1) = S(T,f
1
Lemma 2. (Corollary 5 in [8]). L e t f be a transcendental m e r o m o r p h i c f u n c t i o n , and n a positive integer. L e t Q l ( f ) and Q 2 ( f ) be t w o differential polynomials in f
n o t vanish identically. Suppose t h a t F i s a differential polynomials in f defined by
F := f n Q i ( f )+ Q 2 ( f ) If F
0, then
rQz
are t h e degree and weight of Q 2 , respectively. where Y Q ~and Lemma 3. ([4]). Suppose t h a t f i s a n o n c o n s t a n t m e r o m o r p h i c f u n c t i o n . If ao(2)f".) R ( f )= bo(z)f q ( 2 )
+ + ... +ap(.) + b l ( z )f q - - ' ( z ) + ... + b q ( z ) U l ( 2 )f " - ' ( 2 )
i s a n irreducible rational polynomial in f w i t h c o e f i c i e n t s being small f u n c t i o n s of f , and ao(z)bo(z)2 0, then
T ( r ,R ( f1) = m 4 P , q)T(r,f ) + S(7-1f
1.
Lemma 4. L e t f be a transcendental m e r o m o r p h i c f u n c t i o n , and
P = a, f2"
+ U,-l
p - 2
+ ... + a1 f 2 + ao,
where ao, ...,a, are s m a l l f u n c t i o n s of f , and a,
# 0. T h e n either
+ bn-2gnP2 + ... + b i g + bo, where bo, ...,bn-2 are polynomials in a0 ,...,a,. If bn-2gn-2 + ... + b l g + bo G 0 then ([ll])holds. If b,_2gnp2 + ... + b l g + bo 2 0 , then by Lemma 2 , we have P = angn
22
which implies (10). Lemma 5 . Let f be a transcendental meromorphic function, and P =a n y where
a2,
+
fn-1
U,-l
+ ... + a 2 f 2 + 1;
...,a, ( n > 3) are small functions of f , and a,
0. T h e n either
or P satisfies one of the following two decompositions: (i) There exist a small function QO of f and a positive integer p = n/2 such that
P = a,(f2
(ii) There exist a small function X I , XZ such that n = p1 p2, Alp2
+
P = a,( f Proof. If
a2
=0
then P = f 3
3T(r,f
1
1.
Definition 1.1. A homeomorphic W12z m a p f : D c R2 + I%' is called quasiconformal with M. A. Lavrentiev characteristics (p(x),0(x)), if it transformes a.e. o n D the infinitesimal ellipses with characteristics (p(x),0(x)) onto infinitesimal circles '. A mapping f : D + Rz is called q-quasiconformal, if ess sup p(x) 5 q. XED
Fix a number sequence
Qn 2 1 for all
{Qn}F=3=1 such that n = 1 , 2 , . . . and Qn
+ co
as n
+ 00.
(4)
"This means that for a.e. x E D the differential d f ( x ) : R2 --t R2 has characteristics (p(x),O(x)).
35
For n = 1 , 2 , . . ., we set
By well-known results of the theory of quasiconformal mappings, there exists a quasiconformal map E = w,(x) of D onto an appropriate domain A, c R2 with the characteristics equal to ( p n , 0) a.e. on D . This domain A, either is R2 if D = R2 or it is a simply connected proper subdomain of R2. We denote by B ( a , r )the disc of radius T > 0 with center at a E R2 and by S ( a ,r ) its boundary. Fix two points ao, a1 E D . By auxiliary conformal transformations of the plane of variables = ( E l , &), we attain that the domains A, are the discs B, = B(0,R,), where 1 < R, < 00, and the maps w, satisfy
0) (3)
The starting point of the theory was the famous
Theorem 0 [Gr], [PI. For every n and all fixed z1, ,x, the exact domain of variability of the functional Gf is the closed unit disk IGfI 5 1
'
(4)
47
This theorem solves the extremal problem IGfI
+ max if we fix the constants
x k and vary the mappings f ( z ) .
It arises the following complementary question: What is the solution of the extremal problem lGfl + max if we conversely fix the mapping f ( z ) E C and vary the systems xk? Surprisingly, the maximal value for 1GfI is not always again 1. We have rnax,, IGfI < 1 for fixed f if and only if the image C of IzI = 1 is a quasicircle [Kul], [PI (Theorem 9.12). More precisely, we have 1 max (Gfl = - , Zk X
(5)
where X = X c 2 1 denotes the Fredholm eigenvalue of C ; cf. [Ku2,3], [S], [Kr3,4]. The Fredholm eigenvalue X is defined (cf., e.g.,[KuG]) as the greatest constant 2 1 for which the inequality
is satisfied with the Dirichlet integral D [...I for all pairs h l , hz (both not constant), where hl is (real and) harmonic in the interior of C , hZ (real and) harmonic in the exterior (including C = co),with continuity and hl = h2 at C. Because of the invariance of Dirichlet integrals under conformal mappings we have after a Mobius transformation of C the same Fredholm eigenvalue. This invariance property has also the so-called reflection coefficient Q c of C. This means the smallest dilatation bound 2 1 in the class of all quasiconformal reflections at C. Basic is the Ahlfors inequality
For the question of equality here cf. [Kr2,3,4], [Ku5]. These cases with equality in (7) are in some sense sparse [KK2]. We have X c = co or QC = 1 only in the trivial case of a circle C . Sometimes, we also write Qf and q f instead of QC and q c . Now we study here also the following subclasses of C (cf. [Ku3,5]).
Definition. Let C ( K ) be the class of those mappings of C which have a schlicht and continuous extension t o the whole plane which is Q = e - q u a s i c o n f o r m a l f o r IzI < 1. - Let C ( K ) be the class of those mappings of C f o r which the image oflzl = 1 is a quasicircle C whose Fredholm eigenvalue Xc satisfies I K.
&
Because of (7) it holds C ( K ) c C ( K ) ;cf. [ K u ~ ]Up . to now a simple criterion (beside this definition) for functions of C to belong to the class C ( K )does not exist. Contrary to this, we have with the Grunsky functional in form of (5) a simple criterion for functions of the class C ( K ) . For every fixed system x k the extremal problem lGfl + max in the class C ( K ) 5 5, [KKl], p.111) in form of the
was solved in [Ku l ](cf. also [Krl],Chapter 4,
48
following Theorem. Theorem 1. For every fixed system 21, . . . ,xk the exact domain of variability of the functional Gf in the class C ( K )is the closed disk defined b y
IGfI I K . (8) To every boundary point corresponds exactly one extremal function. For example, the extremal function w = f ( 2 ) with Gf = K satisfies
Note that we obtain e z i s G f instead of Gf if we replace all x k by ei6xk. Therefore we can restrict ourself in the discussion of equality in (8) to the point with Gf = K . We remark here further that, contrary to the class C ( K ) ,in the classical Grunsky case for the class C (cf. Theorem 0), corresponding to the limit case K + 1, to a boundary point of the domain of variability (4) can exist more than one extremal function. This fact was already mentioned by H. Grunsky itself [Gr] (last page) but is today not well-known or mostly forgotten or treated stepmotherly; cf. also remarks in [PI (p.61), [KKl] (end of p.109 and p.108/109). In the literature there is missing a complete discussion of those parameter systems xk for which the corresponding extremal function in the class C is unique. Contrary to the class C ( K ) ,there is hitherto only a small number of extremal problems which are completely solved in the class C ( K ) .The reason is that a variational formula does not exist for this class. What concerns the extremal problem lGfl + max in the class C ( K )we have of course by (5) for every fixed system xk the estimate lGfl 5 K . Our aim is now to add here that it is impossible f o r every fixed system xk to improve this estimate. Observe the subtle distinction to the assertion
Theorem 2. For every fixed system 2 1 , . . . , xk the exact domain of variability of the functional Gf in the class C ( K ) is again the closed disk defined b y
IGfl
I: 6 .
(10)
Again equality holds f o r the mapping (9) and its modifications by replacing all xk by ei6xk. The image of IzI = 1 under these mappings is always a closed analytic Jordan curve C with = qc = K .
&
It remains here as an open question: Are there further extremal mappings ? The proof of Theorem 2 uses an analysis of the mappings (9). Theorem 2 also immediately yields some new examples of quasicircles C for which we can give the
49
exact value of the reflection coefficient and of the Fredholm eigenvalue; cf. older collections of such examples, e.g. in [Kr3,4], [Ku4,6,7], [W]. Finally, in Section 4 we add the analogous theorem for Golusin’s functional. (What concerns the question of uniqueness of the extremal functions there are analogous remarks possible, as in the case of Grunsky’s functional; cf. also [KKl], p. 108/109.) 2. Proof of Theorem 2
First we observe that indeed, by (5), for every system xk the inequality (10) is true. Then we remark that, by Theorem 1, every boundary point of the disk (10) is attained by the mappings (9) and the mentioned modifications. These mappings are not only contained in C ( K )but also in C ( K )because of (7). It remains t o show that the mappings (9) transform the unit circle IzI = 1 onto a closed analytic Jordan curve. Namely, because the corresponding quadratic differential
has at the inside of C only zeroes of even order it would follow by [Ku5] (cf. also [Kru3,4]) that we have for these mappings ~f = q f , which will end the proof. With the abbreviation
the equation (9) becomes the form cp(.)
- Kcp(l/Z)
for
I4 2 1,
@(w) =
( 12)
cp(z) - ~ c p ( z ) for IzI
5
1.
(By the way, icp(l/Z) corresponds to the complex eigenfunctions of C; cf. [ K u ~ ] , Satz 1.) In (12) the left-hand side @(w) is a polynomial of degree n because we can assume x, # 0. As the image of the whole w-plane appears a Riemann surface R with n sheets. Therefore this R also appears as image of the whole z-plane by the right-hand side of (12). Now we show that for every given zo with lzol = 1 the mapping (9) transforms all sufficiently small arcs of IzI = 1 with center at zo onto an analytic arc. For this purpose we start with the development of p(z) in the neighborhood of zo :
cp(z) = a0
+ a,(z
-
~
0
+ .. ., )
~a,
#0,
50
with some rn 2 1. We denote here and in the following by . . . higher powers in the corresponding power series. This yields
@(w) =
(a0
- KZ)
+ [(a,
[a0
+ a,(z
-
i
-
~ ~ ( - - z : ) - ~-] z( z ~ + .). . ~ for ~ IzI
+ .. .]
zg),
-
~ [ a+ o a,(z
- ZO),
+ . . . ] for
2 1,
IzI 5 1 . (13)
Here we have
la,
-
K.a,(-zt)--ml
2 laml
J G l=
-K
(1 - K)IU,[
> 0,
therefore a , - K G ( - Z :#) -0.~This means that, roughly speaking, the half of the neighborhood of zo in IzI 2 1 transforms onto one half of the m-sheeted neighborhood of a0 - KZ. The same holds for the other half in IzI 5 1. We therefore obtain, altogether, a complete m-sheeted neighborhood of a0 - KK,thus also by the mapping @(w).This means an equation of the form, e.g. for 1x1 2 1, (a0- K Z ) +b,(w
with some b,
- wo)m(l+ . . . ) = (a0 - K
# 0. It follows with
some C
+(a, -
Z )
Kia,(Z;)y)(Z
w
-
+ . . . ) = C1’,(z
wo = C l / ” ( Z
-
+. . .) (14)
#0
(w - wo)*(1+ . . . ) = C ( z - zo),(l
(w - wo)(l
- zo)m(l
-
zo)(l
+ . . .),
+ . . .),
z o ) ( l + . . .).
This means, as desired, that this mapping is indeed in a sufficiently small neighborhood of zo analytic with a derivative # 0. Therefore, we have indeed for the mappings (9) the desired equality ~f = 4s. This means that all boundary points of the disk IGf I 5 K are attained by mappings of the class E(K). It is then trivial that also all interior points of this disk are attained. To see this we have only to consider the mappings (9) after replacing K by a smaller value. (This idea is called in another context “Grotzsch’s argument”.) 3. Examples of mappings (9)
The representation (9) of the extremal functions contains unknown parameters, namely some coefficients b k . These are hidden in the Faber polynomials. (This phenomenon of such unknown “accessory” parameters is a circumstance which often can be observed in Geometric Function Theory; cf., for example, also the SchwarzChristoffel formulas, or the representation of extremal functions obtained by the variational method. For “general reasons”, in our case by Theorem 1, it is obvious that there is always a solution for these accessory parameters.) In our case, for a given fixed set x k , the determination of these accessory parameters has to use the fact that we must obtain (if x, # 0) the same n-sheeted Riemann surface R (mentioned also in the proof of Theorem 2), therefore also the
51
same ramification points, by both sides of (9) '. Of course, a concrete discussion is practicable only in some simple cases because in (9) polynomials are involved. We will give here such examples up to the concrete determination of the corresponding quasicircles C , with = qc = K . As usual, the corresponding extremal quasiconformal (= moglichst konforme) reflection at C is contained in (9); cf. [ K u ~ ] . (i) The simplest case n = 1 leaves us with the well-known mapping -
w = {
Ke-2iaL
for IzI
2 1,
z for IzI
< 1.
- Ke-2ia-
(15)
We obtain for every real u: an ellipse C with semi-axis 1 + K and 1 - K , and with 1 - 4c = K . (ii) In the case n = 2 we can now assume 1 ~ =1 1. We will restrict ourself to the case 1x1 I 1. (The remaining case 1x1I > 1 needs a more complicate discussion.) This means that
-Bo j=1
(49)
a? - 47r 3
and equality holds for the annulus. It follows from the general procedure above Theorem 17.(Dittmar, Hantke 2006) For the eigenvalues of problem (38) in R it holds for any n
Equality holds if R is an annulus. Example Let the domain D be the unit disk U1 furnished with a straight slit from xo,0 < 2 , < 1, to 1. This simple connected domain is closely related t o the Grotzsch ring domain. Let C1 be the unit circle (without z = 1) and C2 the straight slit. Ing the eigenfunctions has been constructed and a formula for the eigenvalues was given. We get
2
= -r2
3
17 + 82, + -2; + O(x:). 2
References 1. C. Bandle, Isoperimetric Inequalities and Applications. Pitman Publ., London 1980. 2. B. Dittmar, Isoperimetric inequalities f o r the s u m s of reciprocal eigenvalues. In: Progress in partial differential equations. Pont-8-Mousson 1997. Volume 1 (eds. H. Amann, C. Bandle, M. Chipot, F. Conrad and I. Shafrir), Pitman Research Notes in Mathematics Series 383,78 - 87, Addison Wesley Longman Ltd., Harlow, Essex 1998. 3. B. Dittmar, Sums of Reciprocal Eigenvalues of the Laplacian. Math. Nachr. 237 (2002), 45 - 61. 4. B. Dittmar, S u m s of Reciprocal Stekloff Eigenvalues. Math. Nachr. 268 (2004(, 44 49. 5. B. Dittmar, S u m s of free membrane eigenvalues. J. d’Anal. Math. 95 (2005), 323 332. 6. B. Dittmar, S u m s of free membrane eigenvalues II. (to appear) 7. B. Dittmar, Eigenvalue problems and conformal mapping.In: Handbook of Complex Analysis: Geometric Function Theory, Vol. 2,(ed. R. Kiihnau) Amsterdam etc., Elsevier 2004.
65
8. B. Dittmar, M. Hantke,Robin function and eigenwalue problems. Preprint MartinLuther-Universitat Halle-Wittenberg 2006. 9. B. Dittmar, R. Kuhnau, Zur Konstruktion der Eigenfunktionen Stekloffscher Eigenwertaufgaben. Z. angew. Math. Phys. 51 (ZOOO), 806 - 819. 10. P. Duren, Robin capacity. In: Proceedings of the Third CMFT Conference (eds. N. Papamichael et al.) Singapore, New Jersey, Hong Kong: World Scientific 1997, 177190. 11. M. Hantke, S u m m e n reziproker Eigenwerte. Dissertation Martin-Luther-Universitat Halle-Wittenberg 2006. 12. A. Henrot, Extremum problems f o r eigenvalues of elliptic operators. Birkhauser Verlag 2006. 13. J. Hersch, L. E. Payne and M. M. Schiffer, S o m e inequalities f o r Stekloff eigenvalues. Arch. Rat. Mech. Anal. 57 (1974), 99 - 114. 14. P. Kroger, Upper bounds f o r the N e u m a n n eigenvalues o n a bounded domain in euclidean space. J. F‘unct. Anal. 106 (1992), 353 - 357. 15. P. Kroger, O n upper bounds f o r high order N e u m a n n eigenwalues of conwex domains in euclidean space. Proc. Amer. Math. SOC.127(6) (1999), 1665 - 1669. 16. R. S. Laugesen, Eigenwalues of Laplacians with mixed boundary conditions, under conformal mapping. Ill. J. Math. 42 (1998), 19-39. 17. R. S. Laugesen, Eigenvalues of the Laplacian o n inhomogeneous membranes. Amer. J. Math. 120 (1998), 305-344. 18. R. S. Laugesen, C. Marpurgo, Extremals f o r eigenvalues of Laplacians under conformal mapping. J. of Functional Analysis 155 (1998), 64-108. 19. U. Lumiste, J. Peetre, Edgar Krahn 1894-1961 a Centenary Volume, 10s Press Amsterdam, Oxford, Washington DC, Tokyo 1994. 20. N. Nadirashvili, Conformal maps and isoperimetric inequalities f o r eigenwalues of the N e u m a n n problem. Israel Mathematical Conference Proceedings 11 (1997), 197-201. 21. G. Pblya, Patterns of Plausible Inference. Princeton, University Press, Princeton New Jersey 1954. 22. G. P6lya, O n the characteristic frequencies of a symmetric membrane. Math. Zeitschr. 63 (1955), 331 - 337. 23. G. P d y a , O n the eigenwalues of vibrating membranes. Proc. London Math. SOC.11 (1961), 419 - 433. 24. G. P6lya and G. Szego, Isoperimetric Inequalities in Mathematical Physics. Princeton University Press 1951. 25. G. P6lya and M. Schiffer, Convexity of functionals by transplantation. J. d’Anal. Math. 3 (1954), 245-345. 26. G. Szego, Inequalities f o r certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3 (1954), 343 - 356. 27. H. F. Weinberger, An isoperimetric inequality f o r the N-dimensional free membrane problem. J. Rat. Mech. Anal. 5 (1956), 633-636. 28. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mat einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71 (1912), 441-479.
66
GEOMETRY OF THE GENERAL BELTRAMI EQUATIONS B. BOJARSKI Warszawa
Although the quasiconformal mappings theory-QC-theory for shortoriginated in the context of geometric methods of pure complex analysis and holomorphic 2-d mappings ([28], [as],[32]) from the beginning it was connected with elliptic partial differential equations. These are the Beltrami equation (0.1) and the general Beltrami equation (0.2) which we write in the complex form wz - p(z)w, = 0
(0.1)
and
p(z)ww,- vzz= 0
w y -
(0.2)
where 20,
= z(w, 1
-
iw,),
wz = % 1 (W,
+ iw,).
Systems (0.1) and (0.2) are elliptic (at the point z ) if
1&)1
< 1 or IL4z)I + I).(I
< 1.
The ellipticity is uniform in a subdomain D of the complex plane constant lc
I&)I
+I).(.I
(0.3)
C if for some
I< 1
(0.4)
for all z E D (v = 0 for the equation (0.1)). For convenience of the reader not used to the complex operators and we recall that the Beltrami equations (0.1) and (0.2) have the real form, for w = u f i v ,
&
+ pu, + au + bv + e, = yu, + 6u, + cu + dv + f
vy = QU,:
-v,
w i t h a = b = c = d = e = f =O. The ellipticity condition takes the form
System (0.5) is uniformly elliptic if
&
67
and the coefficients a , p, y, S in (0.5) are uniformly bounded. The discussed theory is also directly connected with second order linear equations for one unknown function 4(x,9 ) :
A4,,
+ 2B4,, + C&y + E4, + F 4 , + G4 = H ,
A > 0,
(0.8)
and the non-symmetric divergent form equations
a
a
-
The proper Beltrami moreover
a6
+ Pu,) + -(Tux 8Y equation (0.1) (v = 0) -
+ Su,) = 0. is obtained if
p
y in (0.5) and
p2 = 1 ( a ,p, y uniformly bounded).
The general real system (0.5) can be written in the complex form WZ
-~
w
Z -
uEiZ = AW
+ BTii + C
(0.10)
which is a linear differential equation over the field of reals. Locally homeomorphic “solution” of the equation ( O . l ) , for some p satisfying the inequality (0.4), is by definition a quasiconformal mapping, QC-mapping for short or, more precisely, K-QC-mapping, with K defined by
kz- K - 1
(0.11) K+1‘ K is then called the dilatation of the QC-map and the coefficient p in the Beltrami equation (0.1) satisfied by the mapping function W ( Z ) is traditionally termed the complex dilatation of the mapping. Also coefficients ( p , v ) in the equation (0.2) will be called complex dilatationscomplex characteristics-of the mapping. So far we used the term “solutions” of (0.1) and (0.2) in a rather loose way. To be more precise we state now that in the following we restrict our considerations to equations (0.1) and (0.2) with measurable coefficients p , v compactly supported on the complex plane C.This means that when considered on the whole complex plane C the equations (0.1) and (0.2) reduce to the Cauchy-Riemann equations near 00 and the considered solutions are holomorphic functions of z for / z ( big enough. “Solutions” of (0.1) and (0.2) are understood then in the generalized sense of Sobolev classes Wlyz(C); thus a priori discontinuous solutions are admitted. It is a fundamental fact, first established in 1954 and published in [13], that, under the uniform ellipticity assumptions (0.4), the generalized or weak solutions of Beltrami equations (0.1) and (0.2) are actually in W,::, for some p > 2 , depending on k only, and thus, by Sobolev imbedding theorem, continuous, more precisely, even Holder continuous with exponent a = 1 - 2P . We recall the formula J-W(.)
=
l W z l 2 - IWZI
2
68
for the Jacobian of the mapping (0.12)
w = w(z)
usually considered on an open subset D of the complex plane @. In the following we shall restrict our considerations to the orientation or sense preserving mappings meaning that at almost every differentiability point of the function w = w(z) in the domain D the Jacobian J w ( z ) i s positive
> O or
Jw(z)
/w,I
> O a.e. in D.
(0.13)
1. Generating the Beltrami equations
Let us remark now that the elliptic Beltrami equations (0.1) and (0.2) naturally arise in the context of general theory of sense preserving transformations. Indeed, if the mapping w = w(z) satisfies J w ( z ) > 0 a.e. then the differential identity
is meaningful a.e. with some measurable function p = pw satisfying the ellipticity condition (0.3) and produces the Beltrami equation (0.1) for w = w(z) such that J-w = jw,12(1-
Ip12)
> 0.
Assume now that we have a pair w = w(z) and w = w(z) of sense preserving mappings, admitting the generalized derivatives w, wZ,w,,w, a.e., satisfying moreover the condition
A = 2 i 1 m ( w z ~ , )# 0 a.e.
(1.2)
Again the differential identity
with p and u defined by the formulas
implies the differential equation (0.2) for the functions w = w(z) and w = w(z). Relations (1.3) and (1.4) can be considered as defining equations of the Beltrami equation (0.2). The natural question arises: what are the conditions on the pair w(z), w(z) assuring the ellipticity of the system (1.3)? The following propositions give the answer.
Proposition 1. The complex equation (1.4) is uniformly elliptic (k-elliptic) i f and only if the linear family w%P =
aw
+ Pv,
a , /3 real,
(1.5)
69
e.
is a family of K-QC-mappings, k = (1.4) is equivalent with the inequality
I n other words, the k-ellipticity of system
I WF’P z I -< k I w y I o n a subset of full measure independent of
cy,
(1.6)
D.
For the proof see Lemma 12.1 in [27]. Analogously holds
Proposition 2. The equation (1.4) is elliptic (I&)\ pings welo are orientation preserving ( i e . lwglPI
1. 2. Principal homeomorphisms’ of the Beltrami equations These are global, i.e. defined on the whole complex plane C,solutions of the equations (0.1) or (0.2) with compactly supported coefficients p and v p
= v =0
for IzI
> R,
(2.1)
properly normalized at 00. Usually we shall consider the case
-
at z
~ ( z )a z
+ 00
(2.2)
with some complex constant a # 0. Strictly following the analytical ideas and formulas developed in [13], [43], [44], we represent the considered homeomorphic solutions of (0.2) or the K-quasiconformal mappings, in the form
(2.3) for some complex constant a and w E LP(@).Here dot is the Lebesgue measure on @ or 1 dot = --d t A dE.
2i
‘I. Vekua 1441 calls them “basic” homeomorphisms (a= 1).
70
The density w ( t ) in (2.3) is assumed to be in Lroc,for some p > 1. In general this means that we are considering the solutions in the Sobolev spaces W;:(@). Then (2.3) may be differentiated and gives the formulas
with
Here Sw may be understood as principal value singular integral operator in the plane. It is classically meaningful for Holder continuous densities w ,but makes sense also for w E L P ( @ ) for all p > 1, as a bounded operator S : LP -+ LP
llswll~p5 A,. Moreover for p
=2
IIwIILP.
(2.6)
Sw is an isometry IISwJIL2= IIwIIp,A2 k.Ap 0) mapping of the complex plane. In fact the following proposition of fundamental importance for the theory of two-dimensional quasiconformal mappings (QC-maps) and the theory of elliptic equations in the complex plane holds.
Proposition 3. The formulas (2.3) with the density w satisfying the integral equation (2.8) define a K-quasiconformal homeomorphism of the complex plane. It is a generalized solution of the Beltrami equation (0.1) of the Sobolev class W,',b(C). 2 The mapping w = W ( Z ) is differentiable a.e. and its Jacobian J w ( z ) = 1 2 ~ -~ /wzl 1 ~ is positive a.e. Moreover, the LP n o r m of w has a n estimate 5 C with the constant C depending on k , p and the LP n o r m of h in (2.8) only.
1 1 ~ 1 1 ~ ~
The inverse mapping z = z(w)also may be represented by the integral formulas of the type (2.3) in the complex plane of variable w and satisfies the corresponding Beltrami equations with coefficients ,G, i; satisfying inequalities (0.4) with the same k .
71
The coefficients jl., 5-the complex dilatations of the inverse map-in the complex plane w are expressed by simple formulas which may be found in [15] and [44] and in numerous references afterwards. For the most important simplest case of the classical Beltrami equation ( O . l ) , i.e., when the coefficient u z 0,
wz - p(z)wz = 0,
(2.9)
the equation (2.8) reduces to
w-pSw=h
(2.10)
which is linear over the complex field @. The equation (2.8) has to be considered over the real field R only. However in both cases the ellipticity conditions (0.4) ensure the unique solvability in the form of convergent Neumann series of the nonhomogeneous equations (2.8) and (2.10). All these facts have been discussed and used in [13], 1141, [15] and are fundamental in the literature on quasiconformal mappings and planar elliptic partial differential equations. We stress explicitly that the remarkable geometric properties of the principal solutions like infinitesimal quasiconformality, positive Jacobian and global (homeomorphism) properties of the mappings (2.3) are the result of the interplay of the form of (2.3), the asymptotics (2.2) and the equations (2.8), (0.2). If the integral equation (2.8) is removed, the representation (2.3) holds for any solutions of the non-homogeneous Cauchy-Riemann equations (2.11)
and the asymptotics (2.2) and these do not imply none of the mentioned geometric properties of ~ ( z whatsoever. ) The representation (2.3) and the formulas (2.4)(2.8) in the context of Sobolev spaces Wk:(C), p > 2, and Proposition 3 with the numerous other analytical and geometrical consequences first appeared in the papers of I. N. Vekua and the author 1131, [14], [43]. L. Ahlfors in the paper [a], published somewhat later in 1955, independently came to the formulas (2.3)-(2.5) and (2.8) (for the classical complex Beltrami equation (O.l), with u = 0), but, since he was working only in L2 or Wk:, he finally had to stop the construction of his global homeomorphisms of the form (2.3) at the level of Holder classes H a , p, w E H a ( @ ) , 0 < a < 1, w E C1+a,though, actually almost one year earlier the Bojarski-Vekua theory as described above was available. In the following years the formulas of type (2.3)-(2.8) and their numerous consequences have been often used in the immense and permanently growing literature on topics of QC-maps, univalent functions with QC-extensions, the study of dependence on parameters of families of QC-mappings and conformal structures, Teichmiiller spaces, complex dynamics, harmonic analysis etc. and somehow dispersed in kind of an unidentified folklore or have been, rather occasionally, identified depending on the intentions of the referent and the context.
72
The recalled above amazingly wide range of applications and consequences of the representation formula (2.3) gives it some features of universality in the local and global complex analysis. The vision of this universality of (2.3) has been independently and in full parallel understood by I. N. Vekua and L. V. Ahlfors in their famous papers of 1955 [43], [2]. In the following years both L. V. Ahlfors and I. N. Vekua propagated the use and consequences of the formula in general complex analysis. Therefore in 2007, the year of their parallel centenary, as a due tribute to these great men of science, it is proper and highly justified to relate the names of these two remarkable mathematicians with the formula (2.3) and call it henceforth the Ahlfors-Vekua representation formula. The basic representation formula (2.3) arises as a solution of the nonhomogeneous Cauchy-Riemann equation
aw
(2.12)
-=w a2
with the asymptotic condition (2.2) at infinity. The operator S(w)appears then as the transformation dW aw dz = s(z>
(2.13)
and the resulting obvious formula
asw az
-
aZW -
-
azay
aw
-
(2.14)
az
implies the crucial isometry property of the operator S : L2 + L2. Analogous facts hold if instead of the whole plane we consider the unit disc K or the upper half plane H : I m z > 0 with the boundary conditions Rew = 0 on the boundary a K , IzI
=
1
Imw = 0 on the boundary d H : real line z = 2.
(2.15) (2.16)
For the problem (2.12), (2.15) the representation formula reads =
-I](* + t-z
ll
1-zt
(2.17)
K
and the singular integral operator
satisfies (2.14) which implies, together with the boundary condition (2.15), the L2isometry of S1. As in the case of the principal solutions, based on the Ahlfors-Vekua representation (2.3)) if in the formula (2.17) we use as densities w ( t ) the solutions of the corresponding singular integral equations of type (2.8), we obtain solutions of
73
the boundary value problem (2.15) with the normalizing condition w(1) non-homogeneous Beltrami equations (0.2)
=
0, for
(2.19) in the Sobolev spaces W1+’(K)with the precise estimates for the W 1 , p ( Knorms ) of the solutions in terms of the ellipticity constant k , and the LP norm of h. Because of (2.7) we obtain also global Holder exponents for the solutions of t h boundary problems (2.15) and (2.16). We can also use the formula (2.20) to represent solutions of the Beltrami equation (0.1) satisfying the boundary conditions (2.15) and the conditions w(O) = 1, Imw(1) = O
(w(1) = 0)
(2.21)
as QC-mappings of the disc K on the half-planes. The operator F 1 in (2.20) is obtained from TI by adding obvious linear functionals, so that (2.21) holds. Moreover,
Thus, if the density w ( z ) in (2.20) satisfies the corresponding singular integral equations of type (2.8), ~ ( zin) (2.19) will be the solution of the Beltrami system (O.l), (0.2) mapping the disc K onto the half-plane Rew > 0. Again we immediately obtain also the WlJ’ estimates for the homeomorphic solutions and consequently the Holder estimates. 3. Structure theorem for general Beltrami equations
The existence theorem of a principal homeomorphism of the Beltrami equation (0.1) immediately implies the following structural theorem. Theorem 1. Let x = x ( z ) be a homeomorphic solution of the Beltrami equation (0.1) in a planar domain G . Then any other generalized solution w = ~ ( z of) the equation (0.1) in G may be represented in the form N.2) =
f (x(z))
(3.1)
where f (x)is holomorphic in the image domain x ( G ) . Conversely, given a solution x = x ( z ) of ( O . l ) , not necessarily homeomorphic, and an arbitrary holomorphic function f ( x ) in x ( G ) the formula (3.1) gives a solution of (0.1).
74
Corollary. Given two arbitrary Jordan regions G1 and G2 of the complex plane there always exists a solution w = w ( t ) of the Beltrami equation (0.1) homeomorphically mapping G1 onto Gz. Such a solution may be extended to the closed regions GI + Ga as a homeomorphism. It is uniquely determined e.g. by the assignment of three arbitrary points on the boundary dG1 or in some other way precisely as in the theory of conformal mappings. This is the famous Riemann mapping theorem for quasiconformal mappings with the given (fixed) complex dilatation p ( z ) ,satisfying the uniform ellipticity condition (0.4). Since our only assumption on the dilatation p ( z ) is its measurability, it is called the measurable Riemann mapping theorem. The homeomorphism x = x ( z ) in (3.1) may be chosen in a variety of ways. If it is the principal homeomorphism of the Beltrami equation (0.1) then it has a univalent conformal extension to the complex plane over the boundary of G and then all the singularities of the behavior of the solution w(z) are transferred to the boundary behavior of the holomorphic component f ( x ) near the boundary of the image domain x(G). If G is e.g. the unit disc D and x = x o ( z ) is the quasiconformal mapping of D onto D , normalized by fixing the origin x(0) = 0 and x(1) = 1, then the following a priori estimate holds ([14], [15]). Proposition 4. With the normalization above the W1iP(D) norms of ~ ( z are ) estimated IXZlLP(D), llXZIlLP(D)
Ic
(3.2)
with the constant C depending on the ellipticity constant k in (0.4) only ( f o r any admissible p , p > 2, in agreement with (2.7)). In particular the homeomorphism x = x ( z ) and the inverse homeomorphism x-'([) satisfy uniform Holder condition with exponent cx = 1 - ?, depending on P the ellipticity constant. These a priori estimates are crucial for the extension of the measurable Riemann mapping theorems to general Beltrami equations (0.2). The representation formula (3.1) holds also for the solutions of the general Beltrami equation (0.2) and is a direct consequence of Theorem l since we admit measurable dilatations: indeed, we can write
0 =WZ
-
pwZ - vZO,
3
-
wz - ~
w
Z
with
which is pointwise legitimate in the class of measurable coefficients p only. Nevertheless, the crucial estimate (0.4)
IE(z)I 5 holds with the same constant k .
Ipl
+ 1 ~ 51 IC < 1 a.e.
75
However, because of (3.3), the homeomorphism x = x ( z ) depends on the represented solution w = w(z), whereas for the Beltrami equation the component x ( z ) was possible to be taken universal, depending only on the coefficient p. It is of fundamental importance that the a priori estimates (3.2) hold with the same constants. This fact and the estimates of Proposition 3 are crucial in the quite not trivial proof of the general measurable Riemann mapping theorem. Also the uniqueness questions in the mapping theorem for equation (0.2) are much more difficult than for the Beltrami equation (0.1). Assume now that we have a fixed k-elliptic real Beltrami equation (0.2).
Theorem 2. Let F ( x ) be a given, in general meromorphic, function in the unit disc D . T h e n there exists a QC-homeomorphism x = x ( z ) of the (closed) disc x : D 4 D normalized by x ( 0 ) = 0 , x(1) = 1, such that the function
is the solution of the given general Beltrami equation. The QC-homeomorphism x ( z ) may be normalized in some other standard way used in the theory of holomorphic conformal mappings. Theorem 3.2 is in some sense inverse to the structure Theorem 1 for measurable real Beltrami equations. The transition from the function F ( x ) to W ( Z ) is a highly non-linear operation. Theorem 3.2 was first published in this generality in [14] in 1955, based on the tools elaborated in 1953-54 by I. N. Vekua and B. Bojarski and shortly described above. As a corollary we get
Theorem 3. For any uniformly elliptic real Beltrami equation (0.2) in the unit disc D , any fixed oriented triple of boundary points of 8 D and any given Jordan domain G with a fixed equally oriented triple of boundary points o n dG, there exists a QC-homeomorphic solution of the equation mapping (D, z 1 , ~ 2z,3 ) onto (G,wlr W Z ,w3). Proof. Take a conformal univalent map f ( x ) mapping ( 0 1 z 1 , z 2 , z 3 ) onto (G, w1 , w2, ws)and “change” the variables in the disc D by Theorem 3.2. The described procedure clearly applies to the construction of solutions of uniformly elliptic real Beltrami equations mapping simply connected domains more general than Jordan domains. The first direct self-contained proof of the measurable Riemann mapping theorem was obtained in the framework of analytical methods and concepts elaborated by I. N. Vekua and the author in the early fifties of the last century. Earlier work, going back even as far as to F. Gauss in the nineteenth century, was done with various smoothment assumptions on the coefficients and as a rule locally with classical methods of potential theory. The highlight here were the bea.utifu1 and deep works of L. Lichtenstein [36] from the time he worked in Krakbw, Poland,
76
in the A. Zaremba group, and published in 1916 in the Bulletin of the Cracow Academy of Sciences. Lichtenstein’s work stopped at the level of Holder continuous coefficients of complex Beltrami equations (0.1) in the real symmetric form (0.5). Though later often rather formally referred to (C. B. Morrey, L. Bers et al.) the Lichtenstein theory does not seem to have been incorporated into a modern, selfcontained presentation of the topic. Important issue of distinction between local and global character of these results wait for a clear and concise presentation. References in loose style in various places to the general Koebe-Poincark uniformization theory are rather unsatisfactory. It may be also that the precise conformally invariant estimates for measurable real and complex Beltrami equations as presented above in Propositions 3 and 4 open the shortest and most natural theory to the general planar uniformization theory of Poincark-Koebe-R. Nevanlinna. This is certainly a challenging problem of the planar geometric function theory (though rather expository, taking into account all the a priori estimates and other strong tools available at the present moment in the general analytical theory of QC-maps). The paper [39] of C. B. Morrey (1938) presented new important methods to discuss the local integral type weighted energy estimates for the general planar uniformly elliptic differential equations (0.5) with measurable bounded coefficients. These estimates imply the universal Holder estimates for continuous homeomorphic solutions of (0.5), (0.2) in the Sobolev spaces Wlk: in terms of the ellipticity constants and L2 bounds of the coefficients. Relying on these C. B. Morrey performed the limiting process in the Riemann mapping and boundary value problems for smooth, Holder continuous coefficients theory of (0.5) or (0.9) and thus extended the L. Lichtenstein existence results to the general measurable Beltrami equations. L. Bers, in several of his papers from late fifties and in Bers-Nirenberg [la] heavily relies on the results of L. Lichtenstein [36], M. Lavrentiev [32]and C. B. Morrey [39]. It is to be regretted that the evidently sketchy presentation in [la] has never been followed by a detailed paper, promised in [12] and elsewhere. Looking at the literature of the topic from the perspective of the later years, and maybe even present, it is rather clear that the methods used by C. B. Morrey and the concepts and methods propagated by L. Bers were rather far from the universality and flexibility of the methods and analytical concepts of I. N.Vekua and his school from the early fifties of the 20th century, partially described above. Also the highly publicized paper [4] on Riemann mapping theorem “for variable metrics” in fact in a rather direct and explicit way relies on the analytic apparatus and estima.tes used several years earlier in the I. N. Vekua school. It is to be regretted that in a part of mainly American origin literature the role of I. N. Vekua’s school in the analytic theory of general measurable Beltrami equation is, gently speaking, at least slightly, underestimated by being referred to as an “interesting observation”. Most of the topics discussed in the present paper are directly related with the results and concepts of Vekua’s school from the years 1954-1957, see [43], [44], [13], ~ 4 1 ~, 5 1 ~, 6 1 .
77
As a contribution to this point of view let us mention also the new elements in the present paper: 1) the formula (2.20) gives a direct explicit expression for the general QC-mapping of the unit disc K onto the half-plane Rew > 0 for arbitrary measurable complex dilatations ( p ,u ) of the equations (0.2) in complete analogy to the generality of the principal QC-homeomorphisms for general Beltrami equations. 2) The discussed below existence theorem for primary pairs, solving Conjecture 1 in [27]. 4. Primary solutions of the general Beltrami equations
Primary solutions of the general Beltrami equation (0.2) allow to recover the dilatations p and u. They are thus crucial in discussing various structural properties and inverse type problems for equations (0.2). The important role of primary solutions was recognized in [27], [20], where the following definition was introduced.
Definition. A pair (p,$) of global homeomorphic solutions (i.e., realizing the homeomorphic orientation preserving mapping of the complex plane p : C + C, p(m) = ca) of (0.2) is called a p r i m a r y pair if the condition (1.2) is satisfied. The fundamental problem was the existence of primary pairs for the general equation (0.2). It was stated in [27] as Conjecture 1 and proved for equations (0.2) satisfying the condition
k
-
E
hi I Fi VI hl
VI hl k
,o
*
h
hl
v
2 B
2
N
G
Y
13
+ 6 II Y
6 5 'S
9
5
N
I/
6
I
N Y
II
c.l
m
h
v
I
c\1
I F?
N P
h
N -
N I
d
v
i
I
i
I
P
Y
I
d
13
I
i
I
Q
6
i
II
I
P
6 s
4
99
where n-1-2n
p
cc
Cnlcpp1 =
p=o
X=O
1
(K-p-l)!(K-p+l)!
and
2[
n-2n L - p - 1
1=
-
(K
n-1-K-p-X
1
1 - p - 2 ) ! ( K - p)!
n-2-n-p-A I‘-p-2
(
(:I:)
Lemma 2’ For
cc
n-1-2n
cnl =
u=o
p
1
(p+l)(p+2)
S X + l ( Z , < ) ( l - IZ12))”+2
X=O
holds
Lemma3 F o r O < p < ~ - 2
where n-1-2n CKK-p
2 =
c
p=o
p
C2 X=O
x (n-p)!(K-p+1)!
and n-2n Cnn-p-12
=
c X=O
Lemma 3’ For
h (K -
p
-
l)!(K - p)!
100
holds i n-2n
Corollary 1 For C,l = C,,
1
+ C,1
-2
follows
where
and
Also
Lemma4 F o r O < a < n - l
(azaz)“cnn= c,,-,+;=I
+
(azaz)‘-r
[cnn-r(l - Iz12)n--2r],
+
where C,,-, = Cnn-,1 Cnn-u2,Cnn-g = CnK-,l Cn,-,2 a n d (dz&)u-r [C,n-r(l - Iz(2)n--2r]according t o t h e definitions in l e m m a s 2 a n d 3 i s a proper
101
linear combination of
U-r-V
U-T
U-T
v=o
U--7-V
p=o
V
Ec=o
V
U-T-V-p
U-T-V-0
L=o
(nl;:;:,) (:)
xC (' T= :'>
+ + 1)([gXfnf2(Z7 C)
(":")
- g&K+l(z7
(y)
(n-r;u-f
C)] (l- IZ12)n-u-r-'
+2(n - CT - 7 - L)gx+,+l(Z,, x=o >, p=O
n-1-26 -
c
fi=O
\
, , ,
(/$-I)!&!
(I+","-')
(1 - I q ) p + 2 K - - 1
1
("+","')
( K - l)!K!
n-26
12-1-6-A
1
X=O n-2-K-A - (6
- 2)!K!
(y) )]gx+l(;,F)(l
(
6-2
- IZ12)n-2.
This is ( 2 5 ) (ii) Proof of Lemma 3 As before without loss of generality p = 0. Rewriting C K n 2 = ReCKK3. With
p=o
X=C
then n-1-2&
p
lp+K-A-l\
112
After some rearrangements and using for 2 5 p
it follows
a,cKK, =
-
n-1-2K
p+l
u=o
X=O
cc
n-2&
Differentiating again gives
n-2~ +
c
X=O
x
(6 - l ) ! K !
.
x (K
- l)(K
/n-l-~-l\
+ l)!
113
Thus
cc
2a,azcKn2 =
p=o
n-2n +
X=O
X=O
A
(K-
l)!K!
x (K-
1)!K!
This is (26). (iii) Proof of Lemma 7 Again p = 0 is no restriction. It holds
and
Using for 2
5p
and K-1
K-1
-
K-1
IF,
(fi+ZK -l)
-
(JL+2K-l) n
114
then
n-2n
+
c X=l
~
1
(K - l)!K!
This is (27).
References 1. H. Begehr: Complex analytic methods for partial differential equations. An introductory text. World Scientific, Singapore, 1994. 2. H. Begehr: Orthogonal decompositions of the function space La(B;C).J. Reine Angew. Math. 549(2002), 191-219. 3. H. Begehr: Combined integral representations. Advances in Analyis, Proc. 4th Intern. ISAAC Congress, Toronto, 2003, eds. H. Begehr et al., World Scientific, Singapore, 2005, 187-195. 4. H. Begehr: Boundary value problems in complex analysis. I, 11, Bol. Asoc. Mat. Venezolana 12 (2005), 65-85; 217-250. 5. H. Begehr: The main theorem of calculus in complex analysis. Ann. EAS, 2005, 184210. 6. H. Begehr: Six biharmonic Dirichlet problems in complex analysis. Preprint, FU Berlin, 2006. 7. H. Begehr: Biharmonic Green functions. Preprint, F U Berlin. 8. H. Begehr, G. Harutyunyan: Robin boundary value problem for the Cauchy-Riemann operator. Complex Var., Theory Appl. 50 (2005), 1125-1136. 9. H. Begehr, G. Harutyunyan: Robin boundary value problem for the Poisson equation. J. Anal. Appl., 4 (2006), 201-213. 10. H. Begehr, G.N. Hile: A hierarchy of integral operators. Rocky Mountain J. Math. 27(1997), 669-706. 11. H. Begehr, A. Kumar: Boundary value problems for the inhomogeneous polyanalytic equation. I, Analysis 25 (2005), 55-71; 11, to appear. 12. H. Begehr, D. Schmersau: The Schwarz problem for polyanalytic functions. ZAA 24 (2005), 341-351.
115
13. H. Begehr, C.J. Vanegas: Iterated Neumann problem for the higher order Poisson equation. Math. Nachr. 279 (2006), 38-57. 14. H. Begehr, T.N.N. VU, Z.X. Zhang: Polyharmonic Dirichlet problems. Preprint, FU Berlin, 2005. 15. H. Begehr; E. Gaertner: A Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane. Preprint, FU Berlin, 2006. 16. E. Gaertner: Basic complex boundary value problems in the upper half plane. Ph.D. thesis, FU Berlin, 2006; http://www.diss.fu-berlin.de/2006/320.
116
FINELY MEROMORPHIC FUNCTIONS IN CONTOUR-SOLID PROBLEMS T. ALIYEV AZEROGLU Department of mathematics, Gebze Institute of Technology, Gebze, 41410 Kocaeli, Turkey
[email protected] P.M. TAMRAZOV Institute of mathematics of National Academy of Sciences of Ukraine, Tereshshenkivska str., 3, 01601, Kiev, MSP, Ukraine
[email protected] r
We establish contour-solid theorems for finely meromorphic functions taking into account zeroes and the multivalence of functions.
1. Introduction
In' the purely fine contour-solid theory for finely holomorphic and finely hypoharmonic functions was established. That theory contains refined, strengthened and extended theorems for the mentioned classes of functions in finely open sets of the complex plane with preservable majorants (from the maximal classes of such majorants for the mentioned function classes). On the other hand, in2 ,3 ,4 ,5 ,6 certain contour-solid theorems for analytic functions from earlier authors' works were extended onto meromorphic functions and strengthened with taking into account zeroes and the multivalence of functions. In the present work we extend and strengthen some results of' related t o finely holomorphic functions. The generalization is related to considering finely meromorphic functions instead of finely holomorphic, and strengthening is connected with taking into account zeroes and the multivalence of functions. We need to recall a number of definitions and notations from' . Let !TI be the class of all functions p : (0, +m) + [0, +m) for each of which the set Ifi := {x : p ( x ) > 0) is connected and the restriction of the function log p ( z ) to Ifi is concave with respect to log z. Let !TI* be the class of all p E !TI for which Ifi is non-empty.
117
For p E f131* let us denote by ~c! and ~c: the left and the right ends of the interval I P , respectively. Obviously, 0 5 z! 5 5 f03. When < the mentioned concavity condition is equivalent to the combination of the following conditions: the function log p ( z ) is concave with respect to log z (and therefore continuous) in the interval (x! , and lower semicontinuous on I”. For p E f131 the limits
1c7
1c7,
1c7)
exist, and we have Po
2 pa,
Po
>-
M,
Po3
< fw.
In particular, if ~ c > t 0 (analogously, if ~c: < +03), then po = +03 (pa = -03, respectively). When PO < +a,define the integer mo by the conditions mo-1 < PO 5 mo, and when pa > -03, define the integer ma by the conditions ma 5 pa < ma 1. For every fixed a E R, ,b E (0, +03), the function p ( z ) := ,bza belongs to tM*, and for it we have po = pa = a . If, moreover, a is an integer, then mo = ma = a . Let be the compact Riemann sphere. We refer to7 ,* ,’ concerning the fine topology and related notions such as thinnes, the fine boundary and the fine closure of a set, fine limits of functions, fine superior and fine inferior limits of functions, finely hypoharmonic, finely hyperharmonic, finely subharmonic, finely harmonic, finely meromorphic functions, the generalized harmonic measure, the Green’s function for a fine domain and so on. Let E C Denote by g the standard closure of E in and by the in which E standard closure of a set E c C in @. The set of all points II: E is not thin is called the base of the set E in and is denoted by b(E). The set E := EUb(E) is called the fine closure o f t h e set E in Clearly, c @, . Denote by 6 E the fine boundary of E in Let d f E := C n G E , ( E ) i := E \ b(E) , ( E ) , := E \ ( E ) i . Points II: E ( E ) , and ~cE ( E ) i are called regular and irregular points, respectively, of the set E . For a set E c let us denote \ E =: F E , and for a set E c @, denote also @\E =: C E . Let G c be a finely open set, the set FG is non-polar and z E G. A set Q c FG will be called nearly negligible relative t o G if for every finely connected component T of G the set Q n d f T contains no compact subset K of the harmonic measure wT(K)> 0 at some (and therefore at any) point z E T . This requirement is equivalent to the following alternative: either FG is polar (and then Q is also polar), or for every finely connected component T of G both d f T is non-polar and Q is a set of inner harmonic measure zero relative to T and any point z E T . If a set E c FG is such that for every finely connected component T of G it contains no compact subset K c d f T of logarithmic capacity Cap K > 0, then E is nearly negligible relative to G.
+
e
e.
e
e.
e
e
e
e, e
e.
118
In particular, any set E c FG of inner logarithmic capacity zero is nearly negligible relative to G. For any finely open set G c C with a non-polar complement C \ G, and w E E, E G,w # there exists Green's function gG(w, Let D be a fine domain in i. e. a finely open, finely connected set. Then fi = 5. In particular, then D is finely separable from a point z E if and only if it is separable from z in the standard topology. So in such a situation we may speak of separability not specifying in what sense. Let G be a finely open set in and z E dfG. Given any functions u : G + [-m,+cc] and h : G -+ we introduce the following notations for fine superior limits of functions:
0 there exists a fine neighbourhood U of w for which lp(z)l t 1q(z)1 b' z E X n U , then we write
5
p ( z ) = fine o ( q ( z ) ) ( z + w,zE X ) .
p ( z ) = fine o ( q ( z ) ) ( z -+ w,zE X ) .
e
Let G c C be a finely open set, h : G -+ be a finely meromorphic function, and p E !JX. Consider the following conditions: (A, cc) m E b(CG) and for every finely connected component T of G with cc E b(T) there holds hoo,T,f< 00;
+
119
(B,m)
co # b(CG), p, > -
and
03
h( 2 . Let the following conditions are satisfied (i) for a given 70 E A, limsup C+z,
-
f (01= 0(P1(IZ - "iol)),
E
(aD\Q),
+ 70, ( 3 )
CED
(ii) for all
VY E A f \ {YO} limsup 14")
- f(O1 = O($((I. - "iol)),
z E (aD\Q>, z
-+
70.
C+z,
(4) Then 4( 0
or v
E
v< E D '
0. If v is not constant, it takes minimum at the point
0 (I< - 11"') quently v
C
= 1 and is
there, as well. This contradicts Hopf's lemma statement. Consew
= 0. But v E 0 implies q5 = f .
References 1. Daniel M. Burns and Steven G. Krantz, Rigidity of homorphic mapping and a new schwarz lemma at the boundary, Journal of the AMS 7 (1994), no. 3, p.661-676. 2. Dov Chelst, A generalized Schwarz lemma at the boundary, Proceedings of the American Mathematical Society, Volume 129, Number 11, Pages 3275- 3278 3. P.M. Tamrazov, Holomorphic functions and mappings in the contour-solid problem, Dokl. Akad. Nauk SSSR 279 (1984), no. 1, English transl. in Soviet Math. Dokl. Vol. 30 (1984), no. 3.
130
4. D. Gilba,rg and N.S. Trudinger, Eliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. 5. P.M. Tamrazov and T.G. Aliyev, A "contour-solid" problem f o r meromorphic functions, taking into account zeros and non-uniaalence, Dokl. Akad. Nauk SSSR 2 2 8 (1986), no. 2; English transl. in Soviet Math. Dokl. Vol. 33 (1986), no. 3. 6. T.G. Aliyev and P.M. Tamrazov, A contour-solid problem f o r meromorphic functions, taking into account nonuniaalence, Ukrainian Math. Zh. 39 (1987), 683-690 (Russian). 7. T.G. Aliyev, Irregular boundary zeros of analytic functions in contour-solid theorems, Bull. SOC.Sci. Lettres Lodz 49 Ser. Rech. Deform. 28(1999), 45-53. 8. W.K. Hayman and P.B. Kennedy, Subharmonic Function, Academic Press, London, 1976
131
SINGULAR PERTURBATION PROBLEMS IN POTENTIAL THEORY AND APPLICATIONS MASSIMO LANZA DE CRISTOFORIS
D i p a r t i m e n t o di M a t e m a t i c a P u r a ed Applicata, Universitd d i P a d o v a V i a Trieste 63, 35121 P a d o v a , Italy.
[email protected] K e y w o r d s : Nonlinear boundary value problem, singularly perturbed domain, Laplace operator, real analytic continuation in Banach space.
1. Introduction. This paper is devoted to present applications of a functional analytic approach to the analysis of nonlinear boundary value problems on a domain with a small hole. We consider a bounded open connected subset 1" of Rn of class C1@with unit outward normal uo and with 0 E II", and with Rn \ cl1" connected, and we assume that the boundary value problem
Au = 0 in I", B" ( t ,u ( t ) ,g(t)) = 0 V t E dII", admits at least a solution U E C1)a(clIIo).Here a is a given element of 10, I[, clI" denotes the closure of I", Bo is a given function of dI[" x R2 to IR, and Cm~cu(clIo) denotes the space of real-valued m times continuously differentiable functions of clII" to R with a-Holder continuous m-th order derivatives, for each natural m 2 0. Problems such as (1) are known to have a solution for a large class of BO's. Here we mention the work of Carleman [5], Nakamori and Suyama [21], Klingelhofer [7], [8],[9], [lo], Efendiev, Schmitz and Wendland [6]. We also mention the contribution of Begehr and Hsiao [3], Begehr and Hsiao [4], Begehr and Hile [a] for problems in the plane. Next we make a hole in the domain I". To do so, we consider another bounded open connected subset IIi of Rn of class C1'" with unit outward normal v i and with 0 E I I Z , and with Rn \ clIi connected, and we consider the annular domain
A(€) E I"\ €ClIIi. for all
I E ~ 5 € 0 , with €0 > 0 sufficiently small so that eclIIi C I[". Obviously, dA(€) = €aniu 81".
132
Next we introduce the boundary conditions on the boundary &Xi of the hole eclIIi by assigning a function Biof 10, to[xdIi x Rz to R,and we consider the boundary value problem
au =0 in A(€), Bi ( t , t E - l , U ( t ) , g + ( t ) )= 0 V t E t d P , V t E dII", Bo ( t , u ( t )g(t)) , =0
(2)
for each E €10, to[. Under suitable conditions on the data, our first goal is to identify a family of solutions { u ( E. ,) } t t ~ O , E O [ of ( 2 ) which in a sense approaches the given solution U when E approaches O+. One could try to solve (1) or (2) by representing the solutions as a sum of a simple and of a double layer potential, and by solving the corresponding integral equations. Here we shall assume that one can do so by using a simple layer potential, and we illustrate in the next section an example where one can actually do so at least for a certain class of nonlinear boundary conditions. We denote by T n the function of 10, +m[ to R defined by 1logr
V r €10, -too[, if n = 2 , r2-, V r €10, +GO[, if n > 2 ,
(3)
where s, denotes the ( n - 1) dimensional measure of a&. We denote by S, the function of IW" \ (0) to R defined by Sn("(dIo), and equation M[O,0, p] = 0 corresponds to the integral equation relative to problem (I) for 6. Under suitable conditions on B", Bi, we may be able to solve locally around ( O , O , p ) equation ( 7 ) , and to prove that the set of solutions of (7) is the graph of a nonlinear operator E H ( E [ E ] , R [of E ]] )- E O , E O [ to C0,"(dIi) x C0>"(dIo) for a possibly smaller € 0 . If B", Biare real analytic, one would expect that (I?[.], I?[.]) is also real analytic. Then by setting ~ ( tt ) ,E
Lo
S,(t
-
s ) R [ E ] (dos s) +
L
S,(t - S E ) E [ E ]dos (S)
V t E c l A ( ~ ) ,(8)
one obtains a solution of (2) which converges in the C1)u-norm on the compact subsets of clIo \ (0) t o U as E tends to 0. Once the family of solutions { u ( E ,.))tE]O,Eo[ is established, two questions appear as natural.
4
0 be a bounded open subset of I['\ (0) such that 0 cl0. What can be said on the map 10, E O [ ~E ++ U ( F , t),clfi E C',"(clfi) around E = 0? (jj) What can be said on the map 10, E O [ E~ H € [ E ] E J D u ( Et)I2 , d t E R around E = O? (j) Let
SA(.,
Problems of this type with linear boundary conditions have long been investigated with the methods of Asymptotic Analysis, which aims at giving complete asymptotic expansions of the solutions in terms of the parameter E. It is perhaps difficult to provide a complete list of the contributions. Here, we mention the work of Kozlov, Maz'ya and Movchan Ill],Maz'ya, Nazarov and Plamenewskii [19],
134
Kuhnau [la], Movchan [20], Ozawa [22], Ward and Keller [27]. For nonlinear problems on domains with small holes far less seems t o be known, we mention the results which concern the existence of a limiting value of the solutions or of their energy integral as the holes degenerate to points, as those of J. Ball [l],Sivaloganathan, Syector and Tilakraj [23], and the literature of homogenization theory. We also mention the computation of the expansions in the case of quasilinear equations of Ward, Henshaw and Keller [25], Ward and Keller [26], Titcombe and Ward [24]. The goal of asymptotic analysis for problem (2) would be to write asymptotic expansions for the maps in (j), (jj). Thus for example an expansion of the form
for suitable coefficients a j . Our goal is instead to represent E E]O,to[ by means of
U ( E , .)lclfi
and I [ €for ]
(a) real analytic maps defined on a whole neighborhood of t = 0; (b) possibly singular at 6 = 0, but known functions of E (such as t p l , logt, etc. . . . ). We observe that our approach does have its advantages. Indeed, if for example we can prove that there exists a real analytic real valued function 3(.) defined in a whole neighborhood of 0 such that F[t]= & I t ] for E ~ ] O , t o [ ,then we know that an asymptotic expansion such as (9) for all T would necessarily generate a convergent series Cj”=,a j & , and that the sum of such a series would be €[E] for E > 0. Now let fi be as in (j). Under conditions in which R[.])is real analytic, the map Ufi of ] - € 0 , E n [ to C1,a(clfi) defined by
(&?[.I,
U,[t] E
Luo
Sn(t - s ) R [ E ] (do, s)
+
Sn(t - SE)E[E](S) dg,
V t E clfi ,
LUi
for all t E] - t O , t O [ offers a real analytic continuation for the map E u(~,.)~~,fi, which is defined only for E E ] O , E ~ [Thus . in this case the answer for question (j) is that
In order to analyze question (jj), we write
135
and we note that
- s), and Now we note that for n > 2, we have Sn(e(t - s)) = (2 - n)s,r,(c)S,(t that S2(t(t- s)) = T ~ ( E -t)&(t - s ) . Hence, we can prove that there exist two real analytic operators Fl,F2 of ] - € 0 , €01 to R such that
&[O]
= 0. Next we note that
v " ( t ) ' D&(t
- S E ) E [ € ] ( S )do,
1
dot.
Hence, one can prove that there exists a real analytic operator such that
F3 of ] - €0,€01 t o R
By (12) and (14) we conclude that €[el = (&[el
+ E ~ [ +ETn[~]Fz[e] ])
€]O,eo[,
(15)
an equality which answers question (jj). We note that our conclusions (lo), (15) are relative to the various simplifications we have made so far to carry on the elementary presentation of this introduction, such as that the solutions of (1) and of ( 2 ) can be represented as simple layers, and that those boundary value problems are solavable exactly when the corresponding integral equations are solvable, and that one can solve locally equation (7) and obtain an implicitly defined operator ( E ( . ]&[.I). , All such circumstances are not always present. Hence, under different circumstances, formulas (lo), (15) may have a different form. This is the case for example when both B', Bi correspond t o the linear Dirichlet boundary conditions, as shown in 1131, 1161, [171, [187.
136
One could extend the ideas exposed above in order t o consider also pertubations of 812, 81"by representing aili, dJI" by global parametrizations say 42, 4" defined for example on the unit sphere dBn of Rn, and analyze the corresponding singular perturbation problem in the complex of variables ( E , 42, gY) considered as a point in the Banach space R x (C'~"(811i))" x (C1>"(dl"))",as done for linear problems in [16] - [18]and for problems related to the Riemann mapping in [13], [14]. In this case the difficulties would increase. Thus for example the analysis of an equation such as (7) is more complicated and the presentation is necessarily longer, and the corresponding treatment is not illustrated here. 2. A concrete case: the case of the nonlinear Robin boundary conditions
The material of this section is entirely based on the paper [15], where the special case Bi ( c , t ~ - ' , u ( t )s(t)) , = -s(t), B" ( t , u ( t )g(t)) , = - G"(t,u(t)) has been considered. Here Go is a continuous map of 81"x R to iW. We first have the following Theorem, which asserts the existence and local uniqueness of the family of solutions U ( E , .). Theorem 1. Let a E]O,l[. Let I i , JIo be bounded open connected subsets of Rn of class Let Rn \ clIi, Rn \ clI" be connected. Let 0 E I i , 0 E I". Let Go E Co(dII"x R) be such that the operator T G of ~ Co~"(dl")to Co(dJIo)defined by
g(t)
T G ~ [ WE] (G~")( t , v ( t ) )
V t E dII",
VW E Co'"(dJIo)
map Co>"(dl")real analytically to itself and map bounded sets of Co~"(dIIo) to bounded sets of Co~"(dIIo). Assume that there exists a solution fi C1."(cllIo) of problem
AU = 0 in I", ( t )= G"(t,u ( t ) )V t E 81°, such that
Then the following statements hold. , ~ ,solutions [ u(E;) E (i) There exists E' E]O,co[ and a family { ~ ( c , . ) ) ~ ~ l Oof C'>"(clB(E)) of problem
{ ---=O
A(€), on E W , =(t) = G"(t,~ ( t 'dt ) ) E dI",
Au=O
in
(3)
such that lim,,ou(e, .),clfi = fi,,,fi(.) in C'@(clfi) for all bounded open subsets
fi of H" \ (0)
such that 0 $ clfi.
137
(ii) If {cj}jtw is a sequence of 10, -too[ converging to 0 and if {uj}jEw is a sequence of functions such that Cl@(Clh(€j)), (3) for E = ~j , uj = U in C’,cy(clfi)for all bounded open subsets fi of 1”\ (0) such that 0 clfi , uj E
uj solves
limj+,
4
then there exists j o E
N such that uj(.)= u ( q ,.) for all j 2 j o .
Then we answer questions (j), (jj) of section 1 by means of the following. Theorem 2. Let the assumptions of Theorem 2 hold. Then the following statements hold. such that 0 4 clfi. Then there exists E” E]O,.5’[ and a real analytic operator U, of ] - E ” , E ” [ to C1,a(clfi) such that clfi C A(€)for all E E] - E ” , E ” [ and such that
(i) Let
fi be a bounded open subset of IIo \ (0)
Moreover, U,[O] = 2LIClfi. (ii) There exists a real analytic operator 3 of ]
s,
3[€] =
-
ID@(€,t)I2d t
d’,E ” [ to lR such that
v.5 €10, ?[.
6)
Moreover, 3 [ 0 ]=
sI0IDtG(t)12d t .
Acknowledgments
The author is indebted to Prof. B. Dittmar and to Prof. E. Wegert for pointing out a number of references concerning the existence of solutions for problems (l), (l),and to Prof. A.B. Movchan, and to Prof. J. Sivaloganathan, and to Prof. M.J. Ward, for pointing out a number of references on nonlinear singular perturbation problems on domains with small holes. References 1. J.M. Ball, Discontinuous equilibrium solutions a n d cavitation in nonlinear elasticity, Philos. Trans. Roy. SOC.London Ser. A, 306, (1982), 557-611. 2. H. Begehr and G.N. Hile, Nonlinear R i e m a n n boundary value problems f o r a nonlinear elliptic s y s t e m in the plane, Math. Z., 179, (1982), 241-261. 3. H. Begehr and G.C. Hsiao, Nonlinear boundary value problems f o r a class of elliptic systems, Komplexe Analysis und ihre Anwendung auf partielle Differentialgleichungen, Martin-Luther-Univeristat, Halle-Wittenberg, (1980), 90-102. 4. H. Begehr and G.C. Hsiao, Nonlinear boundary value problems of Riemann-Hilbert type, Contemporary Mathematics, 11, (1982), 139-153. 5. T. Carleman, Uber e i n e nichtlineare Randwertaufgabe bei der Gleichung Au = 0, Math. Z., 9 , (1921), 35-43.
138
6. M.A. Efendiev, H. Schmitz and W. Wendland, O n s o m e nonlinear potential problems, Electron. J. Differential Equations, 1999, (1999), 1-17 . 7. K. Klingelhofer, Modified Hammerstein integral equations and nonlinear harmonic boundary value problems, J. Math. Anal. Appl. 28, (1969), 77-87. 8. K. Klingelhofer, Nonlinear harmonic boundary value problems. I, Arch. Rational Mech. Anal. 31, (1968)/(1969), 364-371. 9. K. Klingelhofer, Nonlinear harmonic boundary value problems. II. Modified H a m m e r stein integral equations, J. Math. Anal. Appl. 2 5 , (1969), 592-606. 10. K. Klingelhofer, Uber nichtlineare Randwertaufgaben der Potentialtheorie, Mitt. Math. Sem. Giessen Heft 76, (1967), 1-70. 11. V.A. Kozlov, V.G. Maz’ya and A.B. Movchan, Asymptotic analysis of fields in multistructures, Oxford Mathematical Monographs, The Clarendon Press, Oxford university Press, New York, 1999. 12. R. Kiihnau, Die Kapazitut dunner Kondensatoren, Math. Nachr., 203, (1999), 125130. 13. M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan d o m a i n with a small hole, and relative capacity, in ‘Complex Analysis and Dynamical Systems’, edited by M. Agranovsky, L. Karp, D. Shoikhet, and L. Zalcman, Contemp. Math., 364, (2004) pp. 155-167. 14. M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan d o m a i n with a small hole in Schauder spaces, Computat. Methods Funct. Theory, 2, 2002, pp. 1-27. 15. M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a nonlinear Robin problem f o r the Laplace operator in a d o m a i n with a small hole. A functional analytic approach, submitted, 2006. 16. M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of t h e Dirichlet problem f o r the Laplace operator in a domain with a small hole. A functional analytic approach, submitted, 2004. 17. M. Lanza de Cristoforis, A singular domain perturbation problem f o r the Poisson equation, submitted, 2005. 18. M. Lanza de Cristoforis, A singular perturbation Dirichlet boundary value problem f o r harmonic functions o n a domain with a small hole, Proceedings of the 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, Tokyo July 27-31 2004, edited by H. Kazama, M. Morimoto, C. Yang, Kyushu University Press, (2005), 205-212. 19. V.G. Mazya, S.A. Nazarov and B.A. Plamenewskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, I, 11, (translation of the original in German published by Akademie Verlag 1991), Oper. Theory Adv. Appl., 111,112, Birkhauser Verlag, Basel, 2000. 20. A.B. Movchan, Contributions of V.G. Maz’ya t o analysis of singularly perturbed boundary value problems, The Maz’ya anniversary collection, 1 (Rostock, 1998), Oper. Theory Adv. Appl., 109, Birkhauser, Basel, 1999, pp. 201-212. 21. K. Nakamori and Y. Suyama, O n a nonlinear boundary problem f o r the equations Au = 0 and Au = f(x,y ) (Esperanto) Mem. Fac. Sci. Kyusyu Univ. A,, 5 , (1950), 99-106. 22. S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, (1983), 53-62 . 23. J. Sivaloganathan, S.J. Spector and V. Tilakraj, T h e convergence of regularized minimizers for cavitation problems in nonlinear elasticity, SIAM J. Appl. Math., 66, (ZOOS), 736-757.
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24. M.S. Titcombe, M.J. Ward, S u m m i n g logarithmic expansions f o r elliptic equations in multiply-connected d o m a i n s with small holes, Canad. Appl. Math. Quart., 7,(1999), 313-343. 25. M.J. Ward, W. Henshaw, and J . Keller, S u m m i n g logarithmic expansions f o r singularly perturbed eigenvalue problems, SIAM J. Appl. Math., 53, (1993), pp. 799-828. 26. M.J. Ward, J . Keller, Nonlinear eigenvalue problems u n d e r strong localized perturbations w i t h applications t o chemical reactors, Stud. Appl. Math., 8 5 , (1991), 1-28. 27. M.J. Ward and J.B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J . Appl. Math., 53 (1993), 770-798.
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RESIDUES O N A KLEIN SURFACE ARTURO FERNANDEZ ARIAS Dpto. de Matematicas Fundamentales Facultad de Ciencias UNED C/Senda del Rey s / n Madrid 28040 Spain
[email protected] JAVIER PEREZ ALVAREZ Dpto. de Matemciticas Fundamentales Facultad de Ciencias U N E D C/Senda del Rey s / n Madrid 28040 Spain
[email protected] T h e reconstruction of a Riemann surface starting from the meromorphic function field K , comes from Dedekind and Weber who developed an algebraic function theory in one variable over an algebraically closed field k . Alling and Greenleaf present a counterpart t o this approach starting from a real algebraic curve. From this point of view, the residues theorem is a classical result which depends strongly on the algebraically closed character of the base field. In this paper, via the complex double, we translate this fact to the case where we start from a function field in one variable over R.
Keywords: Klein surface, valuation ring, residue.
1. Klein surface of a real function field
Let K be a field and A be a subring of K ; we shall say that A is a valuation ring of K if for each II: E K then x E A or x-l E A; it holds that A is a local ring: if it has two different maximal ideals p l and p a , then, taking a E p1\ p2 and b E p ~ \ p l , we have a / b 4 A and b / a 4 A
Definition 1. Let R be the field of real numbers and K be a finite extension of transcendence degree 1 over R in which -1 is not a square. We shall call Klein surface of the field K , the set S of valuations rings of K which contain R. In the same way, the set S, of valuation rings of K [ i ]which contain C shall be called the double cover of S. We shall denote a point P of S by A, when we intend to indicate the ring that it represents, denoting then by p or m, its only maximal ideal. Let us fix an element z E K \ R, let n be the degree of the extension R(z) 9K and let us denote with 0 the integral clausure of R[z] in K , that is, the subring
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of K formed by the elements which verify a monic polynomial with coefficients in
RbI.
For each element X = XI
+ i X 2 in K[i], let us write X = XI
- iX2.
Proposition 1. 0 i s a noetherian ring. Proof. We shall see that 0 is a free R[z]-module of rank n. If t is a primitive element of the extension R ( z ) L) K , we can suppose, multiplying the minimal polynomial o f t over R[z] by an adecuate element of R[z], that t E 0. In this way, if y E 0, we can write y=
+ a l t + a2t2 + ... + an-1tn-',
ai E
R(z)
If (ai):=' are the R(z)-isomorphisms of K in the minimal normal algebraic extension L of K over R(z), we obtain the n following equations: a i ( y ) = a0
+ alai(t)+ a m ( t ) 2 + ... + an-lai(t)n-l
(1 5 i
< n).
g,
In this way aj = where D is the Vandermonde determinant of the system and A, is a polynomial with integral coefficients in a i ( t ) ,ai(y). Since D 2 is invariant by (ai)y==,and integral over R[z], we have D 2 E R[z]. As on the other hand D2aj = DAj is an integral element of R(z) over R[z], we get D2aj E R[z], hence 0'0 c (1, ...,tn-l)R[rlC 0, and we are done.
Proposition 2. Every prime ideal p
#0
of 0 i s maximal.
Proof. We shall see that for each maximal ideal m c 0, the local ring 0, has dimension 1;this is a consequence of the dimension theory for local noetherian rings, since the dimension of 0, is the maximun number of parametres in m which are algebraically independent over R. See [a]. In this way, for every point P of S , the quotient ring A,/p is a finite algebraic extension of R, whereby A , / p = R or A,/p = C . Therefore we shall call degree of the point P of S , the integer g r ( P ) = [ A p / p: R] . The free group generated by the points of S shall be called group of divisors on S. In this way, given a divisor D = x n i p i on S , we shall call degree of D the integer g r ( D ) = C n i g r ( p i ) . Let us define the maps o : S, + S,, 7r2 : S, -+ S, such that to each valuation ring V, of K [ i ] a(V,) =
=
7r2(VY)=
{X : x E V,} V, n K .
Definition 2. Given f E K and P E S , we shall say that f has a zero of order n E N at p if f E pn \ pn+' in A, and that f has a pole of order s E N at p if l / f has a zero of order s at p . If f i s invertible in the local ring A, then f has neither a zero nor a pole at P.
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From now on we shall denote with S p e c ( 0 ) the set of prime ideals of 0. It is easy t o see that the valuation rings which contain z , contain also 0 [2] and that those are the local rings 0, for p E S p e c ( 0 ) . On the other hand if we denote with 0’ the integral clausure of R[l/z] in K , it is clear that the valuations rings of K that are not in S p e c ( 0 ) are those points in Spec(0‘) which contain 1/z in its maximal ideal; since these are the prime ideals of 0’which appear in the factorization of ( l / z ) 0 ‘ , they are a finite set; particularly every element of K ha,s finitely many zeros and poles in S, and in a language that we shall not use again, we can exhibite S as a scheme structure:
S = Spec(0) uSpec(0’). Proposition 3. The fiber by 7r2 of a point p of S has one or two points depending o n whether A p / p = R or A,/p = C. Proof. See [5]. The following lemma is easy to prove.
Lemma. Let P be a point of S with A p / p = C ; let t be a parameter i n p and A,, Am(,)the points of the fiber b y 7r2 of P , then (t)A, = m,, and (t)Av(,)= mn(,). 2. Abelian Differentials We shall call derivation of K over R any application D : K
D ( a + b) = Da + Db, D(ab) = aDb DX = 0 if X E R.
--+K
verifying
+ dDa
Let us denote with D e r R ( K ) the set of derivations thus defined. If D , is the derivation such that Of(.) = 1 and on any element t E K , D,(t) is such that g z ( z , t ) + g t ( z , t ) D z ( t )= 0,
where g ( z , t ) is the irreducible polynomial o f t over R(z),then any other derivation D’ verifies D’ = D’(z)D,. Thereby we have
Proposition 4. D e r R ( K ) is a R-vector space of dimension 1. Let us also define D e r c ( K [ i ] and ) conclude similarly that this set is a C-vector space of dimension 1.
Proposition 5. Every derivation D de K over R extends t o a derivation D x of K[i] over C. Proof. It suffices to write D * ( i ) = 0.
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Definition 3. W e shall adpot the following notation
RR(K) = HomR(DerR(K),R), R c ( K ) = H o r n c ( D e r c ( K [ i ] )C) , and these shall be called the spaces of Abelian differentials o n S and S, respectively. We define the map K + RR(K) by z e d z where d z is such that d z ( D ) = D z ( D E D e r R ( K ) ) . In this way, by Proposition 4, RR(K) = K d z ; thereafter any element of Q R ( K )shall be considered as a differential on K [ i ]via the extension of scalars
Let w be an Abelian differential on S. For every point P of S such that A,/p = R, let us consider a generator t in p and let us write w = f d t ( f E K ) . If f has in p a pole of order s then g = t “ f E A,. There exists ups E R such that g - a _ , E p , whereby we can write g - a _ , = t X o , A0 E A,. Repeating the process with Xo, and so on, we have the following expansion in a power series in A, ups
a-1
f =+ . .. + t S t + a0
+ U l t + a2t + . . .
(1)
an easy computation We shall define the residue of w in P as the coeficient [3] proves that this definition does not depend on the parameter chosen in the ring A,. If P’ E S is such that Apt/p’ = C, let us choose a parameter z E p’ and let us write w = h d z , there exists a power of z such that z’h E A,,. If Q‘ E 7rT1(P’)the previous lemma allows us to write the following series expansion
In that way, as K’ of a-1.
a-1
E K [ i ] ,we shall define the residue of w in P’, ‘the part in
Definition 4. Let w 6 f l ~ ( K we ) , shall call residue divisor of w , the divisor Pi E S
where Pi is a pole of w with residue ai. Theorem. T h e degree of the residues divisor of a n Abelian differential o n S is 0.
Proof. First of all we shall check that the definition of residue of an ahelian differential w on S at a point P‘ does not depend on the chosen point of the fiber
144
of P' for the projection 7r2 : S, + S. Making use of the previous notations, if a ( & )E i.a'(P)\{Q}, and w = hdz on S , we have the following series expansion b-1
+
f = 7 . .. +
+ bo + b i z + b2z2 + . . . in A u ( q r ) . (3) z z Let us see that the coefficients of the series expansions ( 2 ) and ( 3 ) are conjugate (in K [ i ] )In . fact, in A,,we have z ' f = a-l + 2x0, whereby in Au(,t) we shall have zlf
b-1
~
= .(zlf)
= .(a-l)
+ za(Xo),
whence b-l = a ( a - l ) , and following with tjhe same argument, we have bi = ~ ( a i ) , vi 2 -1. Now, let us determine the degree of the divisor D r ( w ) = CPitS aiPi. If P is a point of S of degree 1 and Q = 7rT'(P) it is clear that the reside of w considered as a differential on S, is real, whereby it coincides with the residue of w in P. If P' E S is such that 7ra1(P')= { & ' , a ( & ' ) } in which w has a residue r E K , then 2r = res(w,Q ' )
+ res(w,
.(&I)).
In this way, by the classical fact
r e s ( w , Q ) = 0, QtS,
we conlude that
References 1. Alling, N. L. and Greenleaf, N. Foundations of the theory of Klein surfaces. Lecture Notes in Mathematics 212 (1971). 2. Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. AddisonWesley (1969). 3. Bujalance, E. Gamboa, J. Gromadzki, G. Automorphism groups of compact bordered Klein surfaces. Lecture Notes in Mathematics, 1439. 4. Lang, S. Introduction to Algebraic and Abelian Functions. Graduate Texts in Mathematics 89 (1972). 5. Iwasawa, K. Algebraic Functions. Translations of Mathematical Monographs, 118 AMS (1993). 6. Gamboa, J. M. Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves. Memorias de la Real Academia de Ciencias de Madrid. Vol XXVII (1991).
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COMBINATORIAL THEOREMS OF COMPLEX ANALYSIS YU.B. ZELINSKII Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Street, Kiev, Ukraine
[email protected] Combinatorial theorems of real convex and complex analysis are considered. We give some generalizations of classical Caratheodory and Helly theorems.
Keywords: Euclidean space, simplex, convex envelop, Mayer-Victoris exact cohomological sequence, complex line, hyperplane.
We consider classical combinatorial theorems of convex analysis under point view of generalization and application to complex analysis. The main goal of the first result is to show that under some additional restrictions on the set, the number of points in Caratheodory’s theorem which defines convex envelop can be decreased.
Theorem 1. Let E be subset
of n-dimensional Euclidean space Rn which consists
of not more than n connected components. T h e n a n y point of convex envelop of E can be presented as convex combination of not more than n points of E . Proof. Let we suppose that the theorem is not true. Than there exists a point x in convex envelop of E , which by Caratheodory’s theorem can be represented as convex combination of n 1 points X I ,2 2 , . . . , x,+1 of E . Those points defined n-dimensional simplex A. According to our supposition the point z can not be convex combination of any n points of E . We consider all ( n- 1)dimensional planes across point x and every subset of ( n - 1) vertex of simplex A. Every hyperplane of this family contains convex cone with apex x generated by (n- 2)-dimensional side of A. Let we consider symmetrical relatively t o point x cones to cones defined above. It is easy to see that no one of the last cones can not contain points of the set E . If the last is not true and let 1c1 be point in one of cited cone, let C1,then point x can be represented as convex combination of point X I and ( n- 1) vertexes of ( n- 2)-dimensional simplex opposite to cone C1. But this is impossible by reasons of theorem. The union of cones Ci, i = 1,.. . , divides Euclidean space Rn into ( n 1) parts, every of which contains one vertex of simplex A (part of the set E ) . The last contradicts to supposition that E has no more than
+
+
146
n connected components. The classical Helly theorem does not admit to obtain information concerning a family of convex compact sets in the Euclidean n-dimensional space if it is known that only subfamilies consisting of k elements, 0 < k 5 n, possess nonempty intersections. Below we consider the variant of Helly theorem for this case and also investigate behavior of generalized convex families. Theorem 2. L e t A = {Ai} be a f a m i l y of convex compact sets in Rn f o r which every subset f r o m k e l e m e n t s has a c o m m o n point. T h e n either every subset f r o m k + 1 e l e m e n t s h a s a c o m m o n p o i n t o r there exist k 1 compacts A t , i = 1 , 2 , . . . , k + 1 f o r which H " ' ( U ~ ~At) ~ # 0 (where H"'(*) is ( k - 1 ) - d i m e n s i o n a l cohomology group). The proof follows from two technical lemmas.
+
Lemma 1. L e t { A i } i = 1 , 2 , . . . ,m, . . . be a f a m i l y of convex compact sets f o r which every subset f r o m k e l e m e n t s h a s a c o m m o n point. T h e n
H j ( A 1 u A2 u . . . u A, n A,+, n . . . n A k ) = 0 f o r every j , m, 0 5 m 5 k .
+
Lemma 2. L e t { A i } i = 1 , 2 , . . . , k 1 , .. . be a f a m i l y of convex compact sets for which every subset f r o m k elements h a s a c o m m o n point. T h e n
H j - 2 [ ( A , U A2 U . . . U Aj) n Aj+l n . . . n Ak+l] x Hj-l[(A, u A2 u . . , u A j
u A j + l )n Aj+2n . . . n Ak+,], 2 2 j 5 k .
Proof of two lemmas follows from Mayer-Victoris exact cohomological sequence.
Definition 1. A set E c C" is called linearly convex if for every z E exists a complex hyperplane L such that z E L c C" \ E .
Cn \ E there
Definition 2. A set E c C" is called C-convex if for every complex line y sets y n E and y \ y n E are connected. Definition 3. Linearly convex set E C C" is called C-quasiconvex if for every complex line y intersection y n E is simpleconnected. Theorem 3. T h e class of C-quasiconvex s e t s i s closed relatively t o t h e intersection of subsets. Proof. Let K 1 , K2 be C-quasiconvex compacts. For an arbitrary complex line y, y n K1 (1K2 # 0 we consider Mayer-Victoris exact cohomological sequence
~ l ( y n ~ , ) ~ ~ l -( + y n~ ~l (, y) n ~ ,+ n ~~ ~~ () y n ( ~ ~ n ~ ~ ) The first element of sequence is trivial because K1, K2 are C-quasiconvex. The last one is trivial because y n (Kl n K2) is a proper compact subset of really two dimensional complex line y.
147
As conclusion follows triviality of middle element and simpleconnectedness of intersection. References 1. Yu. Zelinskii Multivalued mappings an analysis, Kyiv: Naukova dumka, 1993, 264 p. [in Russian]
148
GEOMETRIC APPROACH IN THE THEORY OF GENERALIZED QUASICONFORMAL MAPPINGS ANATOLY GOLBERG Department of Mathematics, Bar-Ilan University, R a m a t - G a n , 52900, Israel golbera@math. biu.ac.il and H o l m Institute of Technology, 52 Golomb St., P.O.Box 305, Holon 58102, Israel
[email protected] The paper presents a geometric approach for studying properties of mappings. We establish new conditions provided the analyticity of continuous functions of complex variable. We also extend this method for investigation of mappings with finite mean dilatations in Rn. Keywords: Bohr’s and Menshoff’s theorems, local univalent functions, quasiconformal mappings, normal neighborhood systems.
1. The Cauchy theorems for univalent functions. In this paper, we present a somewhat new geometric a.pproach which provides the analyticity of local univalent functions as well as of arbitrary continuous functions. In the particular cases these results generalize the classical theorems of Bohr, Menshoff and Trokhimchuk. Of course, all such theorem can be regarded as the Cauchy theorem: Theorem A. If a function f (2) of the complex variable z is continuous and monogenic in a domain G C @, then it is analytic in G . We recall that a function f ( z ) is monogenic if there exist the limit
which is called the derivative o f f at the point z . In fact, Cauchy had proved this theorem under assumption that the derivative f ’ ( z ) is continuous in the domain. Later Goursat [5] showed that the assumption of continuity of f ’ ( z ) can be omitted. The further attempts t o generalize the Cauchy theorem have been naturally related to replace the rather rigid assumption of monogeniety of f ( 2 ) by a condition as weak as possible.
149
Note that the monogeneity of f ( z ) is equivalent to existence of the both limits:
which geometrically means the independence of stretching in the given direction and
which means preserving the angles at the points, where f ' ( z ) # 0. The next natural step is to find the characterizations of analytic functions either only in terms of stretching (1) or only in terms of preserving the angles (2). The first step in this direction was the following theorem of Bohr [l]: Theorem B. If w = f ( z ) is a continuous univalent mapping of a domain G , for which a finite limit ( I ) exists and differs f r o m 0 at almost every point of G , then either the function f ( z ) or the conjugate function f ( z ) is analytic in G . The Bohr example ~
shows that for functions which are not univalent this theorem is, in general, false. The next important result is the following theorem of Menshoff [8] based on the second fundamental property of monogenic function (preserving the angles). Theorem C. If a mapping w = f ( z ) is continuous and univalent in a domain C and if at almost every point of C , finite limit (2) exists, then the function f ( z ) is analytic in G . Of course, it suffices to require in both theorems the local univalence of the functions. Using the quasiconformal mappings, Menshoff has obtained in [9] another generalization of the Bohr theorem. Namely, let us consider a continuous and locally univalent mapping w = f ( z ) of a domain G of the z-plane onto a domain G* of the w-plane. For an arbitrary point zo E G, we take the circle C ( z 0 , r )= { z : Iz - 201 = r } c D and put ff(zo,r)=
max If (2') I z' -z0 I =T min If(z")
-
It''-20 I=T
f(.o)l
-
f
(z0)l.
We say that the continuous univalent function f ( z ) maps the infinitesimal circle C(z0,r ) into a n infinitesimal circle, if lim H ( z o , r ) = 1.
r+O
Obviously, the last condition is more general than the constancy of (1).The important Menshoff generalization of Bohr's theorem is the following result.
150
Theorem D. L e t a f u n c t i o n f ( z ) be continuous and locally univalent in a d o m a i n G , and let m a p t h e infinitesimal circles C ( z , r ) i n t o infinitesimal circles for almost all p o i n t s z E G. T h e n either f ( z ) o r f (2) is analytic in G . ~
2. Main result for locally univalent functions. We shall use the following notations. Let z be an arbitrary point in C.Assume that some closed neighborhood Gt(z) of z is defined for any t E (0,1]. We say that a set of the neighborhoods Gt(z) of the point z constitutes a normal system, if there exists a continuous function 'u : @. + IR such that ~ ( z=) 0, v(C) > 0 for any 5 # z . Here & ( z ) = {C E C : v(A*(z)< 00. To give the corresponding geometric description for generalized quasiconformal mappings we consider also certain set functions. Let dj be a finite nonnegative function in domain G defined for open subsets E of G so that C,"=,@ ( E k ) 6 @ ( E )for any finite collection {Ek}T=2=1 of nonintersecting open sets E k c E . We denote the class of such set functions CP by F.
154
The upper and lower derivatives of a set function @ E F at a point x E G are defined by
@(Q) W ( x )= lim sup h+od(Q)
+co)
4 5 ,Y) E [-GO, -too)
m , n 2 2.
157
(c) for some p
>0
there is a function u E Lg,(R) such that u 5 u.
Then u is subharmonic. Though the cited result of Armitage and Gardiner includes our Theorem A, and in fact their result is even "almost" sharp, we present below in Theorem 1 a generalization to Theorem A. This is justified because of two reasons. First, our LFoc integrability condition, p > 0, is, unlike the condition of Armitage and Gardiner (I), very simple, and second, our generalization to Theorem A is stated for quasi-nearly subharmonic functions, and as such, it is very general, see 2.1. below. 1.2. Functions subharmonic in one variable and harmonic in the other. An open problem is, whether a function which is subharmonic in one variable and harmonic in the other, is subharmonic. For results on this area, see e.g. [WZ91], [CS93] and [KT961 and the references therein. We consider here a result of Cegrell and Sadullaev, Theorem B below, and a result of Kolodziej and Thornbiorson, Theorem C below.
Theorem B. ([CS93, Theorem 3.1, p. 821) Let R be a domain in Rm+", m , n 2 2. Let u : R + E% be such that (a) for each y E
R" the function
is subharmonic, (b) for each x E Rm the function
is harmonic, (c) there is a nonnegative function p E Lfo,(R) such that -p 5 u. Then u is subharmonic. Cegrell and Sadullaev use Poisson modification in their proof. In [Ri062]we give a new proof, which avoids the use of Poisson modification, and is based simply on mean value operators and on Theorem 1 below. Cegrell and Sadullaev state also a corollary [CS93, Corollary, p. 821 to their result, just choosing p = 0. In Theorem 2 below we give a similar counterpart to
Cegrell's and Sadullaev's Corollary for quasi-nearly subharmonic functions. Again our result is very general. Kolodziej and Thorbiornson gave the following result. Their proof uses, among other things, the above result of Cegrell and Sadullaev, see [CS93, proof of Theorem 3.2, p. 831. Theorem C. ([KT96,Theorem 1, p. 4631) Let Let u : R + R be such that
R
be a domain in Rm+", m , n 2 2.
158
(a) for each y E Rn the function
is subharmonic and C2, (b) for each x E Rm the function
is harmonic. Then u is subharmonic and continuous. Below in Theorem 3, Theorem 4 and Corollary we give generalizations to the above result of Kolodziej and Thornbiornson. Instead of the standard Laplacians of C2 functions we use generalized Laplacians, that is the Blascke-Privalov operators. 2.
Definitions and notation
2.1. Our notation is rather standard, see e.g. [Ri89], [RiOO],[Ri061],[Ri062] and [He71]. Let D be a domain in the Euclidean space RN,N 2 2. A Lebesgue measurable function u : D -+ [0, +m] is quasi-nearly subharmonic, if u E C:,,,(D) and if there is a constant K = K ( N ,u,D ) > 0 such that
for any ball B N ( x , r ) c D . For the Lebesgue measure in R N , N 2 2, we use both m and m N . (Below m will be used also for the dimension of the Euclidean space Rm, but this will surely cause no confusion.) This function class of quasinearly subharmonic functions is natural, it has important and interesting properties and, at the same time, it is large, see e.g. [Pa94], [RiOO] (where they were called pseudosubharmonic functions), [PR05], [Ri061]and [Ri062].We recall here only that it includes, among others, nonnegative subharmonic functions, nonnegative nearly subharmonic functions (see e.g. [He71]),functions satisfying certain natural growth conditions, especially certain eigenfunctions, and polyharmonic functions. Also, any Lebesgue measurable function u : D t [m,M I , where 0 < m 5 M < +co,is quasinearly subharmonic. Constants will be denoted by C and K . They will be nonnegative and may vary from line to line. 2.2. As a counterpart to nonnegative harmonic functions, we recall the definition of Harnack functions, see [Vu82, p. 2591. A continuous function u : D + [0, fco) is a Harnack function, if there are constants X E ( 0 , l ) and C = C(X) 2 1 such that z Emax B (z ,AT)
u(z)
min ztB(x,Ar)
u(z)
159
whenever B ( z , r ) C D . It is well-known that for each compact set F in D there exists a smallest constant C ( F ) 2 C depending only on N , A, C and F such that for all u satisfying the above condition, maxu(z) 5 C ( F ) minu(z) ZEF
Z€F
One sees easily that Harnack functions are quasi-nearly subharmonic. Also the class of Harnack functions is very wide. It includes, among others, nonnegative harmonic functions as well as nonnegative solutions of some elliptic equations. Also, any continuous function u : D + [m,MI, where 0 < m M < +m, is a Harnack function. See [Vu82, pp. 259, 2631.
0 such that for all 0 5 s 5 t . 2C
See also [PR05, Lemma 1 and Remark 11. Recall that a function p : [0, +m) -+ [0, f m ) satisfies the &-condition, if there is a constant C = C(p) 2 1 such that 4 % ) 5 C p ( t )for all t E [0, +m). 3.
Separately subharmonic functions
3.1. The counterpart to Theorem A is:
Theorem 1. Let R be a domain in Rm+n, m,n Lebesgue measurable function such that (a) for each y E Rn the function
R(Y) 3 II: is quasi-nearly subharmonic, (b) for each II: E R" the function
R(z) 3 Y
t+
2 2.
Let u
:
R -+ [ O , + c o ) be a
4 5 , Y ) E [O, +m)
4 2 ,Y
) E [O, +m)
is quasi-nearly subharmonic, (c) there exists a non-constant permissible function $ : [0,+m) that $ 0 u E &(R).
+ [0,+m) such
Then u is quasi-nearly subharmonic. Proof. Since permissible functions are continuous, $ o u is measurable. To see that $ 0 u is locally bounded, take ( a ,b) E R and R > 0 such that Bm+n((u,b ) , R ) c R.
160
We show that there is a constant K = K ( m ,n, u,$, 0) > 0 such that
$(u(Zo,Yo)) 5
K
Rm+n
/'
$(.(Z,
d m m + n ( z ,Y )
((a,b),R)
2)
for all ( ~ 0 , y o E) B m ( u , x B n ( b , f).For this purpose choose (zo, yo) E B m ( a ,f )x Bn(b, arbitrarily. Then
2)
Using then properties of the permissible function $ one sees easily that also u is locally bounded above, thus locally integrable. Proceeding then as above, but now $ replaced with the identity mapping and choosing ( 2 0 ,yo) = ( a ,b ) , one sees that u satisfies the condition ( 2 ) , and is thus quasi-nearly subharmonic on R. 3.2. Remark. Unlike in Theorem A , the measurability assumption is now necessary. Indeed, with the aid of Sierpinski's nonmeasurable function, given e.g. in [Ru79, 7.9 (c), pp. 152-1531, one easily constructs a nonmeasurable, separately quasi-nearly subharmonic function u : C2 -+ [ I , 21. Indeed, let Q = (0) x Q x (0) c R x R2 x R, where Q c R2 is the set of Sierpinski, see [Ru79, pp. 152-1531. Then the function ~ ( 2 12 ,2 ) = ~ ( 2 1 y1,22, , y2) := ~ ~ ( ~ 1 ~ 12 is2 clearly ) nonmeasurable, but still separately quasi-nearly subharmonic.
+
4.
The result of Cegrell and Sadullaev
Then a counterpart to Cegrell's and Sadullaev's Corollary of their result, Theorem B above, see [CS, Corollary, p. 821: Theorem 2. Let R be a domain in Rm+n, m, n 2 2. Let u : R t [0,+m) be such that
161
(a) for each y E Rn the function 3x
* 4x1 Y) E (0,-too)
is quasi-nearly subharmonic, (b) for each x E R" the function
R(x) 3 Y
* 4 2 ,Y) E [O, fool
is a Harnack function. Then u is quasi-nearly subharmonic. Proof. It is well-known that u is Lebesgue measurable. Let ( a ,b) E 0 and R > 0 be such that Bm+n((u,b), R ) c 0. Choose (z0,yo) E Bm(u,f)x Bn(b,f ) arbitrarily. Since u(., yo) is quasi-nearly subharmonic, one has K 420,YO) 5 R U(Z,YO) dmrn(x). (TIrn
J'
B"(zo,
$1
On the other hand, since the functions u ( z ,.), x E Bm(u,+), are Harnack functions in Bn(b,$), there is a constant C = C ( n ,A, CX,R ) (here X and CXare the constants in 2.2) such that
for all x E Brn(u,f ) .See e.g. [ABROl, proof of 3.6, pp. 48-49]. Therefore U(Z0,YO)
C.K C u ( x , b )dm,(x) = 7
5R
" J '
(4)" B"
(4)"
(~o,?)
-
J'
U(x,b ) dmrn(2)
B" ( a ,
KJ' R"
u ( x ,b ) dm,(x)
< 00.
Bm(a,q)
Thus u is locally bounded above in Bm(u,%) x Bn(b,f),and therefore the result follows from Theorem 1 above.
5. The result of Kolodziej and Thornbiornson 5.1. In our generalization to the cited result of Kolodziej and Thorbiornson, we use the generalized Laplacian, defined with the aid of the Blaschke-Privalov operators, see e.g. [Sa41], [Ru50], [Sh71] and [Sh78]. Let D be a domain in RN,N 2 2, and f : D + R,f E LiOc(D). We write
162
If A*f(z) = A,f(z), then write Af (z) := A*f ( x ) = A, f ( x ) . If f E C 2 ( D ) ,then
the standard Laplacian with respect to the variable z = (XI,.. . ,ZN).More generally, if z E D and f E t $ ( x ) ,i.e. f has an C1 total differential at z of order 2, then A f (z) equals with the pointwise Laplacian of A f at z, i.e. N
A . f ( x )=
c
Djjf(X).
j=1
Here
Djj f
represent a generalization of the usual
@, j
= 1 , . . . , N . See e.g. [CZ61,
p. 1721, [Sh71, p. 3691 and [Sh78, p. 291.
Recall that there are functions which are not C2 but for which the generalized Laplacian is nevertheless continuous. The following function gives a simple example:
f ( z )=
{
< 0,
-1,
when
0,
when X N = 0,
XN
when X N > 0. 1, If f is subharmonic on D , it follows from [Sa41, p. 4511 (see also [Ru50, Lemma 2.2, p. 2801) that A*f (z) = A,f(z) for almost all z E D . Below the following notation is used. Let R is a domain in Bm+n,m, n 2 2, and u : R -+ B.If y E Bn is such that the function
R(y)3 z
++
f ( z ):= u ( 2 , y ) E R
is in Cioc(R(y)), then we write A I u ( x , y ) := A*f(z), Al*u(z,y) := A*f(z), and
A I U ( Z , Y:=) Af(2). 5.2. Then a generalization to [KT96, Theorem 1, p. 4631:
Theorem 3. Let R be a domain in Bm+n,m, n 2 2. Let u : R t B be such that for each (z0,yo) E R there is T O > 0 such that Bm(zo,ro)x B n ( y o , r o ) c R and such that the following conditions are satisfied:
(a) for each y E B n ( y o ,T O ) the function Bm(Z0,To)
3z
I+
u(z,y) E R
is continuous and subharmonic in Bm ( xo ,T O ) , (b) for each z E Bm(zO,T O ) the function B"(Y0,To) 3
Y
*U(X,Y) E B
is continuous and harmonic in B n ( y o ,T O ) , (c) f o r each y E Bn(yo,ro) one has Al*u(z,y) < +cc for all z E B m ( z o , ~ g ) , possibly with the exception of a polar set in Brn(x0,T o ) ,
163
(d) there is a set H function
c B"(y0, T O ) , dense in Bn(yo, T O ) , Brn(xo,ro)3
2
such that for each y E H the
++ A ~ u ( z , YE) JR
is defined and continuous, (e) for each yo E Bn(yn,rO), for almost all 20 E B r n ( x 0 , r ~and ) , for all sequences xj E Brn(xo,r0),j = 1 , 2 , . . . , such that xj 4 530, the sequence A1,u(zj,yn) has a convergent subsequence, converging to Alru(2n, yo). Then u is subharmonic. Since the proof is too long to be presented here, we just refer t o [Ri062]. 5.3. Another variant of the above result is the following, where the assumption (e) is replaced with a certain "continuity" condition of w(.,.) in the second variable.
Theorem 4. Let R be a domain in E%rn+n, m, n 2 2. Let u : R 4 JR be such that > 0 such that Brn(xo,rn) x B n ( y o , r o ) c R and f o r each (x0,yo) R there is such that the following conditions are satisfied: (a) for each y E Bn(yO,rn) the function
Brn(zO,ro)3 z is continuous and subharmonic in B" (b) for each x E Brn(ll:O, T O ) the function
F+
u(z,y) E R
( 2 0 ,ro),
B"(Yo,rn) 3 Y
* ~ ( z , YE )IR
is continuous and harmonic in Bn(yo,T O ) , ( c ) for each y E B"(y0,ro) one has Al,u(x,y) < +co for all 11: E B'"'(zo,ro), possibly with the exception of a polar set in Brn(x0,ro), (d) there is a set H C B n ( y g , r o ) , dense in Bn(yo,rO), such that for each y E H the function Brn(ZO,To)
3
17:
k-+
AlU(Z,Y) E R
is defined and continuous, (e) for each 11: E Brn(x0,T O ) the function Bn(Yo,ro) 3 Y
++
.(.,Y)
:=
s
GBm(zO,rg)(x,Z)Al'ZL(Z,Y)dm(z) E IW
is continuous. Then u is subharmonic. For the proof we refer again to [Ri062]. 5.4. The assumptions of Theorems 3 and 4 above, especially the (e)-assumptions, are undoubtedly somewhat technical. However, just replacing Kolodziej's and
164
Thornbiornson’s C2 assumption of the functions u(., y) by the continuity requirement of the generalized Laplacians A,u(., y ) , we obtain the following concise corollary to Theorem 3:
Corollary. Let R be a domain in Rrn+”) m, n (a) for each y E R” the function O(Y) 3
IL:
is continuous and subharmonic, (b) for each II: E IW” the function
R(z) 3 Y is harmonic, (c) for each y E R” the function O(Y) 3
II:
-
2 2 . Let u : R
U(IL:,Y)
E
R be such that
R
u(II:,v)E
* AlU(II:,Y)
E
R
is defined and continuous. Then u is subharmonic.
References [ABROl] Axel, S., Bourdon, P., Ramey, W. “‘Harmonic Function Theory” , SpringerVerlag, New York, 2001 (Second Edition). [AG93] Armitage, D.H., Gardiner, S.J. “Conditions for separately subharmonic functions to be subharmonic”, Pot. Anal., 2 (1993), 255-261. [Ar66] Arsove, M.G. “On subharmonicity of doubly subharmonic functions”, Proc. Amer. Math. Soc., 17 (1966), 622-626. [Av61] Avanissian, V. “Fonctions plurisousharmoniques et fonctions doublement sousharmoniques”, Ann. Sci. Cole Norm. Sup., 78 (1961), 101-161. [CZ61] Calderon, A.P., Zygmund, A. “Local properties of solutions of elliptic partial differential equations”, Studia Math., 20 (1961), 171-225. [CS93] Cegrell, U., Sadullaev, A. “Separately subharmonic functions”, Uzbek. Math. J., 1 (1993), 78-83. [He711 Herv, M. “Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces”, Lecture Notes in Mathematics 198, Springer-Verlag, Berlin, 1971. [KT961 Kolodziej, S., Thorbiornson, J. “Separately harmonic and subharmonic functions”, Pot. Anal., 5 (1996), 463-466. [Le45] Lelong, P. “Les fonctions plurisousharmoniques”, Ann. Sci. Cole Norm. Sup., 62 (1945), 301-328. [Le61] Lelong, P. “Fonctions plurisousharmoniques et fonctions analytiques de variables relles”, Ann. Inst. Fourier, Grenoble, 11 (1961), 515-562. [Le69] Lelong, P. “Plurisubharmonic Functions and Positive Differential Forms”, Gordon and Breach, London, 1969. [Pa941 PavloviC, M. “On subharmonic behavior and oscillation of functions on balls in R””, Publ. Inst. Math. (Beograd), 55 (69) (1994), 18-22. [PRO51 PavloviC, M., Riihentaus, J. “Classes of quasi-nearly subharmonic functions”, preprint, 2005.
165
[Ri89] Riihentaus, J. “On a theorem of Avanissian-Arsove”, Expo. Math., 7 (1989), 69-72. [RiOO] Riihentaus, J. “Subharmonic functions: non-tangential and tangential boundary behavior” in: Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA’99), Proceedings of the Syote Conference 1999, Mustonen, V., RAkosnik, J. (eds.), Math. Inst., Czech Acad. Science, Praha, 2000, pp. 229-238 (ISBN 80-85823-42-X). [Ri061] Riihentaus, J. “A weighted boundary limit result for subharmonic functio~is”, Adv. Algebra and Analysis, 1 (2006), 27-38. [Ri062] Riihentaus, J. “Separately subharmonic functions”, manuscript, 2006. [Ru50] Rudin, W. “Integral representation of continuous functions”, Trans. Amer. Math. SOC.,68 (1950), 278-286. [Ru79] Rudin, W. ReaZ and Complex Analysis, Tata McGraw-Hill, New Delhi, 1979. [Sa41] Saks, S. “On the operators of Blaschke and Privaloff for subharmonic functions”, Rec. Math. (Mat. Sbornik), 9 (51) (1941), 451-455. [Sh71] Shapiro, V.L. “Removable sets for pointwise subharmonic functions”, Trans. Amer. Math. Soc., 159 (1971), 369-380. [Sh78] Shapiro, V.L. “Subharmonic functions and Hausdorff measure”, J. Diff. Eq., 27 (1978), 28-45. [Vu82] Vuorinen, M. “On the Harnack constant and the boundary behavior of Harnack functions”, Ann. Acad. Fenn., Ser. A I, Math., 7 (1982), 259-277. [Wi88] Wiegerinck, J. “Separately subharmonic functions need not be subharmonic”, Proc. Amer. Math. SOC.,104 (1988), 770-771. [WZ91] Wiegerinck, J., Zeinstra, R. “Separately subharmonic functions: when they are subharmonic” in: Proceedings of Symposia in Pure Mathematics, vol. 52, part 1, Eric Bedford, John P. D’Angelo, Robert E. Greene, Steven G. Krantz (eds.), Amer. Math. SOC.,Providence, Rhode Island, 1991, pp. 245-249.
166
HARMONIC COMMUTATIVE BANACH ALGEBRAS AND SPATIAL POTENTIAL FIELDS S. A. PLAKSA Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Street, Kiev, Ukraine
[email protected] For investigation of equations with partial derivatives we develop a method analogous to the analytic function method in the complex plane. We have obtained expressions of solutions of elliptic equations degenerating on an axis via components of analytic functions taking values in a commutative associative Banach algebra. Keywords: Laplace equation; harmonic commutative Banach algebras; monogenic function; axial-symmetric potential; Stokes flow function.
1. Introduction Analytic function methods in the complex plane for plane potential fields inspire searching of analogous methods for spatial potential solenoid fields. The problem to construct such methods for spatial potential solenoid fields was posed by M.A. Lavrentyev [1,p. 2051). Besides, even independently of relation with applications in mathematical physics, for a long time a variety and an effectiveness of the analytic function methods stimulate developing of analogous methods for equations with partial derivatives. Apparently, Hamilton made the first attempts to construct an algebra associated with the three-dimensional Laplace equation
in that sense that components of differentiable functions taking values in this algebra satisfy Eq. (1). However, after constructing quaternion algebra he had not studied a problem about constructing any other algebra (see [ 2 ] ) . In the paper [3] I. Mel’nichenko considered the problem to construct commutative associative Banach algebra such that monogenic (i.e. differentiable in accordance with Gateaux) functions taking values in this algebra have components satisfying Eq. (1).It is obvious that the specified problem is appeared as an attempt to generalize the fundamental relation between the algebra of complex numbers and the two-dimensional Laplace equation. As it is well known, this relation means that, on the one hand, analytic functions of complex variable satisfy the two-dimensional
167
Laplace equation, and, on the other hand, plane harmonic functions conjugate with Cauchy-Riemann conditions are components of certain analytic function of complex variable. Because monogenic functions taking values in a commutative Banach algebra form a functional algebra, note that a relation between these functions and solutions of Eq. (1) is important for a development of effective methods for constructing mentioned solutions. It is quite natural that on such a way a quantity of fulfilled operations will be minimal in an algebra of third rank. But in the paper [3] it is established that there does not exist commutative associative algebra of third rank with the main unit over the field of real numbers in which monogenic functions would satisfy Eq. (1).At the same time, for commutative associative algebras of third rank over the field of complex numbers in the papers [3, 4)I. Mel’nichenko developed a method for extracting bases such that hypercomplex monogenic functions constructed in these bases have components satisfying Eq. (1). However, it is impossible to obtain all solutions of Eq. (1)in the form of components of monogenic functions taking values in commutative algebras of third rank that were constructed in the papers [3,4]. In particular, for each mentioned algebra there exist spherical functions which are not components of specified hypercomplex monogenic functions. Below, we consider an infinite-dimensional commutative Banach algebra F over the field of real numbers and establish that any spherical function is a component of some monogenic function taking values in this algebra. Thus, monogenic functions taking values in IF form the widest of known functional algebras associated with Eq. (1). 2. A problem about extracting harmonic triad of vectors Let A be a commutative associative Banach algebra over the field of real numbers R with the basis { e k } t = l , 3 5 n I 00. Consider in A the linear subspace Em generated by vectors e l , e 2 , . . . , e m , where m 5 n. Let G be a domain in Em. We say that a function CJ : G -+ A is monogenic in the domain G if Q, is differentiable in accordance with Gateaux in every point of G, i.e. for every [ E G there exists an element (a’([) E A such that lim E-o+o
[a([ + ~
h-)a([)]
& C 1
= ha’([)
‘dh E E m .
For a domain Q of the three-dimensional space R3 consider the domain Qc := {[ = z e l ye2 ze3 : (z, y, z ) E Q} c E3 which are congruent to Q . Note that if there exists a twice differentiable in accordance with Gateaux function @ : Qc 4 A which satisfies Eq. (1) and the inequality a”(1'
-
1 2
dUk+2(x,
Y,'),
dX
k = 4, 5 , . . ,
,
be satisfied in Q , and that the following relations be fulfilled:
The proof of Theorem 2.1 is similar to the proof of the corresponding classical theorem in the theory of analytic functions of complex variables. Note that the conditions (4) are similar by nature to the Cauchy-Riemann conditions for monogenic functions of complex variables. It is clear that if the Gateaux derivative @' of monogenic function : Qc + F,in turn, is monogenic function in the domain Qc, then all components Uk of expansion (3) satisfy Eq. (1) in Q in consequence of condition (2). At the same time, the following statement is true even independently of relation between solutions of the system of equations (4) and monogenic functions.
Theorem 3.2. If the functions uk : Q R have continuous second-order partial derivatives in the domain Q and satisfies the conditions (d), then they satisfies Eq. ( I ) in Q . --f
170
Note that the algebra F is isomorphic to the algebra F of absolutely convergent trigonometric Fourier series
-
with real coefficients ao, u k , bk and the norm
11g11F :=
la01 +
-
C ( l a k l + 1bk.i). In k=l
this case, we have the isomorphism e2k-1 i"' cos ( k - 1)r,e2k ik sin k r between basic elements. Let us write the expansion of a power function of the variable E = xe1+ ye2 + z e 3 in the basis {ek}r=l, using spherical coordinates p, 8, q5 which have the following relations with x,y, z :
z = psinOcosq5. (7) In view of the isomorphism of the algebras IF and F, the construction of expansions
x
= pcos19,
y = psinesinq5,
of this sort is reduced to the determination of relevant Fourier coefficients. So we have
sin mq5 ezm,
+ 2 5 n! m=l ( n m)!
+
+ cos mq5 ~ w + I ) ) ,
(8)
where n is a positive integer, P, and P," are Legendre polynomials and associated Legendre polynomials, respectively, namely:
+
1 linearly independent spherical functions of the n-th power are Thus, 2n components of the expansion (8) of the function 5". Using the expansion (8) and rules of multiplication for basic elements of the algebra IF, it is easy to prove the following statement.
Theorem 3.3. Every spherical function
where a,,~,an,ml bn,m E R,is the first component of expansion of the monogenic function
+
in the basis { e k } r ? l , where := x e l ye2 with the spherical coordinates p, O,q5.
+ ze3, and x,y, z
have the relations (7)
171
4. Monogenic functions associated with axial-symmetric potential
fields
c uke2k-l r x
Now, let us consider a subalgebra
:= { u =
00
: Uk
E R,
lUkl
< a}
k=l of the algebra F.In the paper 151 I. Mel'nichenko offered the algebra W for describing spatial axial-symmetric potential fields. A spatial potential solenoid field symmetric with respect to the axis O x is described in meridian plane xOr in terms of the axial-symmetric potential p and the Stokes flow function $J satisfying the following system of equations: k=l
+
where r2 = y2 2'. As in the papers [6, 71, consider a comlexification IH[c := W @ i H
= { c = a + ib
:
M
a , b E W} of the algebra IH[ such that the norm of element g :=
cke2k-l E wc is k=l
c Ickl. Consider the set c4
given by means the equality 11g(/w, :=
10:= { g E Hc :
k=l 03
00
E(-l)'"(Rec2k--l
-
-
k=l
+
Imczk) = 0, ~ ( - l ) ' ( R e c ~ k Imczk-1) = 0} which is a k=l
maximum ideal of the algebra IHIc. Let f i , : MI@ C be the linear functional such that 10is its kernel. Consider the Cartesian plane p := {< = zel re3 : z, r E R}. For a domain D c R2 we consider the domains D , := { z = z f ir : ( z , r ) E D } c C and Dc := {[ = x e l re3 : ( x ,r ) E D } c y which are congruent to the domain D . Wc the Let A be the linear operator which assigns to every function : DC function F : D , --f C by the formula F ( z ) := f i , ( @ ( C ) ) , where z = x ir and = zel reg. It is easy to prove that if CD is a monogenic function in D c , then F is an analytic function in D,. In the paper [6] we established necessary and sufficient conditions for monogenety of function @ : Dg + We in the form similar to (4) - (6). We established also relations between monogenic functions taking values in the algebra IHIc and solutions of the system (9) in so-called proper domains. We call D, a proper domain in C, provided that for every z E D , with Im z # 0 the domain D , contains the segment connecting points z and 2. In this case DC is also called a proper domain in p .
+
-
+
0.
Acknowledgments
The author would like to thank the foundation Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, (No. 17540163). References 1. I. N. Baker and P. J. Rippon, Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser. A 1 Math., 9(1984), 47-77. 2. R. L. Devaney, Complex dynamics and entire functions, in Complex Dynamical Systems, Proceeding of Symposia in Applied mathematics 4 9 (American Mathematical Society, Providence, 1994), 181-206. 3. S. Morosawa, Fatou components whose boundaries have a common curve, Fund. Math. 183(2004), 47-57. 4. S . Morosawa, Y . Nishimura, M. Taniguchi, and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, (Cambridge University Press 2000).
178
O N POTENTIAL THEORY ASSOCIATED T O A COUPLED PDE ALLAMI BENYAICHE Uniuersite' Ibn Tofail, Faculte' de Sciences, B.P: 133, Ke'nitra-Morocco In this paper, we give some results concerning the Martin boundary and the restricted mean value property for harmonic functions associated with a harmonic structure given by a coupled partial differential equations. In particular, we obtain such results for biharmonic functions (i.e: A2cp = 0) and for A2cp = cp equations.
Keywords: Martin boundary, mean value property, biharmonic functions, coupled partial differential equations.
1. Introduction
Let D be a domain in IRd, d 2 1 and let Li; i=1,2, be two second order elliptic differential operators on D leading to harmonic spaces ( D ,X L i ) with Green functions Gi (see"). Moreover, we assume that every ball B c B c D is a Li-regular set. Throughout this paper we consider two positive Radon measures p1 and p2 such that K g = S,Gi(.,y)pi(dy) is a bounded continuous real function on D ; i=1,2, . 11 K g I(m< 1. We consider the system: and 11 K g
Note that if U is a relatively compact open subset of D , p1 = Ad, where Ad is the Lebesgue measure, pa = 0, and Ll = La = A, then we obtain the classical biharmonic case on U . In the case when p1 = pa = Ad, and A d ( D )< 00,we obtain equations of A2'p = 'p type. In this work, we give some results concerning the Martin boundary and the restricted mean value property for harmonic functions with respect to the balayage space given by ( S ) .The interested reader can see2i3for more results. Let us note that the notion of a balayage space defined by J. Bliedtner and W. Hansen in4 , is more general than that of a P-harmonic space. It covers harmonic structures given by elliptic or parabolic partial differential equations, Riesz potentials and biharmonic equations (which are a particular case of this work). In the biharmonic case, a similar study can be done using couples of functions as presented in1l51l2 .
179
2. Notations and preliminaries
For j=1, 2, let X j = D x {j}, and let X = X I UXz. Moreover, let i j and mappings defined by
~j
the
We denote also by 7r the mapping from X to D such that 7r IxJ= T , Let Uo be the set of all balls B such that B c B c D , U, be the image of UOby i,, j=1,2 and U = U1U U z .
Definition 2.1. Let v be a measurable function on X. For U E U l , we define a kernel Su by:
suv = (H:l(u)(w 0 ill) 0
+ (K:;(.)(w
0 i2)) 0 T l .
For U E U 2 , we define a kernel SU by:
Suv = (H:2(u)(w Where
0 i2)) 0 7r2
+ (K:z(q (v 0 il)) 0 ~
2
.
j=1,2, denote harmonic kernels associated with ( D ,E L , ) and
K : : ( & 4 = SGrl'U'(.,s)w(Y)~,(dli) , i = 1 , 2 . Where w is a measurable function on D and Gr"u) is the Green function associated with the operator L, on 7riT2(U). Let G,, j = 1 , 2 , be the Green kernel associated with L, on D . The family of kernels (Su)uEu yields a balayage space on X as defined in4)' . Let * ' R ( X ) denote the set of all hyperharmonic functions on X , i.e.
* N ( X ):= { w E B ( X ) ; v i s 1.s.c and suv 5 w
vu E U } .
Where B ( X ) denotes the set of all Bore1 functions on X . Let S ( X ) be the set of all superharmonic functions on X , i.e.
S ( X ) := {v E * N ( X ) ;(SUW)luE C ( U ) vu E U } , and let N ( X ) be the set of all harmonic functions on X :
N(X):= { h E S ( X ) ; S u h= h vu E U } . Denoting W := * N + ( X ) ,the space ( X ,W ) is a balayage space (See4>').
Theorem 2.1 (3). Let w be a function on X such that K Z ( v o ik); j # k ; j , k E (1, a } , i s a finite function. Then, the following properties are equivalent 1 . u i s harmonic on X , 2. v1 := v o il - K F (w o iz) and v2 := v o i2 - K g (v o i l ) are Ll-harmonic and L2-harmonic function on D , respectively.
180
Remark 2.1. (1) Note that if w is a positive harmonic function on X then K g ( w o i k ) , j # k ; j , k{1,2},isafinitefunction ~ (2). (2) If w E X(X), then the couple (w o i l , w o i2) is a solution of (S). (3) The above theorem still hold if we replace the "harmonicity" by the "hyperharmonicity" ( See3) +cc
fcc
In the following, we denote Q :=
C ( K g K g ) n(resp. T
:=
C (KZKg)n)
n=O
n=O
which coincides with ( I - K g K E ) - ' (resp. ( I - K E K g ) - l ) on Bb(D), where ( I - K g K g ) - ' (resp. ( I - K E K g ) - l ) is theinverseof the operator ( I - K g K g ) ( r e s p . ( I - K g K g ) ) on & ( D ) , and & ( D ) denotes the set of all bounded Borel measurable functions on D . We have the following equalities:
( K g K g ) T = T ( K g K g ) ,( K F K g ) T + I = T , K F Q = T K g and K g T = Q K g Remark 2.2. We note that if cp is a finite positive Borel measurable function on D such that KgKEcp is bounded, then Qcp < +co. 3. Martin boundary associated with (S)
Let us fix x0 E D and set for all x, y E D
g2(2,y) :=
{ ;;2(1.n,yj GZ(z,Y)
if x # 2 0 OT y # if 2 = y = 2 0 .
20
Let A1 = {g'(z, .), IC E D},Aa = {g2(z,), z E D } and A = A1 U A2. As in,6 we consider the Martin compactification D of D associated t o A. The boundary A = 5 \D of D is called the Martin boundary of D associated to (S). The function gk(x,.), k = 1 , 2 , x E D can be extended, on 5, to a continuous function denoted gk((z,.), 5 = 1 , 2 , II: E D as well. Definition 3.1. (1) A positive Lj-harmonic function h on D is called Lj-minimal if for any positive Lj-harmonic function u on D, u 5 h implies u = a.h with a factor (Y > 0; j = 1 , 2 (2) A positive harmonic function h on X is called minimal if for any positive harmonic function u on X, u 5 h implies u = ah with a factor cy > 0. Denote for j = 1, 2, Aj = {y E A : g j ( . , g ) i s Lj y E A, the function gj(.,y) is Lj-harmonic on D .
-
minimal}. Note that, for all
181
Theorem 3.1 (3). L e t u be a positive m i n i m a l h a r m o n i c f u n c t i o n defined o n X s u c h that t h e f u n c t i o n KgK:(w o ik), j # k , j , k E {l,2}, i s bounded. T h e n , there exist t w o real n u m b e r s a a n d ,B s u c h t h a t
v = QW or v = ,Bs, W h e r e w and s are t w o positive h a r m o n i c f u n c t i o n s defined o n X by :=
{
(Qgl(., Y)) 0 TI on XI, Y E Ai, (KEQg'(.,Y)) o r 2 on X z , Y E A,,
s :=
{
( Q K g g 2 ( . , y ) ) O ~ on I Xi, Y E ( T g 2 ( . , y ) )o r 2 on X2, Y E A2.
w and
A2,
We prove in3 that the set B := { h E %!'(X) : ( h o i l ) ( ~ ,+ ) ( h o i 2 ) ( ~ , ) = l},
2,
E
D.
is a compact base of the cone % ! + ( X ) Let . E ( B ) denote the set of all extreme points
of %!+(X)belonging to B (see6). Using theorem 3.1, we have
E(B) = &1(B)u E Z ( B ) Where
and
Theorem 3.2. If g j ( x , .); x E D , separates Aj, t h e n for a n y positive h a r m o n i c f u n c t i o n u o n X s u c h t h a t t h e f u n c t i o n K z K E (u o i k ) ; j # k j , k E {1,2} i s bounded, there exist t w o u n i q u e measures v1 a n d v2 supported respectively by A, and A2 s u c h t h a t u c a n be represented o n XI by
and o n X2 by:
Proof. If v = 0, we have v1 = v2 = 0. If v # 0, we may assume without loss of generality that u E B . Consider the mapping:
.:{
A,u A, Y
* .(Y)
4
&(B)
182
Where Q(y) is defined by If y E A, :
Q(y) :=
( Q K g g 2 ( . , y ) )0 ni on XI (Tg2(.,y))on2 on X2
The mapping XD is bijective because g l ( x , .) and g2(x,.) separate A, and A2 respectively. P! and its inverse Q-l are continuous because g' and g2 are continuous on A x D . Then there exist, from Choquet's representation (11), a unique measure u supported by A,U A2 such that:
V U E B ,U
=
J'
*(Y)dU(Y).
AlUA2
Let u j , j = I, 2 be the restriction of the measure u on on XI as
Aj.Then, w may be written
and on X z as
4. Restricted mean value property
Let D be a domain in Rd, d 2 1, and r be a numerical positive function on D such that the closed ball B ( x , r ( x ) )of center x and radius r(x) is contained in D , for any x E D . A numerical function f on D is called r-median if A,,,(,,(f) = f(x), for any x E D . Where X is the Lebesgue measure on Rd and
A,,(%)
:=
( W ( ~ > r ( 4 ) ). X)Br( Zi , T ( % ) ) . A .
If f is a harmonic function on D, then f is r-median. Several authors are interested t o the converse question : If f is r-median, under what conditions, f is harmonic ? For a survey of the history we can see2,8 . In this section, we study the restricted mean value property for solutions of the system ( S ) for Lj = A; j = 1, 2 using the Cornea - Vesely approach Let r" be a positive function on X = X I U X2 such that B ( z ,r" o i j ( z ) )c D , for
183
any x E D.For a measurable function f defined on X . We say that f satisfy the restricted mean value property if ~ ~ , ? ( ~ )=( ff )( z ) .Where
1
F(Z)
PZ,F(Z)(f)
:= 7
S d - l ~ ~ ~ ( ~ ( ~ , ~ ) ) ( f ) (for Z ) zd s= ,
(x,j);j = 1,2;
2
ED
0
In other words a function f defined on X satisfies the restricted mean value property if and only if
for any 2 E D . Where H B ( ~ is , ~the ) classical harmonic kernel associated to the ball B ( x ,s ) .
Remark 4.1. In an obvious way; for any couple of functions ( f l , f 2 ) on D , we can define the restricted mean value property for such couple if we replace f o i j by f j and F by r in the previous equalities. In the following , we assume as in7 that r > 0 and defined on D . Moreover, there exists E > 0 and a lipschitzian function p on D , with Lipschitz constant 1, such that: p < d i d ( . , a D ) and E P 5 r 5 (1 - ~ ) p For . any measurable and positive function g on D , we consider the kernels
and
Remark 4.2. 1) The function F will be defined on each X, by r o 7rj 2 ) The restricted mean value property for a function f on X is equivalent to T f = f . 3) We have, Kgg(z) - N(Kgg)(z) f o r K g g < +co f o r K 2 g = +a.
184
Moreover, by definition of Rj, we have R j g 2 0, for any positive measurable function g . Hence the function K 2 g is a N - supermedian function. Therefore, from [7, theorem 1.2.(f)], K g g is either locally A-integrable or identically 00. Definition 4.1. A measurable positive function f on X is called T-supermedian if T ( f ) 5 f . A positive function on X such that f o i j ; j = 1 , 2 are locally Aintegrable is called T-invariant if T f = f . Theorem 4.1 (2). L e t s be a T - s u p e r m e d i a n f u n c t i o n o n X . T h e n s o ij i s either identically 03 or locally A-integrable. Definition 4.2. We consider, for z, y E D
4(z, Y) = ( A ( B ( z ,r(z))))-l. X B ( Z , T ( Z ) ) ( Y ) 4l(z,Y ) = 4(z, Y) V + l ( z ,Y ) =
Jn
dn(X,
04( 0. Theorem 1. Let ICO E R", yo E Rm and a = (50,yo) E Rn+m.Let D be a domain in Rn+m,a E D , and F : D + Rm be a continuous function of the Sobolev class
a
Wll,l,Jc(D).
Suppose that I(F'(x,y ) ( l 2 C a.e. in the domain D for some constant C > 0 and detA # 0 for all A E N,(F, a).Then there exist p > 0 and a unique continuous
188
mapping
G : B"(xo,p) + Rm, G(zo) = yo, such that F ( x , G(z)) = F ( z 0 ,yo) for all x E Bn(zo,p ) . Proof. In the proof we used the theorem on the the radius of injectivity for mappings with bounded distortion [ 5 , 81. Let R be a domain in R" and P : R + R is a function of the class Ll(R). We shall denote by Ph the mean function of the function P [ 5 , 61. Here h is the parameter (radius) of averaging. In the sequel we shall need the following lemma. Lemma 2. Let A be a mapping defined in R" and taking values in the set of real (m, k)-matrices. Suppose that the functions a; - the elements of the matrix A, are locally integrable in Rn. Let Ah be a matrix-valued function obtained by averaging the functions u i . Then IAh(z) I 5 I A l h ( z ) and IIAh(z) I( 5 llAllh(z) for every x E Rn. By condition detA # 0 for all A E N y ( F ,u ) and Clarke results [ 1, 21 there exists d , 0 < d < 1, and r > 0 such that the following conditions are satisfied: for every u E Rm, I u I = 1 there exist w E Rm, IwI = 1, such that the inequality 2 d , z = (z, y ) , holds almost everywhere on the neighbourhood D , = (w,IFy,(z)Iu) F' (.I Bn(xo,r)x B m ( y o , r ) c D . We have w = O-lu for some orthogonal matrix 0 E 2 d we have Mm,m. By the inequality (u, 0 mu) F'Y(Z) ~
where l ( z ) =
v,
IOF I/(.).
k =
-
I(z)u~ 5 kI(z),
(1)
Jm < 1. Using Lemma, from this we get
< kIh(Z), where z E D,p = Bn(xo,r/2) x B"(yo,r/2) and h < r/4. The function I ( z ) is greater than zero almost everywhere on the neighbourhood D,. Hence, I h ( z ) > 0 ( O F h & ( Z ) U- I h ( Z ) U I
everywhere on D,l2. By the inequality (1) we have estimate IFh&b)I5 Ih(.)(k
+ 1).
(2)
Moreover, it follows from (2) that Jh(z)(l - k ) 5 IFhI/(z)uI and we deduce that
Ih(z)(l - k) 5 I det(Fh y'(~))l'/~.
(3)
The inequality (3) shows that for any fixed z E Bn(zo,r/2) the mapping F h ( z ) is a local homeomorphism on Bm(yo,r/2)with respect to y E Bm(yo,r/2). Now we consider the case m 2 3 . From ( 2 ) and (3) we deduce that the distortion coefficient
(E)m,
of the mapping Fh(z) is bounded from above by which does not depend on the choice of h < r / 4 and x E B"(xo,r/2). Since any one of the mappings
189
F h ( z ) , h < r / 4 , z E B"(zo,r/2), is a local homeomorphism on B m ( y o , r / 2 ) ,by the theorem on the radius of injectivity for mappings with bounded distortion [ 5 , 81 a neighborhood Bm(yo,rl) ( T I < r/2) of the point yo can be found such that each mapping F h ( z ) , h < r/4, z E B n ( z o , r / 2 ) ,is a homeomorphism on B m ( y o , r l ) . From (3) and Q ( F h ( z ) ) 5 passing to the limit as h tends to zero, we conclude that the mapping F ( z ) , z E D,lz, is not constant and is a quasiconformal homeomorphism on B m ( y o , r l )for any fixed z E Bn(zo,r/2). Now we consider the case m 5 2. Fix arbitrary point z E B n ( z o , r / 2 ) .Let 21 = (z,y1), 2 2 = (z,y2) E D,p = Bn(zo,r/2) x Bm(yo,r/2) and y1 # YZ. Set
(s)m
U =
21 - 2 2
, zt
121 - z21
= z1
+ t(z2
- Z'),
t
E
[O, 11.
Then by (2) we obtain
Integrating we find
and
By the inequality
si Ih(zt)dt 2 5 we conclude that IFh(Z,Y2)
- Fh(z,Yl)I
2 Ply2 - Y l l ,
(4)
C(1-k)
where P = 7Y , Y S ,E~ B ~ " ( Y O , ~ / II:~ E ) ,B n ( z 0 , r / 2 ) .If IY then for arbitrary point y1 E dBm(yo,r/2) we have IY - Fh(z,Yl)I2 Hence
min
IFh(z,Y')
-
Fh(2,YO)I
-
IY - Fh(z,Yo)I
-
Fh(z,yo)l
r
> BT
r
-
B,
< B$, r
= B,.
IY - F h ( z , y ) I 2 attains at some point y* E Bm(yo,r/2) and
Y EB" ( Y o , T / 2 )
(IY - F h b , Y ) I 2 ) : , ( Y * ) = Fh:,(2,Y*)(Y - F h ( X , ! / * ) )= 0. Since det(Fhk(z,y*))# 0 we have Y = Fh(z,y*) and Bm(Fh(z,y0),P$) C F ( z , B m ( y o , 7 - / 2 ) )From . (4) passing to the limit as h tends to zero, we conclude that the mapping F ( z ) ,z E D,/Z, is a homeomorphism on B"(y0, r/2) for any fixed 5 E B"(zo,r/2). For sufficiently small 7-0 we have Bm(Yo,ro) c F ( z , B m ( y o , r l ) ) (Yo = F ( z 0 , y o ) for all z E Bn(zo,ro)in both case m 2 3 and m 5 2. Consider the mapping CI, : D,, + Rn+mdefined by (2, Y)
3 ( X ,Y) =
(2,
F ( 2 ,Y)),
(z, y) E D,, = Bn(zo,T O ) x B m ( y o , T O ) . From above the map and @(DTo)3 B"(z0,ro) x Bm(Yo,ro).
is a homeomorphism
190
The mapping @ had been defined such that its inverse map has the form y = g ( X ,Y). Next we observe that
(X, Y ) = W - W , Y ) )= ( X ,F ( X ,six, Y ) ) )
x =X , (5)
and F ( X ,g(X, Y))= Y . We put G ( x ) = g ( X ,Yo).By (5) we now find F ( x ,G ( x ) )= YO= F(zo,YO) and G(xo) = d Z o , Yo) = s ( X o , Y o ) = YO. Uniqueness of the map follows from the bijectivity of @. w References 1. Clarke F. H. O n the invers f u n c t i o n theorem, Pac. J. Math., Vol. 64, No 1, 1976, p. 97-102. 2. Clarke F. H. O p t i m i m i z a t i o n and n o n s m o o t h analysis, "Nauka", Moscow, 1988. (In Russian). 3. Hiriart-Urruty J. B. Tangent cones, generalized gradients a n d mathematical programm i n g in B a n a c h spaces, Math. Oper. Res., 4, 1979, p. 78-97. 4. Cristea M. A generalization of s o m e theorems of F.H. Clarke and B.H. Pourciau, Rev. Roumanie Math. Pures Appl., 5 0 , N 2, 2005, p. 137-152. 5. Reshetnyak Yu. G., Space mappings with bounded distortion, "Nauka" , Novosibirsk, 1982, 278 pp. (In Russian). 6. Sobolev S. L. S o m e applications of functional analysis in mathematical physics, "Nauka", Moscow, 1988. (In Russian). 7. Zhuravlev I. V. S u f i c i e n t conditions f o r local quasiconformality of mappings w i t h bounded distortion, Russian Acad. Sci. Sb. Math., Vol. 78, No. 2, 1994, p. 437-445. R . V. 8. Martio O., Rickman S., Vaisala J. Topological and m e t r i c properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, No 488, 1971, p. 1-31.
191
A RELATION AMONG RAMANUJAN’S INTEGRAL FORMULA, SHANNON’S SAMPLING THEOREM AND PLANA’S SUMMATION FORMULA KUNIO YOSHINO
Department of Mathematics, Sophia University, Tokyo, Japan
1. Introduction
It is well known that Shannon’s sampling theorem is very important in digital signal analysis. On the other hand there is a very curious so called Ramanujan’s Integral formula. But unfortunately Ramanujan’s Integral formula is not always correct. The aims of this paper are (1) to give an interpretation of Ramanujan’s Integral formula, (2) to prove Ramanujan’s Integral formula,
(3) t o clarify the relation between Ramanujan’s Integral formula and Shannon’s sampling theorem. We will do these things by using the theory of Fourier-Bore1 transform and Avanissian-Gay transform of analytic functionals. Finally we will reveal the meaning of Plana’s summation foumula in the theory of analytic functionals. Especially we will obtain the relation between Cauchy-Hilbert transform and Avanissian-Gay transform of analytic functionals by using Plana’s summation foumula.
2. Ramanujan’s Integral formula In this section we will consider following Ramanujan’s integral formula. Ramanujan’s integral f ~ r m u l a ( R a m a n u j a n [ ~ ] )
Now we list up some examples of Ramanujan’s integral formula.
192
Example 1.
f ( z )= 1
Following example 2 is equals to the definition of Euler 1 Example 2. f ('1 = r ( i + 2) Ua-l 0
r - function.
e --u d u = I ' ( a ) =
n=O
7l
r(i
-
u ) sin(7ru)
Following example 3 tells us the necessity of some additional conditions on function f ( z ) . Example 3. f ( z ) = sin(7r.z) In this case right hand side in (2-1) is -7r, but left hand side of (2-1) is 0 . So Ramanujan's integral formula is not valid in this example 3. This means that we need some conditions on f ( z ) in Ramanujan's integral formula.
3. Shannon's sampling theorem
The following formula is called Shannon's sampling theorem. It is very important in digital signal analysis. Theorem 1 (["I , [')I Suppose that entire function f ( z ) satisfies the following estimate : VE > 0,3C, > 0 s.t.
(3 - 1)
If(.)[
I: C, exp(klyl+ E I z I ) ,
(Vz = IC
+ i y E C).
If 0 5 k < rr, then we have
Right hand side is sometime called cardinal series(3).
4. Transformations of analytic functionals 4-1 Fourier Bore1 transform
Let K be a convex compact set in Cn.H ( K ) denotes the space of holomorphic functions defined near K . The element of the dual space H ' ( K ) is called analytic functional carried by K . Let T be an analytic functional carried by compact set K . We define Fourier-Bore1 transform T ( z ) of T as follows :
T ( z ) =< Tt,etz > T ( z ) is an entire function which satisfies the following estimate :
193
V€
> 0,3c, > 0 s.t. (4 - 1) I f ( z ) l 5 C, exp(HK(z)
+EIzI),
(vz = z
+ iy E Cn)
Ezp(Cn : K ) denotes the space of entire functions which satisfiy the estimate (4-1). Following theorem gives a characterization of Fourier-Bore1 image of analytic functionals. Theorem 2 (Polya-Martineau ["I) Fourier-Borel transform is a topological isomorphism fiom H ' ( K ) to Ezp(C" : K ) .
4-2 Avanissian-Gay transform
Now we introduce Avanissian-Gay transform G T ( w )of analytic functional T . It is called z-transform in digital signal analysis. In2 V. Avanissian and R. Gay introduced Avanissian-Gay transform for analytic functionals with compact carrier. Later, ins M. Morimoto and K. Yoshino extended Avanissian-Gay transform for analytic functional with non-compact carrier. If K is contained in {t E C : J l r n t ( < T } , then we can define Avanissian-Gay transform GT(w)of T E H ' ( K ) as follows:
G T ( w )=< Tt,
~
1 1 - wet
>.
We have the following proposition. Proposition 1 ( [ 2 ] , [4] , ['I) Suppose that compact convex set K is contained in {t E C! : llrn tl T E H'(K). (1) G T ( w )is holomorphic in C \ e z p ( - K ) . (2) G T ( w )= CF=oT ( n ) w n (3) limlwl+oo G T ( w )= 0 (4) (inversion formula (I))
< T } and
where C is a path surrounding K . (inversion formula (2))
where C' is a path surrounding ezp(-K).
For the details of Avanissian-Gay transform for analytic functional with noncompact carrier, we refer the reader to8 .
194
5. A proof of Ramanujan's integral formula
Let f (2) be a holomorphic function defined on a right half plane and satisfies the following estimate : VE > 0, VE' > 0, 3CE,,l > 0 s.t.
Theorem 3 Suppose that holomorphic function f ( z ) satisfies (5-1). If 0 5 k then Ramanujan's Integral formula always valids in the following sense.
< 7r,
where T ( z ) = f ( z ) . (Proof) There exists an analytic functional T carried by { t = u + iv : u 2 a , IvI 5 b } such that ? ( z ) = f ( z ) ( 7 ) . By inversion formula (4) in Proposition 1, we have
1 f ( z ) = T ( z )= 27ri ~
s,
GT(e-")e"'du.
+
+ + [a - E , cm).
where C = a - E + i [ - b - E , b + E ] U -i(b E ) + [a - E , cm) U i(b E) Deforming the integral path C to 7 r i (-co,cm)U -7ri (-co,cm),
+
GT(-e-")enizdu +
+
27ia
GT(-e-U)e-nizdu
Putting z = e-", we obtain Ramanujan's integral formula. 6. A relation between Ramanujan's Integral formula and Shannon's sampling theorem
In section 5, we derived Ramanujan's formula from the inversion formula. Now our conclusion is as follows : Conclusion Suppose that f ( z ) E Ez p ( C;K ) and K = [-ik, ilc]. I f O < k < 7 r , then we can derive Ramanujan's integral formula and Shannon's sampling theorem b y deforming the integral path in the inversion formula (4) in Proposition 1.
195
7. Plana's summation formula Following formula is known as Plana's summation formula. Plana's summation foumula [ 5 ] Suppose that K c [-a,a] f E Exp((C : K ) . If 0 5 b < 27r and IIm sI + b < 27r, then we have
+ i[-b,b]
and
For the details of Plana's summation formula, we refer the reader to [ 5 ] and ["]. By using Plana's summation formula, we can show the following theorem. Theorem 4. P u t K = [-a, a] + i [ - b , b]. Suppose that f ( z ) satisfies following conditions (1) and ( 2 ) . (1) f ( z ) E Ew(@.: K ) ( 2 ) f ( Z ) = 0, If 7r 5 b < 27r, then we have
-
f ( z ) = sin7rzS(z), where S E H ' ( [ - a , a] + i[7r - b, 7r + b ] ) (Proof) See
[lo] .
For another proofs of theorem 4, we refer the reader to3 and4 .
8. The meaning of Plana's summation formula in the theory of analytic functionals
Cauchy-Hilbert transform C H ( T ) ( w )of T E H ' ( K ) is defined by 1 C H ( T ) ( w )=< Tt, -> t-w
Suppose that compact convex set K is contained Proposition 2 (["I , ['I , ["I) in { t E C : Ilm tl < n}. Then we have
(1) C H ( T ) ( w )=
ice
p(~)e-~~dz,
196
where C is the contour surrouding K .
By proposition 2 we have
Since we have following isomorphism
( K c W, W :open) we can conclude that GT(e-t) - C H ( T ) ( t )E H ( W ) Now problem is
:
What i s GT(e-t) - C H ( T ) ( t )?
Plana’s summation formula gives an answer to this problem
GT(e-t) - C H ( T ) ( t )
where W = { t E
C : Ilrn tl < T )
This is the meaning of Plana’s summation formula in the theory of analytic functionals. References 1. M. Anderson : Topics in Complex Analysis, Springer Verlag(1996) 2. V. Avanissian and R. Gay : Sur une transformations des fonctionnle analytiques et ses applications aux fonctions entieres a plusieurs variables, Bull.Soc.Math.France 103(1975) 341-384 3. R.P. Boas : Entire Functions, Academic Press(1982) 4. C. Berenstein and R. Gay : Complex Analysis and Special Topics in Harmonic Analysis, Springer Verlag( 1995) 5. Erdelyi,Magnus, Oberhettinger and Tricomi : Higher Transcendental Functions Vol. I, Mcgrawhill( 1953) 6. G.H. Hardy : Ramanujan, Chelsea, New York(1978)
197
7. M. Morimoto : A n a l y t i c functionals w i t h non-compact carriers, Tokyo J.Math. l(1978) 72-103. 8. M. Morimoto and K. Yoshino : Uniqueness theorem f o r holomorphic f u n c t i o n s of exponential type, Hokkaido Math.J. 7(1978) 259-270. 9. K. Yoshino : Liouville type theorem f o r entire f u n c t i o n s of exponential type, Complex Variables, 5(1985) 21-51 10. K. Yoshino and M. Suwa : Plana’s s u m m a t i o n f o r m u l a for entire f u n c t i o n s of exponential type and its applications, preprint,
198
ASYMPTOTIC EXPANSIONS OF THE SOLUTIONS TO THE HEAT EQUATIONS WITH GENERALIZED FUNCTIONS INITIAL VALUE KUNIO YOSHINO and YASUYUKI OKA Department of Mathematics, Sophia University, Tokyo, 102-8554, Japan We will drive the asymptotic expansions of solutions of the heat equation with generalized functions initial data. Keywords: Heat Equation,Asymptotic expansion,Tempered Distributions, Distributions of exponential growth and Fourier Hyperfunctions
1. Introduction
In4 , T.Matsuzawa characterized tempered distributions as the initial value of the solutions to the heat equation. Theorem 1. For u f S', V ( X ,= ~) (u* E ) ( z , t )satisfies
U ( z ,t ) E C"(Rd x (0, ca)),
Moreover,
U ( z , t )+ u, (t
0) in S'(Rd),
We analyze (#) more precisely. Namely, we will drive the asymptotic expansions of solutions of the heat equation with generalized functions initial data in this paper. Main Theorem 1.Let U ( z , t )E C"(Rd x (0, ca))satisfy the following conditions:
199
(i) (& - A) U ( z ,t ) = 0 , (ii) 3C > 0, 3v 2 o and ~k 2 (z E Rd, 0 < t < I),
o s.t. Iu(z,
t)l 5 Ct-.(1+
Then U ( z ,t ) has the following asymptotic expansion :
i.e.
+. +
.. . where Az = 82, Corollary 1. For u E S', P u t U ( x , t ) = ( u * E ) ( z , t ) , then < u,cp >= lirnt+o J R d U ( X t)cp(z)dX('dY , E S). We obtain the similar consequences with regard to distributions of exponential growth and Fourier Hyperfunctions by using the results of Suwa5 and K.W.Kim et al3 respectively. 2. Preliminaries
We use the multi-index notations such as
8"
=
a,.1 ...a,.. ,
d 8. - ( j = 1,... , d ) . - dXj
'
Defination 1.We put E ( z , t )= ( 4 7 d - 2 e - S , (X E Rd , 0 < t < +w) , E ( x ,t ) is called heat kernel and have the following properties :
(+)E(z,t)=o,
( X d ,O
d2
whereA=-+...+dXGq
0 s.t. I < u,'p > I i CC,p,+k<mSUPzEwd(l + 1 ~ 1 2 ) k l @ ' p ( ~> )(V'p 1 E S(Rd)). 3. Asymptotic expansions of the solutions to the heat equations
with the tempered distributions initial data Theorem 2. Let U(z, t ) E Cm(Rd x (0, m)) satisfy the following conditions : (i) (& - A) U ( z ,t ) = 0 , (ii)3C > 0, 3v 2 0 and 3k 2 0 s.t. I U ( z , t ) ( 5 Ct-.(l+ (zOk, (z E Rd, 0 < t < l), Then U ( z ,t ) has the following asymptotic expansion :
U ( z ,t ) N
O0
k=O
tk k!
, ( u E S'(Rd) such that u = lim U ( z ,t ) ) .
- A tu
t +o
i.e.
where A, =
+ . . . + 82, .
Lemma 1. Let
'p
be in S ( R d ) and t
> 0. Then
{ l'(18)N8:'p(y
LdeCZ2za is in S(Rd).
+ a z 8 ) d6'
1
dz
201
Since 'p(y) E S(Rd),we have the following estimate, (1+ lY12)v;+"(Y)l
I c7,4.
So we obtain p;+P'p(y
+ J4tzO)I 5 C7,p(1+ Iy + J4tz0l2)-Y.
Thus,
Hence we have
This completes the proof of Lemma 1 . 0
Proof of Theorem 2 : By Matsuzawa's result4 , There exists u E S'(Rd) such that U ( z ,t ) = (u* E ) ( z ,t ) . For any cp E S ( R d ) ,
< U ( 5 , t ) , 'p >= < (u* E ) ( z ,t ) , 'p > = c u z l , E ( z , t )* ' p
>
202
By Taylor's formula,
x { ~ l ( l - d ) N a ~ p ( y + \ / l i Z H )dd
Then we obtain the following equality :
Therefore we have the following equality :
I
dz.
203
We obtain the following estimate by Proposition 1 and Lemma 1. For any n E
N,
This completes the proof of Theorem 2 4. Asymptotic expansions of the solutions to the heat equations with the distributions of exponential growth initial data We introduce Gel'fand-Shilov space S1(Rd) and recall its properties. Definition 4. (') We define S1(Rd) as follows :
and S~,A(%?') is defined as follows :
where (A +
8)" = (A1 +
(A, +
. (Ad+ $ ) a d .
The topology in S~,A(IW~) defined by the above semi-norms makes S ~ , A ( Rthe ~) F r k h e t space and ,!?,(ad) is LF-space. Proposition 2. (I)
For
E
E
IWd, we have the following estimate,
204
where C is some constant and
Proposition 3.(l) Let cp be a function in C"(Rd) equivalent .
. Then the following statements (i) and (ii) are
&.
where aj =
Proposition 4.(l) Let cp be a function in C"(Rd) . Then the following conditions (i) and (ii) are equivalent . (i) cp E SI,A(Rd). (ii) 3A E R$ , n E N~ , VP E Z$ , 3~~ 2 o s.t. ~ @ c p ( x ) 5 ~
Co exp
(-5
a>Ixj1)
, where
a; =
e(Aj+&)
'
j=1
Definition 5. Si (Rd) denotes the dual space of S1(EXd). Proposition 5. Let u be a linear form from S1 to @. Then the following statements (i) and (ii) are equivalent. (i) u E S:(Rd), (ii) VA, 3C > 0 s.t. I < u,cp > I 5 CllcpII~,~ , (Vcp E S~,A(J@)).
Remark : For the details of Si(EXd),we refer the reader to2 and.5
The following Theorem 3 is Main result in this section. Theorem 3. Let U ( z ,t ) E Cm(Rd x (0, m)) satisfy the following conditions : (i) (& - A) ~ ( zt ), = o , (ii) VE > o , 3 ~ 2, 0 , X , 2 0 s.t. l ~ ( x , t ) 5 I C,t-NceEJ+I, (0 < t < 1 , XERd).
Then U ( z ,t ) has the following asymptotic expansion :
- * G A 2 u , (u tk
U(x,t)
k=O
E
Si(Rd) such that u = lim U ( z , t ) ) . t-0
205
i.e.
where Az = a&
+ . . . + d&
.
We need the following Lemma 2 for the proof of Theorem 3. Lemma 2 . ( 6 ) Let cp be in S 1 , ~ ( R dand ) t > 0. Then
Proof of Theorem 3 : By Suwa’s result5, There exists u E S{(IRd)such that U ( z ,t ) = (ti * E ) ( z , t ) .For any cp E &(Rd),
< U ( X ,t ) , cp >= < (u* E ) ( z ,t ) , cp > =
< uly , E ( z , t )* cp >
By a similar calculation to Theorem 2, we have the following equality :
We obtain the following estimate by Proposition 5 and Lemma 4. For any n E N+1
(4.1) 5 Cy,N,A,nt2t
-K
= Cy,N,A,nt’
0 (t 4 0).
4
W,
206
This completes the proof of Theorem 5 0
Remark 1. We obtain the similar result for Fourier hyperfunctions in6 References 1. 1.M.Gel’fand and G.E.Shilov, Generalized Functions, Volume 2, Space of Fundamental and Generalized Functions, Academy of Sciences Moscow, U.S.S.R, (1958) . 2. M.Hasumi, Note on the n-dimensional tempered ultra-distributions, Tohoku.Math.J. 13,(1961),94-104. 3. K.W.Kim, S.Y.Chung and D.Kim,Fourier hyperfunctions as the boundary values of smooth solutions of the heat equation, 4. T.Matsuzawa, A calculus approach t o the hyperfunctions 111, Nagoya Math. J., 118 (1990), 133-153. 5. M.Suwa, Distributions of exponential growth with support in a proper convex cone, Publ, RIMS, Kyoto Univ. 40 (2004),no.2, 565-603. 6. K.Yoshino and Y.Oka, Asymptotic expansions of solutions to heat equations with Generalized Functions initial value, preprint.
207
ON THE EXISTENCE OF HARMONIC DIFFERENTIAL FORMS WITH PRESCRIBED SINGULARITIES EUGENIA MALINNIKOVA Department of Mathematics, Norwegian University of Science and Technology 7491, Trondheim, Norway eugeniamath.ntnu. no In this note we obtain analogues of the Mittag-Leffler and the Weierstrass theorems for harmonic differential forms on noncompact Riemannian manifolds. Keywords: Separation of singularities, harmonic differential forms, Mittag-Leffler theorem, Hodge decomposition
1. Introduction A differential q-form 4 , defined and smooth on an open subset of a smooth oriented Riemannian manifold, is called harmonic if
dq5=O
and
6q5=O,
where d is the exterior differential operator and 6 is its adjointl>' . Harmonic differential forms on Riemannian manifolds can be considered as a generalization of analytic functions of one complex variable or analytic differentials on Riemann surfaces3 . In this note we proof versions of the Mittag-Leffler and Weierstrass theorems for harmonic forms. Construction of harmonic forms with prescribed singularities turns out to be a classical problem, it was discussed for example in the book by B.Rodin and L.Sario4 . We will give another approach to this problem and prove a version of the MittagLeffler theorem in Section 3 . In Section 2 we formulate necessary and sufficient conditions for separation of singularities (in the sense of N.Aronszajn5). In several complex variable separation of singularities corresponds to the additive Cousin problem. This scheme of proofing Mittag-Leffler theorem was applied to harmonic functions on manifolds by T.Bagby and P.Blanchet' , and by P.Gauthier7 ; and to harmonic differential forms on open subsets of R" by the author* . In Sections 2 and 3 the results of Ref. 8 are generalized to forms on Riemannian manifolds. We give a new proof of Lemma 1 based on the Hodge decomposition. Further we formulate the result on separation on singularities for a given form (see Theorem a ) , the last corollary repeats our result for forms on
208
open subspace of Rn. Finally, Theorem 2 is used to obtain a version of the MittagLeffler theorem. In the last section a generalization of the Weierstrass theorem is given. In particular, we construct a harmonic form that vanishes on a prescribed discrete set of points, the form can be chosen to be both exact and coexact. One of the main tools we employ for solving both problems is approximation of harmonic forms by elementary ones (as in the classical theorem by Runge), we adjust the methods used in Ref.9. We use also an abstract result on simultaneous approximation and interpolation due to H.Yamabe.lo For harmonic and analytic functions simultaneous approximation and interpolation can be found in the works by L.Walsh and his coauthors. 2. Separation of singularities for harmonic differential forms
In this article M denotes a smooth (of class C") connected oriented noncompact Riemannian manifold of dimension n 2 3. We use the Hodge-de Rham decompositions for forms (and currents) on M T = HIT
+ H2T + H T
and refer the reader to de Rham's book' . Lemma 1. Let R be an open subset of M and let 1 5 p there exists p E C p l ( R ) such that 6dp = ha.
5 n. For any Q E CF(R)
Proof. Let { K j } j be an exhaustion of R by compact subsets such that R \ Kj has no connected components relatively compact in R. We take I/Jj E D(R) such that 0 5 $ j 5 1and I / J j = 1on a neighborhood of K j , I/Jo = 0. Then 1 = C3.($j-$j-l) = C j$ j , where $ j E D(R) and $ j = 0 on a neighborhood of Kj-1. We define t c j = Hl(q5jcu). Then ~j E C F ( M ) and tcj is exact. Further, 6Kj
= d H l ( 4 j a ) = 6($jf2
-
Hz($jCC) -
H ( $ j Q ) )= 6($jCX).
Now we consider ~j in a neighborhood of Kj-1 where 4j = 0. We claim that ~j can be uniformly on KjP1 approximated by exact in R forms that are coclosed on R. First, suppose that R is relatively compact in M . We will show that ~j can be approximated uniformly on Kj-1 by forms of the form H l ( c ) , where e n = 0. Let T be a current that is orthogonal to all such forms with T C Kj-1. So we have
for all c whose support does not intersect R. (We used that T n c = 0 to move H1,H2 above, see details in the book by de Rham.') Clearly H Z ( T ) is a harmonic form off Kj-1. Last identity implies that HZ(T) vanishes with all it's derivatives at each point of M \ But M \ 0 intersects all components of M \ Kj-1, thus H 2 ( T )= 0 on M \ Kj-1. So we have
a.
209
In general, let R = U l R l and each 01 is relatively compact in 0 , 0 2 1 cc RI+I. As we have seen ~j can be approximated on Kj-l by uj,l = H l ( c j , l )where (cj,l)nn,) = 0.We then approximate uj,l on 01 by a form H l ( c j , 2 ) with ( c ~ J n) 0 2 = 0. And continue the procedure. We get a sequence of forms uj,l that are exact and coclosed on R1. This sequence can be done convergent uniformly on compact subsets of R . Then it follows from the de Rham theorems that the limit form wj is exact and coclosed on R. It provides an'approximation to ~j on Kj-1,
zj(&j
The series w = - wj)is uniformly absolutely convergent on compact subsets of R , for each 1 the sum C j , l ( ~-j w j ) is a harmonic form on a neighborhood of KLand we have
j=1
j=1
on a neighborhood of Kl. Thus Sw = ha. Our last step is to show that w is an exact form on R. For any finite p-cycle in R we get
r
We applied dominated convergence theorem, see ( l ) ,and then used that forms /c.j and w j are exact in 0. Then, by the de Rham theorem, w = d/3 for some
P
E
cpl(fl).
We denote by H q ( 0 )the singular homology spaces of an open subset R of M (we consider smooth singular q-chains with real coefficients, see textbook 2 for details). There is a natural mapping sq : ~
~n R2) ( + Hq(R1) 0 ~@ H q ( o 2 ) ,
defined by s,(r) = (r,-F). Let R be an open subset of M . We denote by Hq(R) the (de Rham) q-cohomology spaces, Hq(R) is the factor space of all (smooth) closed q-forms on R over the subspace of exact q-forms. De Rham cohomology spaces are dual to singular homology spaces. The dual operator to the operator sq is defined as
aq : Hq(Rl)e Hq(R2)+ H q ( R 1 n R2), where a,([wl], [wz]) = [(wl -w2)1a2"a2], here we mean that the difference w1 -w2 is restricted to the intersection of the open sets and then the equivalence class is taken, we omit the restriction sign in what follows. It is easy to see that ker(s,) = (0) if and only if aq is surjective. Now, using Lemma 1, we can prove the theorem on separation of singularities. Theorem 2. Let R1,Rz be two open subsets of M , and R = 01 n 0 2 . Then a
210
harmonic q-form w E C r ( 0 ) can be written as 21 = w1 - 212, where v j E C r ( 0 j )are harmonic q-forms if and only if [w]E Im(a,) and [*w]E Im(anpq). Proof. The necessity of the condition is clear, if w = 211 - v2, then clearly [w]= a,([vl], [m]) and = an-q([wl,[w]). Suppose now that [w]= a q ( [ v l ][w~]). , We have w = u1 - 212 d h in R. Moreover, using Lemma 1, we can find f j such that 6dfj = 6wj in R j then w j = uj - dfj is both closed and coclosed and w = w1 - w2 dg. Now [*(w1 - W Z ) ]= U ~ - ~ ( [ * W [*wz]) ~ ] , and *dg = *w - * ( w 1 - w2) E 1m(unuq).It is enough to show that dg can be written as the difference of two forms each harmonic on the corresponding domain R j . We can always write g = g1 - g2, where gj E Coo(Rj), then *dg = *dgl - *dg2. Since *dg E Im(anPq),we have *dg = 171 - 172 for some forms ql and 172 closed in 01 and 0 2 respectively. (Really we have *dg = s1 - s2 dr = (s1 d r l ) - (s2 d r a ) , where T = r1 - r2 is a Coo-decomposition.) Now we consider
I.*[
+
+
+
K = {
*171 *172 -
dg1 on 492 on
+
+
01 0 2
Clearly, r; is well-defined on R1 U 0 2 . Using Lemma I once again, we find r E C p l such that 6 d r = 6 ~Then . dg = (dgl + d r ) - (dg2 d r ) is the desired decomposition.
+
Corollary 3. A harmonic q-form w E C r ( R ) can be written as 'u = u1 - 212, where v j E C r ( R j ) are harmonic q-forms if and only if there exist q-forms ul,u2 such that u = u1 - u2,uj E C r ( 0 j ) , and
are exact forms on
01 u R2.
Proof. The corollary follows immediately from the theorem and the exactness of the Mayer-Vietoris exact sequence of the pair: ...H P ( R ~ CB ) H P ( R ~ )aP, H P ( R ~n 0,)
%~
u
p + l ( o ~0,) ....
We use that Im(u,) = Ker(dP) and apply this for p = q , n - q. The statement of the corollary does not depend on the choice of the decomposition 'u = u1 - u2. Detailed discussion of the mapping d p can be found in the book by R.Bott and L.Tu." Similar to the result' for harmonic forms on the open subsets of n , we formulate necessary and sufficient condition for separation of singularities, using mappings sp: Corollary 4. Let R l , R 2 be two open subsets of M , and R = 01 n 0 2 . Then every harmonic q-form 'u E C r ( R ) can be written as 'u = w1 - 212, where uj E C r ( R j ) are harmonic q-forms if and only if ker sq = (0) and ker s , - ~ = (0).
211
3. Mittag
-
Leffler theorem for harmonic differential forms
As we have seen earlier,8 even for the case of the Euclidean space there are some obstacles for construction of harmonic forms with prescribed (massive) singularities. On the manifold we will divide the problem into two parts. First, given a compact set e and a form u harmonic on w \ e for some neighborhood of w of el we want to construct a form v harmonic on M \ e that has same singularities on el i.e. such that v - u has harmonic extension to a neighborhood of e. This problem can be reformulated as separation of singularities problem. Let 01 = M \ e, 0 2 = w . We have a harmonic form on 01 n 0 2 and want to decompose it into the sum of two forms, one harmonic in 01 and another harmonic in 0 2 . As we have seen in the previous section it can be done if and only if sq
: %(w
\ e ) + Hq(M \ e ) @ H P ( W )
has trivial kernel for q = p , n - p . For example, a sufficient condition is
\ e ) -+
j,+l %+1(M
%+l(M)
is surjective for q = p , n - p . (To see it one can write the exact homology sequence of the pair (111\ e, w ) . ) The last condition is satisfied if e is a subset of a coordinate chat of M . Another part of the classical Mittag Leffler theorem addresses the question of sewing together a sequence of singularities to one harmonic form. Here the situation is exactly the same as in the theory of functions of one complex variable. Theorem 5. Let e = U j e j be the union of a sequence of compact pairwise disjoint subsets of M . Assume that each compact subset of M intersects only finitely many sets of the sequence. Let hj be a sequence of p-forms such that hj is defined and harmonic on M \ ej. Then there exists a q-form h harmonic on M \ e such that h - hj has harmonic extension in a neighborhood of ej. ~
Proof. Let oj be a relatively compact neighborhood of e j such that aj is a sequence of pairwise disjoint subsets of M and each compact subset of M intersects only finitely many sets of the sequence. Further for each j we chose an open set w j such that ej c w j CC a j . We define v on w = U j w j by u = hj on w j . We want to show that v can be written as the difference of a form harmonic on M \ e and a form harmonic on w . Using the corollary proved after Theorem 2, for each j we get hj = uj - r j , where uj E Cw(M \ e j ) , rj E C m ( w j ) and aj =
d u j on M d r j on w j
\ ej
and ,8j =
d * uj on M d * rj on w j
\ ej
is exact and coexact on M . Moreover, we can choose uj with compact support in a j . Then v = C juj r . and both (u = aj and ,l3 = pj are exact. To see it, we 3 3 apply the de Rham theorem, all periods of a and /? vanish.
c.
212
4. Weierstrass's theorem for harmonic differential forms
We will show that there exists a harmonic form that interpolates given sequence of germs {uj} at a discrete set of points { p j } on M . We say that a smooth differential form 4 has zero of order m at point p if in some coordinate chart all coefficients of q5 vanish with all their partial derivatives up to order m. The following result is simple after it is formulated [Yamabelo]Let Y be a dense linear subspace of the normed linear space X . Then for any L1, ..., L , E X * , any x E X and every E > 0 there exists y E Y such that 1/11: - yI( < E and &(x) =
Lz(y),i= 1,...,71. We will use this result for simultaneous harmonic approximation and interpolation. It implies the following Lemma 6. Let K be a compact subset of M such that M \K has no relatively compact connected components. Let further { p l , . . . , p ~ }be a finite sequence of points of KO and { m l ,..., m ~be} a sequence of positive integers. If u is a q-form that is both exact and coexact on an open set containing K and if E > 0, then there exists a q-form U exact and coexact on M and such that Iu - UI < E on K , U - u has zero of order mj at point p j , j = 1,...,N . Proof. Using coordinate chats we can rewrite the condition U - u has zero of order mj at p j as finite number of conditions Lj,s(U) = Lj+(u),where Lj+ are bounded linear functionals on the space H,C>"(K) of forms exact and coexact in a neighborhood of K . We equip this space with the uniform norm. Applying the methods we used in Section 1, we see that q-forms that are exact and coexact on M are dense in H,C,e(K).Thus the statement of the lemma follows from the claim formulated above.
Theoem 7. Let { p j } j be a discrete sequence of points in M and for each j let u j be a q-form harmonic near p j . Then there exists a harmonic q-form u on M such that u - u j has zero of order mj at p j for each j . Proof. There exists an exhaustion of M by compact subsets, M = U z 1 K j such that M \ Kj has no relatively compact in connected components. For simplicity (and Poincark's lemma without loss of generality) we may assume that p j E K: \ implies that u1 is exact and coexact near p l . Then, applying Lemma ??, we get a q-form U1 that is exact and coexact in M and such that Ul - u1 has zero of order ml at p l . Suppose that we have constructed a q-forms Uj-1 on M that is exact and coexact, and such that Uj-1 - ul has zero of order ml at pl for 1 = 1,...,j - 1. Since p j $2Kj-1 we have d j = ( p j , KjPl) > 0. Let Ej = KjP1U B ( p j , r j ) , where rj < d j / 2 and rj is such that uj is harmonic in B ( p j , 2 r j ) . We consider the following form near Ej
213
Then wj is exact and coexact near E j , there are no compact components of M \ Ej. Then using the lemma we can find a q-form Uj exact and coexact on M and such that U j - wj vanishes to the order s1 at pl for 1 = 1,...,j. In addition we can have llUj - w j l l ~ ,< 6 2 - j . Then
Thus sequence { U j } converges uniformly on compact subsets of M to some harmonic q-form that solves the interpolation problem. Acknowledgments This note is based on the talk given by the author at ICCAPT 2006 and it is my pleasure to thank the organizers of the conference. Thanks go also to A.Nicolau and J.Ortega-Cerda, whose question initiated author’s interest in the interpolation problem. The author was supported by the Research Council of Norway, project no 160192/V30, ”PDE and Harmonic Analysis” References 1. G. de Rham, Differentiable manifolds. Forms, currents, harmonic forms. (SpringerVerlag, 1984). 2. W. F. Warner, Foundations of Differential Manifolds and Lie Groups, (SpringerVerlag, 1983). 3. L. Ahlfors and L. Sario, Riemann surfaces, (Princeton University Press, 1960). 4. B. Rodin and L. Sario, Prznczpal functions, (D.Van Nostrand company, inc., 1968). 5. N. Aronszajn, Acta Math. 65 (1935), 1-156. 6. T. Bagby and P. Blanchet, J . Anal. Math. 62 (1994), 47-76. 7. P. M. Gauthier, Can. J . Math., 50(3), (1998), 547-562. 8. E. Malinnikova, Separation of singularities and Mittag-LeSJEer theorems for harmonic differential forms (preprint, 2006). 9. E. Malinnikova, St.Petersburg Math J., 11 (2000), no.4, 625-641. 10. H. Yamabe, Osaka Math. J . 2 , (1950), 15-17. 11. R. Bott and L. W. Tu,Dzfferential forms in algebraic topology. (Springer-Verlag, 1982).
214
APPROXIMATE PROPERTIES OF THE BIEBERBACH POLYNOMIALS ON THE COMPLEX DOMAINS DANIYAL M. ISRAFILOV
Balikesir University, D e p a r t m e n t of M a t h e m a t i c s Balikesir, 101 00, T u r k e y
[email protected]. t r Let G be a finite smooth Jordan domain of the complex plane C,zo E G, and let ’PP
( z ) :=
7
[cpb
(c)12’p dC,
2
E G,
P
> 0,
20
where w = cp(z) is the conformal mapping of G onto D ( O , r o ) := {w :I w I< r o } with cp ( z o ) = 0, cp‘ ( z o ) = 1. Let also T ~ , ~ ( Zn) = , 1 , 2 , ..., be the generalized Bieberbach polynomials for the pair (G, 20). We investigate the uniform convergence of these polynomials on G and obtain an estimation for the quantity
/I ‘
~ prn,p ~
m gIPP(~) Z E G
-~
n(z)l
in term of the integral modulus of continuity of cpb.
K e y w o r d s : Generalized Bieberbach polynomials; Conformal mapping; Smooth boundaries; Uniform convergence.
1. Introduction and main results
Let G be a finite domain in the complex plane @, bounded by a rectifiable Jordan curve L , and let G- := ExtL. Further let T : = {w 6 @ : (201 = l}, D := IntT and D- := ExtT. By the Riemann conformal mapping theorem, there exists a unique conformal mapping w = cp(z) of G onto D(O,ro) := {w :I w I< r 0 } with the normalization cp ( z o ) = 0,cp‘ ( z o )= 1. The inverse mapping of cp we denote by $. Let also cpo be the conformal mapping of G- onto ID- normalized by
and let
$0
be its inverse mapping. For an arbitrary function f given on G we set p
If the function f has a continuous extension to
> 0.
c we use also the uniform norm:
215
It is well known (see [12, p. 4331 ) that the function Pp ( 2 ) :=
j
[Pb (01”/” 4,
E G,
P
>0
ZO
minimizes the integral
11
f’
G with the normalization f
(p
> 0) in the class of all functions analytic in
0, f ’ ( 2 0 ) = 1. In the literature there are results concerning the region of exponents p which lead to the univalence of ‘pp.In this work we study the approximation of ‘ p p by the extremal polynomials defined below. In fact this problem is a particular case of a more general one, formulated in [14, pp. 318-319, problems 1, 21 and is important for the approximately construction of the conformal mappings. Let 11, be the class of all polynomials p , of degree at most n satisfying the conditions p , (zo) = 0, p k ( Z O ) = 1. Then we can prove that the integral 1) pb p i I :p(G) ( p > 1) is minimized in 11, by an unique polynomial T , , ~ . We call (see a l ~ o : )~ these , ~ extremal polynomials T , , ~ the generalized Bieberbach polynomials for the pair (G,zo). In case of p = 2 they are the usual Bieberbach polynomials T,, n = 1 , 2 , .... The approximation problems for 972 = cp in closed domains with various boundary conditions, where approximation is conducted by the usual Bieberbach polynomials were intensively studied in2>4>7-11)13 . Similar problems for ‘ p p ( p > 1) using the generalized Bieberbach polynomials were investigated in1>516 (see also,*).In the above cited works the rate of convergence to zero of the quantity (20) =
II PP
-rn,p
l l ~
as n tends to 00 , has been estimated by means of the geometric properties of G. One of the interesting problem in this direction is the problem connected with a conjecture due to S. N. Mergelyan, who in” showed that the Bieberbach polynomials satisfy
for every E > 0, whenever L is a smooth Jordan curve and stated it as a conjecture that the exponent $ - E in (I) could be replaced by 1 - E . In7 it has been possible for us to obtain some improvement of the above cited Mergelyan’s estimation (1).For its formulation, it is necessary to give some definitions as follows. We denote by L p ( L ) and EP (G) the set of all measurable complex valued functions such that 1 f Ip is Lebesgue integrable with respect to arclength, and the Smirnov class of analytic functions in G respectively. For a weight function w given on L , and p > 1 we also set LP(L,W)
:= {
f E L1 ( L ) :I f
y
w E L1 ( L ) } ,
EP ( G , w ) := { f E El ( G ) : f E L p ( L , w ) } .
216
Let g E L p (T, w ) and let gh(W) be the mean value function for g defined as:
g h ( w ) :=
l
lh h
O 0, i = 1,2,3.
In spite of the fact that the function pp is defined on G, it has6 a continuous extension to G. Therefore, the uniform norm in the above inequality is well defined. From this theorem in case of p = 2 we have the following result.
Corollary 1 Let G be a domain with a smooth boundary L . T h e n
for every
E
> 0 and with a constant
c1 = c1 ( E )
> 0.
As it follows from Definition 1 the modulus of continuity than R,,j+bl ((p' 0 $0) ($h)4 , i) defined above.
W;+~((P',
i) is simpler
2. Auxiliary results
We shall use c, c1, c2, ... to denote constants (in general, different in different ralations) depending only on numbers that are not important for the questions of interest. One of the important step in the proof of the main result is the following theorem, given here without proof. Theorem 2 Let G be a domain with a smooth boundary L , 1 < p , r
let
be the nth partial sums of the Faber series of pk. T h e n
II for every
E
'PP sn (&,.) -
lIL'(L)
0 and with a constant c = C ( E ) > 0.
For the function
'pp
and a weight w we set
where inf is taken over all polynomials p , of degree at most n and
< co and
218
Corollary 2 Let G be a domain with a smooth boundary L andr>l .Then
for every
E
> 0 and with c = c ( E ) > 0.
Proof. Since
and l/& E Lr( L ) ,for every r
for any p o
2 1, by Holder's inequality
> 1. Now, applying Theorem
we have
2 (in case of r := ~ p o we ) conclude tha.t
Choosing the number po sufficiently close to 1 we finally get
with c = c ( E ) . The proof of the following theorem is similar to that of Theorem 11 from3 . Theorem 3 Let G be a domain with a smooth boundary L and let p
> 1. Then
with a constant c > 0 . The approximation properties of the polynomials L,(G) are given in the following lemmas.
T : , ~ ,n
= 1 , 2 , ..., in the space
Lemma 1 Let G be a domain with a smooth boundary L and let p
for every
E
> 1. Then
> 0 and with c = c ( E ) > 0 .
Proof. For the polynomials qn,p ( z ) best approximating y ~ bin L,(G) we set
Qn,p (2)
:=
.i a
qn,p
( t )d t ,
tn,p (2)
:= Q n , p
(2)
+ [I
- qn,p
( z o ) ]( 2 - ZO) .
219
Then t,,p (zo) = 0 , tL,P( 2 0 ) = 1 and hence by Theorem 3 we obtain
This relation by the inequality
If (Z0)l 5 c Ilf
IILp(G)
,
which holds for every analytic function f with IlfllLPcG, < 00,and by the extremal property of the polynomials T , , ~ implies that
Now applying Corollary 2 (in case of r = p ) we obtain the statment of Lemma 1. W
The proof of the following lemma can be found in2 (for p = 2 ) and6 (in case of 1< p
< m).
Lemma 2 L e t G be a f i n i t e d o m a i n w i t h a s m o o t h boundary L and let p , be a n y polynomial of degree 5 n w i t h p,(zo) = 0. T h e n
1
11 p:, 11~5
C V G F
&
IIL,(G)
,
P = 2;
11 i n IIL,(G) , Z-1+&
c6np
f o r every
II P:,
I( Pn'
P
1 0, and w i t h c > 0 and
> 2;
c6
< = Cg(&) > 0.
The following lemma was proved in [a, Lemma 151 in case of p = 2 treating the Bieberbach polynomials T,, n = 1 , 2 , .... The general case p E (1,co), concerning the extremal polynomials T , , ~ ,n = I,2, ..., goes similarly.
of
Lemma 3 L e t G be a f i n i t e simple connected d o m a i n , and let p , be a polynomial degree 5 n satisfying t h e condition p,(zo) = 0 . A s s u m e t h a t
II P:, I,
i 0. Then dh ( K )= T ~ ( K ) . 2-00
Proof. It is clear that the existence of the usual limit in (10) implies that T~ ( K ,Q) + T ( K ,19) for all Q E C. Then, due to the second condition, we pass to the limit under the integral in (14) and, applying (15), obtain the desired equality. Remark 3.1. The second condition in Theorem 3.2 holds if 0 is an interior point of K . 4. Prolate spheroids Consider the following one-parameter family of compact sets enclosed by confocal prolate spheroids with the focuses ( f l ,0,O):
By (rj,6,'p) we denote the spheroidal coordinates corresponding to the family (17), which are connected with the Cartesian coordinates t = ( t l , t 2 , t 3 ) as follows:
tl
= coshq cos 6, t 2 = sinh r j sin t9 cos p, t 3 = sinh r j sin 6 sin p,
TO and the solution of (1.2) and (1.3) with llyo - zo11 < 61,170 - t o / < 62 satisfying
Let us set 7 = 61,we can obtain
which contradicts with (3.3). Hence (3.4) is valid. Now let 0 < 6: < 61,O < 6; < 62 and €2 < S = min{ST, 6;) such that
243
U2(62) < b2(6).
(3.5)
Then we can prove that
whenever 6; < \/yo- z o l l < 61 and 6 : < 170 - to1 < 62. In fact, if (3.6) is not true, then there would exist tl > t 2 > TO and the solution of (1.2) and (1.3) with 6: < llyo - lcoll < 61,6,* < 170- to1 < 62 satisfying
Let us set 8 = 6 and using (Az), we get
which contradicts with (3.5). Thus (3.6) is valid. Then the solution y(t, 7 0 ,yo) of the system (2.3) through ( T O ,yo) is initial time difference strictly uniformly stable with respect to the solution z ( t - q, t o , 2 0 ) . This completes the proof of Theorem 3.1. Theorem 3.2: Assume that ( A l ) for each 7,O < q~ < p, V, E C[R+ x in z and for ( t , z ) E R+ x S,, and llzll 2 q,
S,,R+]and V, is locally Lipschitzian
(A2) for each 8 , 0 < 0 < p , VQ E C[R+ x S,, in z and for ( t , z ) E R+ x S, and llzll 5 8,
R+]and Vo is locally Lipschitzian
where ~ ( t=) y ( t , q , yo) - z ( t - q,to,zo) for t 2 ~ 0 , y ( t , ~ 0 , y of 0 ) the system (2.3) through ( T O ,yo) and z ( t - 77, t o , ZO),where z ( t ,t o , 5 0 ) is any solution of the system (2.1) for t 2 70 2 0,to E IR+ and q~ = TO - t o .
244
Then the solution y ( t , T O ,yo) of the system (2.3) through (70,yo) is the initial time difference uniformly strictly asymptotically stable with respect to the solution of the system (2.1) for t 2 TO 2 0,to E R+ and q = TO - t o . Proof of Theorem 3.2: We note that (3.9) implies (2.1). However, (3.8) does not yield (2.2). As a result of these, we obtain because of (3.8) only stability of perturbed systems with initial time difference with respect to z(t - q, t o , 20) that is for given any € 1 I p and TO E R+ there exist 61 = S l ( t 1 ,TO) > 0 and 62 = 6 2 ( t l , q ) > 0 such that
llv(t, 70,yo)
-
z(t - q, t o , zo)ll < €1 for t 2 TO whenever llyo - z011< 610 (3.10) and 170 - t o I < 620.
To prove the conclusion of Theorem 3.2 we need to show that the solution y ( t , TO, yo) of the system (2.3) through (TO, yo) is strictly uniformly attractive with respect to z ( t - V , t o , z o ) for this purpose, let € 1 = p and set 610 = & ( p ) and 620 = & ( p ) so that (3.10) yields Ily(t,ro,yo) - z(t - q,t0,zo)11 < p for t 2 TO whenever llyo < 610 and 170 - to1 < 620. Let llyo - zo/I < 610 and /TO- to1 < 620. We show, using standard argument, that there exists a t* E [ T O , T O TI, where T = T ( E , T o>) where 610 and 620 are the numbers corresponding to t1 in (3.10) that is in stability of perturbed systems with initial time difference with respect to z(t - q, t o , 2 0 ) such that lly(t*, 70,yo) - z(t* - q,to,zo)(( < 61 for any solutions of the systems (2.3) and (2.2) with Ilyo - z011 < 610 and 170- to1 < 620. If this is not true, we will have l l y ( t * , T O ,yo) z(t* - q, t o , z0)ll 2 61 for t E [TO, TO TI. Then, q = 61 and using ( A l ) with (3.8), we have
~011
+
(L~l($~$~p)
+
~
5 a1(610,62o) - c1(61,62)T = bl(61) in view of the choice of T . This contradiction implies that there exist a t* E [TO,ro TI satisfying IIy(t*,ro,yo) - z(t* - q , to,zo)II < 61.Because of the stability y @ ,r ~yo), of perturbed systems with initial time difference with respect to z(t - q, t o , zo), this yields that
+
llv(t, TO,YO) - z ( t - 11, t o , z0)11< €1 for t 2 TO which implies that there exists a llW(T0
70
+ T ,7 0 , yo)
+ T 2 t*
< Tl < T such that -
470
+ T - rl, t o , z0)ll = €1
245
Now, for any 612,0 < 612 < 610 and 0 < 612 < 620 we can choose bz(t1) > ~ ~ ( 6 and 2 ) 0 < € 2 < €1 < 612. Suppose that 612 < Ilyo - z0)ll < min{61o,6zo} and 612 < ) , T2 = TI + T . min(blo, 620). Let us define T = b a ( c ~ ~ ~ ~ ; ( r zand
Since, t
-
(TO
€2
such that
1 7 0- to1
0 and t 2 TO,TO E R+, there exist a 61 > 0 such that
If given any
and for every 62
< 61 there exists an
€2
> 0 , 0 < €2 < 62 such that
246
wo
2 Sz
implies m ( t )< €2 for t
2 TO.
Here, u l ( t )and uz(t)are any solutions of (i) in (3.11) and (ii) in (3.11);respectively. Following main result based on this definition that result is formulated in terms of comparison principle.
Theorem 3.3: Assume that (Al) for each q , 0 < q < p, V, E C[R+x S,, R+] and V, is locally Lipschitzian in z and for ( t , z ) E R+ x S, and 11211 2 q ,
( A z ) for each 8,O < 8 < p, Ve E C[R+x S,, R+] and Ve is locally Lipschitzian in z and for ( t , z ) E R+ x S, and ( ( z (5 ( 8,
where gz(t,U) 5 g i ( t , u ) , g i , g z E C[R$,R],gl(t,O) = gz(t,O) = 0 and z ( t ) = y ( t , T O ,yo) - z ( t - t o , zo) for t 2 T O ,y ( t , T O , go) of the system (2.3) through ( T O , Y O ) and ~ (- tq,to,zo), where z(t,to,zo)is any solution of the system (2.1) for t 2 TO 2 0, t o E R+ and q = TO - t o . Then any strict stability concept of the comparison system implies the corresponding strict stability concept of the solution y(t, T O , yo) of the system (2.3) through ( ~ 0 , y o ) with respect to the solution z(t - q,tO,zo) of the system (2.2) with initial time difference where z ( t ,t o , 20) is any solution of the system (2.1) for t 2 To 2 0, to E R+.
v,
Proof of Theorem 3.3: We will only prove the case of strict uniformly asymptotically stability. Suppose that the comparison differential systems in (3.11) is strictly uniformly asymptotically stable, then for any given €1, 0 < €1 < 6,there exist a S* > 0 such that u10 5 S* implies that u l ( t ,T O , u10) < b l ( ~ 1 for ) t 2 TO. For this €1 we choose 61 and 611,such that a1(6T) 5 6* and 6; < €1 where S I = max(b1, &1}, then we claim that
247
If it is not true, then there exist tl and
t2,
t2
> tl > 70 and a solution of
Choosing 11 = 6; and using the theory of differential inequalities we get
which is a contradiction. Here r ( t ,t o , u10) is the maximum solution of (3.11). Hence, (3.14) is true and we have uniformly stability with initial time difference. Now, we shall prove strictly uniformly attractive with initial time difference. For any given 62,
2 t . For these
bl(~2) -
€2
> 0, 62 < 6*we choose Szand Tzsuch that al(62) < 62and
-
62 and
T2,
since (2.11) is strictly uniformly attractive, for any
-
63 < hathere exist
T3
-
and TI and
T2
(we assume
T2
< TI) such that 63 < u10 =
-
u20
< 62 implies
where T ( t , T O ,u1o) and p ( t , T O ,u20) is the maximal solution and minimal solution of (3.11) (i) and (3.11) (ii); respectively. -
Now, for any 63,let b2(63) 2 63. We choose €3 such that using comparison principle (3.11) ( 2 ) and (Al), we have
< €3.Then by
~ ~ ( 6 3 )
248
+
+
which implies that for l l y ( t , ~ ~ , y o )z ( t - q,to,z0)11 < € 3 for t [TO T2,70 TI]. Hence, the solution y ( t , ~ o , y o )of the system (2.3) through ( ~ o , y o ) is strictly uniformly attractive with respect to the solution z(t - 7, t o , 2 0 ) is any solution of the system (2.1) for t 2 702 0, t o E R+.The proof is complete.
References 1. Koksal, S. and Yakar, C., Generalized Quasilinearization Method with Initial Time Difference. (2002): Journal of Engineering Simulation 24 (5). 2. Lakshmikantham, V. and Leela, S. (1969): Differential and Integral Inequalities Vol. 1, Academic Press, New York. 3. Lakshmikantham, V. and Mohapatra, R. N. (2001): Strict Stability of Differential Equations, Nonlinear Analysis, Volume 46, Issue 7, Pages 915-921. 4. Rajalakshmi, S and Sivasundaram, S. (1993): Variational Lyapunov Second Method, Dyn. Sys. and Appl. 2, 485-490. 5. Shaw, M. D. and Yakar, C., Lipschitz Stability Criteria with Initial Time Difference. Journal of Applicable Analysis. (to appear) 6. Shaw, M. D. and Yakar, C. (1999): Generalized Variation of Parameters with Initial Time Difference and A Comparison Result in terms of Lyapunov-like Functions. International Journal of Nonlinear Differential Equations Theory-Methods and Applications. Vol. 5, No: 1&2, Jan-Dec. (86-108). 7. Shaw, M. D. and Yakar, C. (2000): Stability Criteria and Slowly Growing Motions with Initial Time Difference. Journal of Problems of Nonlinear Analysis in Engineering Systems. Volume 6, Issue l(11) . (50-66). 8 . Yakar, C. and Shaw, M.D. (2005): A Comparison Result and Lyapunov Stability Criteria with Initial Time Difference. Dynamics of Continuous, Discrete and Impulsive Systems. A: Mathematical Analysis. Volume 12, Number 6, Pages 731-741.
249
PIECEWISE CONTINUOUS RIEMANN BOUNDARY VALUE PROBLEM ON A CLOSED JORDAN RECTIFIABLE CURVE YU.V.VASIL'EVA Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Street,Kiev, Ukraine kudjavina @mail.ru There are expanded classes of closed Jordan rectifiable curves and given functions in the theory of piecewise continuous Riemann boundary value problem and characteristic singular integral equation with Cauchy kernel that is connected with it.
Keywords: Cauchy type integral; Riemann boundary value problem; closed Jordan rectifiable curve; local centered module of smoothness.
1. Introduction
Let y be a closed Jordan rectifiable curve in the complex plane C , D+ and Dbe, respectively, the interior and exterior domains bounded by y, O E D+,and let T := { a l , a 2 , . . . , a,} be a fixed finite set of points of the curve y. By HTf we denote the set of holomorphic in Dk functions F (having also a limit at infinity in the case of the set H F ) which are extended continuously on y \ T and satisfy the estimation
where the constant c does not depend on z , and VF is a certain number from the interval (0; l ) ,depending on the function F . Consider the piecewise continuous Riemann boundary value problem: to find functions @+ E H$ and @- E HF which satisfy the equality
@'(t) = G ( t ) F ( t )+ g ( t )
V t E y\ T ,
(2)
where G and g are given functions. If g ( t ) $ 0, then the problem is called nonhomogeneous, and if g ( t ) 0, then the problem is called homogeneous. In this paper there are expanded classes of closed Jordan rectifiable curves and given functions in the theory of the piecewise continuous Riemann boundary value problem, which was studied earlier in [l - 91.
=
250
2. Homogeneous problem
For
E
> 0 and X c C denote y,(X)
U { t E y : ( t - 21 5 E } .
:=
If X = {z}, then
XEX
% ( X ) =: %(Z). All integrals over the curve y are understood in terms of their principal value, i.e.
s
cp(t)d t
J’
:= lim E+O
Y
cp(t>d t
1
Y\YE(X)
where X is a finite set of discontinuity points for the function cp. Introduce into consideration the Cauchy type integral
F ( z ) :=
& 1pt (- tz)d t , 27ra
zEc\y.
(1)
Y
If the function p is summable on y or p E HT := H$ holomorphic in C\y. For every point aj E T define the numbers
+ H F , then the function 5 is
1 A p ( a j ) := liminf inf Re T+O In T z ~ q ~ : l t - l=T a,
F(.)
and, moreover, suppose, that the following relation is fulfilled
AP(aj) 5 4,(aj)
+c
b ! ~ jE T ,
(2)
where c is a constant. Relation (2) means, that for all points u j E T there exist the equalities A p ( a j ) = A,(aj) = +00, or the equalities Ap(aj) = A,(uj) = -00, or the numbers AP(aj) and 4,(aj) are finite. If the numbers Ap(uj) and A,(uj) are finite for all aj E T , then the index ae of piecewise continuous Riemann boundary value problem is defined by
z:=caej, m
j=1
where zj:=
if A,(uj) is integer, {A?J(a.i), [A,(aj)] + 1 , otherwise.
If there exists -00 among the values A,(aj), then the values 4,(uj) but -m does not, then z = +co.
EE
= -m. If -too exists among
251
The following theorem is proved using the scheme, proposed in [4, p. 461, and generalizes Theorem 1 from [7]. Theorem 2.1. L e t y be a closed J o r d a n rectifiable curve and G be a f u n c t i o n of t h e f o r m G ( t ) = exp(p(t)), where p E HT a n d , f u r t h e r m o r e , relation (2) i s fulfilled. Then: I ) if t h e relations - X I 5 z < 0 are satisfied, t h e n t h e homogeneous R i e m a n n boundary value problem h a s only trivial solution; 2) if t h e equality z = + X I is satisfied, t h e n t h e homogeneous R i e m a n n boundary value problem h a s a n infinite s e t of linearly independent solutions; 3) if t h e relations 0 5 z < 00 are satisfied, t h e n t h e homogeneous R i e m a n n boundary value problem h a s ae 1 linearly independent solutions and i t s general solution i s given by t h e expression
+
m
a*((z)= exp(g(z))p=(z) n ( z - aj)-,j ,
z E D'
,
j=1
where P, i s a n arbitrary polynomial of degree a t m o s t
E.
3. Nonhomogeneous problem
The nonhomogeneous Riemann boundary value problem is considered in the case where the index is finite and for all points aj E T the following numbers
are finite. We use the following metric characteristic of the curve y (see
[lo]):
O(&) := sup OZ(&), ZE-r
where O,(E) := { t E y : It - Z I 5 E } and denotes the linear Lebesgue measure on 7. For a function q given on y\T and for a point x E y\T we introduce the local centered module of smoothness of the first order (see [7]): sup
R,(q, y,E ) :=
t E 7 : It--2l=e
{o,
( q ( t ) - q(x)(,if { t E y : It - x ) = E }
# 0,
if ( t E y : I t - x ( = E } = Q ) .
In general case, the function 0, ( 4 , y, E ) is not monotone with respect to E as distinct from the classical module of continuity. Therefore, the function R,(q, y, E ) takes into account oscillations of the function q. Notice that a function q is continuous at a point x if and only if R, ( 4 , y, E ) + 0 as E + 0.
252
By F + ( x ) , F - ( x ) we denote the limiting values of function (1)at a point x E y\T taken in the domains D+ and D - , respectively. The following theorem describes the solvability of the nonhomogeneous Riemann boundary value problem with a finite index under minimal assumptions on the coefficient G of the problem.
Theorem 3.1. Let y be a closed Jordan rectifiable curve satisfying the following condition
0 (E) = 0 ( E V ) ,
E
-+
0,
(1)
where 0 < u 5 1. Suppose that the function G is represented as G ( t ) = exp ( p ( t ) ) , where p E H T , and, furthermore, the Cauchy type integral 5 has limiting values F + ( x ) , F - ( x ) o n y\T, and the numbers A,(aj), A ; ( a j ) are finite for all aj E T . Suppose also that th,e function g i s represented as g = g+ + g-, where g+ E HTf, and the function g- is holomorphic in D - , continuous o n \ T and satisfies the following condition SUP zcY\Y6
and estimations m
where the constant c does not depend o n t . T h e n for ze 2 -1 the nonhomogeneous R i e m a n n boundary value problem is solvable, but when 2 < -1 it is solvable if and only if the following - 2 - 1 conditions are satished:
T h e general solution of the nonhomogeneous R i e m a n n boundary value problem is
(2 - a j ) - = j
if if
,
z E D+,
z E D-,
z E D*,
(6)
253
and P, i s a n arbitrary polynomial of degree at m o s t ze when when ze < 0.
E
2 0 , and P,(z) 0
+
Proof. Taking into account the equalities g ( t ) = g+ ( t ) g- ( t ) and G ( t )= exp (P+ ( t )- P - ( t ) ) ,which are satisfied for all t E y \ T , rewrite the boundary condition (2) as m
(a+ ( t )- g+ ( t ) )
n (t
-
a- ( t )
aj)=j
j=1
m
m
-a j p
j=1
-
exp (P+ ( t ) )
n (t
exp (5- ( t ) )
Define F+ ( t ) := exp (-P+ ( t ) )
m
n (t
-
aj)=j
+
g- (t)
n (t
- aj)=j
j=1
exp (P+ ( t ) )
. The function F+ satisfies the
j=1
inequality I F + ( ~5 ) I It - ajlzj-A;(aj)-E, for some E > 0 and for all t E y \ T , which are sufficiently close to aj E T . The consequence of inequalities (3) and (7) is the following estimation
Ig- ( t ) F+ ( t )I 5
- ajIaj+Ej-A;(aj)-E
(7)
(8)
where constants c in (7) and (8) are different, not depending on t. Moreover, for sufficiently small E the inequality aj zej - A g ( a j ) - E > --v is fulfilled and, therefore, the function h ( t ) := g - ( t ) F + ( t ) is summable on y. Then, taking into account condition (a), conclude, that the integral h has limiting values on y\T in the domains D + , D- and Sokhotski formulas are satisfied. Therefore, the function is satisfied the boundary condition (2). Now estimate the function @ , ( z ) in a neighbourhood of a point aj ( j = 1 , ) . For this notice, that for E > 0 in the sufficiently small neighbourhood of this point the following estimation is fulfilled
+
m
lexp ( ~ ( z ) ) l
12- ajl-=j
-1,
,Bj
+ Ap ( a j )
-
A; ( a j ) - 2~
> -1,
which are fulfilled for sufficiently small E , conclude, that the function satisfies the inequality of form (1). Thus, @O is a particular solution of the nonhomogeneous Riemann boundary value problem. Notice, that for
x!
< 0 the function
exp ( g ( z ) )
m
n
(2 - uj)-=j
has
j=1
a pole of an order -x! at infinity and @O is a solution of Riemann boundary value problem only when the conditions (5) are satisfied. To complete the proof notice that in formula (6) the general solution of the nonhomogeneous Riemann boundary value problem is represented as a sum of the particular solution of this problem and the general solution of the homogeneous problem. The following theorem, which is proved analogously to Theorem 3.1, does not contain conditions ( 2 ) , (4) of Theorem 3.1, but it contains additional assumptions on the function G. Theorem 3.2. Let y be a closed Jordan rectifiable curve satisfying the condition ( l ) , where 0 < u 5 1. Suppose that the function G is represented as G ( t ) = exp ( p ( t ) ) , where p E H T , and satisfies the condition of the f o r m (2) and estimations
n m
IG(t)I 2 c
It
-
ujln’
V t E y\T,
n j 2 0;
j=1
where the constant c does not depend o n t , and, furthermore, the numbers A p ( u j ) , A;(uj) are finite f o r all j = l,m (here p ( t ) := lnG(t) is understood as a n arbitrary continuous o n y \ T branch of this function). Suppose also that the function g is represented as g = gf + g - , where gf E H $ , and the function g- is holomorphic in D - , continuous o n \ T and satisfies the inequality m
j=1
Icj
> A;(uj)
-
A p ( u j )+ nj
+ max{nj
- mj
-
1;- u } ,
where the constant c does not depend o n z . T h e n f o r x! 2 -1 the nonhomogeneous R i e m a n n boundary value problem is solvable, but f o r EE < -1 it is solvable if and only if conditions (5) are satisfied. The general solution of the nonhomogeneous R i e m a n n boundary value problem is given by formula (6).
255
As an application of Theorem 3.2 we have obtained a result on solvability of characteristic singular integral equation with Cauchy kernel. References 1. F. D. Gakhov, Boundary value problems, Moscow: Nauka, 1977, 640 p. [in Russian] 2. N. I. Muskhelishvili, Singular integral equations, Moscow: Nauka, 1968, 511 p. [in Russian] 3. R. K. Seifullaev, Riemann boundary value problem on a nonsmooth open curve, Mat. Sb., 112 (1980), no. 2, 147-161. [in Russian] 4. B. A. Kats, An exceptional case of the Riemann problem with an oscillating coefficient, Izv. Vyssh. Uchebn. Zaved. Mat. (1981), no. 12, 41-50. [in Russian] 5. E. A. Danilov, Dependence of the number of solutions of a homogeneous Riemann problem on the contour and the coefficient module, Dokl. Acad. Nauk SSSR, 264 (1982), no. 6, 1305-1308. [in Russian] 6. B. Gonzalez, J. Bory, The homogeneous Riemann boundary value problem on rectifiable open Jordan curves, Science. Mat. Havana, 9 (1988), no. 2, 3 - 9. 7. S. Plaksa, Riemann boundary value problem with an oscillating coefficient and singular integral equations on rectifiable curve, Ukr. Math. J., 41 (1989), no. I, 116-121. 8. K. Kutlu, On Riemann boundary value problem, A n . Univ. Timigoara: Ser. Matematiza-Informatiza, 38 (2000), no. 1, 89 96. 9. D. Pena, J. Bory, Riemann boundary value problem on a regular open curve, J . Nut. Geom., 22 (2002), no. 1, 1 - 17. 10. V. V. Salaev, Direct and inverse estimates for a singular Cauchy integral along a closed curve, Mat. Zametki, 19 (1976), no. 3, 365 - 380. [in Russian] -
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A NOTE O N THE MODIFIED CRANK-NICHOLSON DIFFERENCE SCHEMES FOR SCHRODINGER EQUATION ALLABEREN ASHYRALYEV Department of Mathematics, Fatih University, 34900 Buyukcekmece, Istanbul, Turkey ALI SIRMA Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli, Turkey In present paper the nonlocal boundary value problem
+ AU = f ( t ) , 0 < t < T , ~ ( 0= ) au(X) + p , la1 < 1, 0 < X 5 T 2%
for Schrodinger equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The the second order of accuracy r-modified Crank- Nicholson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. In practice, the stability inequalities for the solutions of difference schemes for Schrodinger equation are obtained. A numerical method is proposed for solving a one-dimensional Schrodinger equation with nonlocal boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.
Key Words: Schrodinger equation; Difference schemes; Stability 1. Introduction. Modified Crank-Nicholson Difference Schemes
It is known that various problems in physics lead to Schrodinger equation. Methods of solutions of the problems for Schrodinger have been studied extensively by many researchers (see, e.g., and the references given therein). In the present paper the nonlocal boundary value problem
{
ig + Au = f ( t ) , 0 < t < T , u(0) = C Y U ( X ) + p, la1
< 1, 0 < X 5 T
(1.1)
for Schrodinger equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. In the papers [6]-[8] the second order of accuracy r-modified Crank- Nicholson difference schemes for the approximate solutions of parabolic equation in a Banach space E with nonsmooth data were presented. The main aim of this paper is to study r-modified Crank- Nicholson difference schemes
257
for the approximate solutions of problem (1.1).It is assumed that 2r second order of accuracy r-modified Crank- Nicholson difference schemes
I
iuk--Z"k-l
I
A + Y('Q + ~ k - 1 ) = pk,
uo = Q ( I + ZloA)~l+ p UO
= aul+ p, r r
UIJ=
a(I
-
< A, 3
+ iZlA)!j(ul +
pk =f(tk -
$),trc = k r , r
5 X.The
+ 15 k 5 N ,
ilgpl, TT 2 A,
E
z+,
+p
UL+~)
-
dlpl,TT
< A, $ @ Z+
(1.2) for the approximate solutions of this nonlocal boundary value problem are presented x [". Z+ denotes here the set (2, ...,n, ...} and 1 = [$I, 10 = A - [,IT, 11 = X - x - $. The stability of these difference schemes is established. In applications, the stability inequalities for the solutions of difference schemes for Schrodinger equation are obtained. A numerical method is proposed for solving a one-dimensional Schrodinger equation with nonlocal boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.
[,IT
2. Theorem on Stability
BY6
>
k Rk
(2.3)
T, = ( I - aB1-TR )-'. Let T r < X and condition
@ Z+.In this case if uo = a ( I
T
2
269
ui = n — 1, n,n + 1. VN-l
To solve this difference equation we have applied a procedure of modified Gauss elimination method for difference equation with respect to n with matrix coefficients. Hence, we seek a solution of the matrix equation in the following form Un = an+1Un+1
••,2,1,0,
(4.5)
where a.j (j = 1, • • •, M — 1) are (N + 1) x (N + 1) square matrices and /3j (j = 1, • • •, M — 1) are (N + 1) x 1 column matrices. Since (4.6)
we have 0 0 0 ... 0 0 0 0 0 ... 0 0 0 0 0 ... 0 0 0 0 0 ... 0 0 0 0 0 ... 0 0
and 0 0
0 0 Using the equality Us = as+iUs+i + 0g+i , (for s = n, n and the matrix equation AUn+l +BUn + CT/ n _i = Dlpn,
we can write [A + Ban+i + Canan+l]Un+i + [B/3n+i + Can(3n+
C/3n] =
The last equation is satisfied if we select A + Ban+i + Canan+i = 0, [B/3n+1 + Can/3n+l + Cj3n] = D