Complex Analysis and Potential Theory: Proceedings of the Conference Satellite to ICM 2006

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

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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<) k(f). This conjecture has been recently proved by R. Kuhnau and the author. We prove that approximation is possible in a much stronger sense and also give applications. One of them is the inversion of Ahlfors and Grunsky inequalities, which solves the old Kiihnau problem on exact lower bound in the inverse inequality estimating k(f) by ~ ( f ) .

Keywords: Quasiconformal, univalent function, Grunsky coefficient inequalities, universal Teichmuller space, subharmonic function, Strebel’s point, Kobayashi metric, generalized Gaussian curvature, holomorphic curvature, Redholm eigenvalues.

1. Grunsky inequalities and Fredholm eigenvalues 1.1.

The technique of Grunsky coefficient inequalities provides a powerful tool in solving various problems. We shall consider the Grunsky coefficients from several points of view. These coefficients give rise to a noninvariant Finder structure on the tangent bundle of the universal Teichmuller space T dominated by the canonical Finder structure of this space. Estimating deviation of these structures from below provides interesting analytic and geometric consequences. The metric generated on ‘IT by this structure is naturally related t o quasiconformal extensions of conformal maps of quasidisks and reflections across their boundaries and allows to establish the bounds for dilatations of such extensions. This geometric approach is somewhat new. The exposition is mainly focussed on the presentation of main ideas and results; for complete details, see the author’s recent papers. The classical Grunsky theorem states that a holornorphic function f ( z ) = z const O ( z - l ) in a neighborhood Uo of z = co is extended to a univalent

+

+

4

holomorphic function on the disk A* = { z E its Grunsky coefficients a,, where a,,

= C U {co} : IzI

satisfy the inequalities

I

> 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 * , where

N*

-+ 0,

=3 ~ ( f ~ , ~ ) / ( 2 h ) .

The approach exploited above can be extended to other important inequalities, for example, to exact estimating of the minimal dilatation k o ( M ) of quasiconformal extensions of M-quasisymmetric homeomorphisms of the circle.

15

References Ah. Di. EKK GL. Gr . KO. Krl. Kr2. Kr3.

Kr4. Kr5. Kr6. KK1. KK2. Kul. Ku2. Ku3. Ku4. Mi.

Min. Po.

Rol.

Ro2. sc.

L. Ahlfors, Remarks o n the Neumann-Poincard integral equation, Pacific J. Math. 2 (1952), 271-280. S. Dineen, T h e Schwarz Lemma, Clarendon Press, Oxford, 1989. C.J. Earle, I. Kra and S.L. Krushkal, Holomorphic motions and spaces, Trans. Amer. Math. SOC.944 (1994), 927-948. F.P. Gardiner and N. Lakic, Quasiconformal Teichmiiller Theory, Amer. Math. Soc., 2000. H. Grunsky, Koefizientenbedingungen fur schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), 29-61. S. Kobayashi, Hyperbolic Complex Spaces, Springer, New York, 1998. S.L. Krushkal, Grunsky coeficient inequalities, Carathdodory metric and extremal quasiconformal mappings, Comment. Math. Helv. 64 (1989), 650-660. S.L. Krushkal, Plurisubharmonic features of the Teichmiiller metric, Publications de 1’Institut Mathkmatique-Beograd, Nouvelle sBrie 75(89) (2004), 119-138. S.L. Krushkal, Quasiconformal extensions and reflections, Ch. 11 in: Handbook of Complex Analysis: Geometric Function Theory, Vol. I1 (R. Kuhnau, ed.), Elsevier Science, Amsterdam, 2005, pp. 507-553. S.L. Krushkal, Schwarzian derivative and complex F i n d e r metrics, Contemporary Math. 382 (2005), 243-262. S.L. Krushkal, Inversion of Ahlfors and Grunsky inequalities, Georgian Math. J. 13 (2006), 495-500. S.L. Krushkal, Rational approximation of holomorphic functions and geometry of Grunsky inequalities, Contemporary Math., to appear. S.L. Krushkal and R . Kuhnau, Quasikonforme Abbildungen - neue Methoden und Anwendungen, Teubner-Texte zur Math., Bd. 54, Teubner, Leipzig, 1983. S. L. Krushkal and R . Kuhnau, Grunsky inequalities and quasiconformal extension, Israel J. Math. 152 (2006), 49-59. R. Kuhnau, Verzerrungssatze und Koefizientenbedingungen v o m Grunskyschen Typ f u r quasikonforme Abbildungen, Math. Nachr. 48 (1971), 77-105. R. Kuhnau, Moglichst konforme Spiegelung an einer Jordankurve, Jber. Deutsch. Math. Verein. 90 (1988), 90-109. R . Kuhnau, W a n n sand die Grunskyschen Koefizientenbedingungen hinreichend fur Q-quasikonforme Fortsetzbarkeit? Comment. Math. Helv. 61 (1986), 290-307. R. Kiihnau, Uber die Grunskyschen Koe@zientenbedingungen, Ann. Univ. Mariae Curie-Sklodowska, sect. A 54 (2000), 53-60. I.M. Milin, Univalent Functions and Orthonormal Systems, Transl. of mathematical monographs, vol. 49, Transl. of Odnolistnye funktcii i normirovannie systemy, Amer. Math. Soc., Providence, RI, 1977. D. Minda, T h e strong f o r m of Ahlfors’lemma, Rocky Mountain J. Math., 17 (1987), 457-461. Chr. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975. H.L. Royden, Automorphisms and isometries of Teichmuller space, Advances in the Theory of Riemann Surfaces (Ann. of Math. Stud., vol. 66), Princeton Univ. Press, Princeton, 1971, pp. 369-383. H.L. Royden, T h e Ahlfors-Schwarz lemma: the case of equality, J. Anal. Math. 46 (1986), 261-270. M. Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149-164.

16

SS. Sh. St. Zh.

M. Schiffer and D. Spencer, Functionals of Finite R i e m a n n Surfaces, Princeton Univ. Press, Princeton, 1954. Y.L. Shen, Pull-back operators by quasisymmetric functions and invariant metrics o n Teichmuller spaces, Complex Variables 42 (ZOOO), 289-307. K. Strebel, O n the existence of extremal Teichmueller mappings, J. Anal. Math 30 (1976), 464-480. I.V. Zhnravlev, Univalent functions and Teichmuller spaces, Inst. of Mathematics, Novosibirsk, preprint, 1979, 23 pp. (Russian).

17

DECOMPOSITIONS OF MEROMORPHIC FUNCTIONS OVER SMALL FUNCTIONS FIELDS CHUNG-CHUN YANG

Department of Mathematics The Hong Kong Univ. of Sci. & Tech. Clear Water Bay, Kowloon Hong Kong mayangQust. hk P I N G LI

Department of Mathematics Univ. of Sci. d Tech. of China Hefei, Anhui, 230026 People’s Republic of China [email protected]. cn Let P denote the polynomial ring of meromorphic functions f , with their coefficients being meromorphic functions of f . In this paper, we shall use Nevanlinns’s distribution theory to discuss the decompositions of functions in P.As an application of the results, we are able to show the non-existence of admissible solutions f and g of a certain class of functional equations of the form: P ( f )= Q ( g ) , where both P ( f ) and Q ( g ) belong to P.

1. Introduction and Results

Throughout the the text, we shall use f ( z ) to denote a nonconstant meromorphic function, and use the standard notations in Nevanlinna’s value distribution theory, such as T ( T , ~N) (, r , f ) and m ( r , f ) .The notation S ( T , ~is) defined to be any quantity satisfying S(T,f ) = o ( T ( r ,f ) ) as T + cm possibly outside a set of T of finite linear measure. For details of the theory and its notations, we refer the readers to [ 2 ] . A meromorphic function a(.) (2cm) is called a small function of f ( z ) provided that T ( Tu, ) = S(T,f ) .It is obvious that the set of all small functions of f is a field of functions. Let P denote the polynomial ring of meromorphic functions f , with their coefficients being small meromorphic functions of f . When the coefficients of a polynomial P ( f ) E P are constants, P ( f ) can be decomposed as a product of linear functions in f . This result is no longer clear when the coefficients are replaced by small functions o f f . However, we shall utilize recent results obtained by Yamanoi ([6] and [7]) regarding the generalized second main theorem and its applications to show that if P ( f ) have few zeros, then P ( f )can be decomposed as a product of linear functions

18

or polynomials of degree two in f over small functions field Theorem A ([5]). Let f ( z ) be a transcendental meromorphic function., and P ( f ) E P with degree n, i.e.,

P (f ) = a, f

+

f ,- ... + a1f

+ a,-1

If aj ( j = 0 , ..., n ) are small functions then there are only three possibilities below.

of

+ ao, a,

20

f and if N ( r ,l/P(f ) )= S ( r ,f ) ,

(i)

(ii)

where cv 2 0 is a small function of f , and p = n/2 is a positive integer. (iii)

P ( f )= a , (f

+

= nun

b1

-

hlcvl)

(f

+--nu,

h2cv1) b2

where a1 2 0 i s a small function of f , p 1 , p2 are positive integers, XI, A2 E C, XI # X2,and p1 p2 = n , Alp1 X 2 p 2 = 0 . Moreover, if f satisfy X ( T f, ) = S(T,f ) , then the first case above must hold. Here we weaken the condition E(T, 1/P(f ) )= S(T,f ) in Theorem (A), and prove the following Theorem 1. Let f ( z ) be a transcendental meromorphic function, and P (f ) E P with degree n, i.e.,

+

P ( f )= a n y If

aj

+

+ a,-1

f

+ ... + a1f + ao, a,

20

(1)

( j = 0 , ...,n ) are small functions of f , then either

holds f o r any c E @, or one of the three cases in Theorem A holds. Theorem 2. Let f(z) be a transcendental meromorphic function, and P ( f ) a polynomial in f defined b y (1). Then either

19

holds for any positive number

E,

or one of the three cases in Theorem A holds.

As applications of Theorem 2, we can obtain the following Corollary 1. Suppose that f and g are transcendental meromorphic functions, n and k two positive integers such that k > 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<

a1

+

+

+ Qg)P.

(13)

of f , positive integers p 1 , p2 and constants = 0 , and

X2p1

f

X1Ql)F1(

+

xi==,f k P 3 + 1. By Lemma ( 2 ) , we have

(5

ak

);

+ N ( r , f ) + N ( r ,f ) + S ( r ,f ) ,

which implies (12). In the sequel, we suppose a2 $? 0. Let Pl = a n f n P 2 a,-l P can be rewritten as P = f2P1 1. By Lemma 2 , we have

+

+

which is equivalent to

Let

a = f'P, and

Then we have

(14)

X2(u1)P2.

+ -21f P ;

-

1 2

P'

-fP1--,

P

f"-3

+ ... + a2. Then

23

a=-

G 2P’

1 P‘ af=-2‘P

(19)

If a 0, then P is a constant. This is impossible. Therefore, a 2 0. It follows from the above equations that the poles of a must come from the zeros of P if they are not the poles of the coefficients of P . Thus N ( r ,a ) 5 N(r,1/P)+ S(T,f ) . On the other hand, from the second equation in (19) and by Lemma 1, we have m ( ~a ), 5 S(T,f ) . Hence Suppose that zo is a simple pole of f , but not a zero or pole of the coefficients of P . At the neighborhood of z0, we have Laurent expansion

where c, d are complex number and c # 0. Simple computation shows that

--(

ff ’((zz)) - z --1 -o

P’(z)

d 1 - -c( z - zo)

+O(z

-

z0l2

+O(z

-

zoy)

-n

- -(1 - B ( z - 2 0 ) P ( z ) z - zo

hold at the neighborhood of

20,

where

Therefore,

From (19), (21) and ( 2 3 ) , we get c =

-A and 2420)

thus

,

24

+ ... + a2 f 2 + 1.Then we have P = a, f " + R. Furthermore, we

Let R = a,-l fn-' have

anf

, ' which shows that the multiple poles of f must be the zeros of $ - n$ as long as they are not the zeros or poles of the coefficients of P.Note from (19) that such multiple poles off are zeros of a , and hence they are zeros of cp. From (17), (18),(19) and ( 2 5 ) , we have

2

a2fy+na= f H

where

P' H := a2P

-

n P' -p12 P

n-2

+ n f ' C Uk+2 f"' + Tn ~ +l 2aa2an-l nun

k=l

-~ a2an

an

(27)

From ( 2 6 ) , we see that any pole of f is a zero of H as long as it is not a pole or zero of the coefficients of P . By Lemma 1, we have m(r,H ) = S (r ,f ) . Therefore,

Let

Clearly, m ( r , H l ) = S( r ,f). From (17) and ( 2 7 ) , we get N(r,H1) = N ( r , 1 / P )

S(r1 f 1. Therefore,

Vr, H1) I

(r,

);

From ( 1 7 ) ,(18),(19) and ( 2 7 ) ,we deduce that

+ S ( r ,f 1

+

25

k-2 k=2

f

I

1 2

- -fP;

1 P’ + -fP12 P

n k-3 k=3

it follows that

3na3

Therefore, any zero of f is a zero of H I as long as it is not the zero or pole of the coefficients a z , ..., a,. Thus, if H 2 0 and Hl2 0, then

which prove (12). In the following we distinguish two cases. Case 1. H 3 0. In this case, from (26) and the first equation in (19), we get

26

, where

2 n

Q2:=-f

1 +-2an-1 f+-. n2an a2

(33)

Substituting PI = C;==,a k f n P 2 and (32) into (18), and from equation (19), we obtain

which is equivalent to

where

From the definition of

Q2

and P , it follows from the above equation that

where a1 = 0. If Q = 0, then from (34) and (36) we get

k t l -ak+l a2

If a,

+

~

2kan-l n2a,

-

2(n - k n

= 0 then it follows from

+ I) a k - l = : O , k = 2 ,..., n - 1 .

(37)

(37) that a2k-1 z 0, k = 1 , 2 , .... Note that if a, 2 0, then n is even. By Lemma 4, we can obtain the conclusion easily. In the sequel, we assume a, 2 O.From (38), we see that there exists a nonzero constant A such that a:-' AaK-,. Since n and n-1 are relatively prime, thus there exists a meromorphic function h which is small with respect to f such that a, = h". Therefore, by (38), we can find constants d k ( k = 2, ...,n ) such that

27

ak

= dkhk, k = 2 ,...,n.

Thus n

P = Cdk(hf)'

+ 1 = d,(hf

- T I )...( hf - r n ) ,

k=2

where

rk,

k = I, ...,n are roots of the equation

n

C dkzk + 1 = 0. If this equation k=2

, by Nevanlinna's second fundamental has at least three different roots rl,r 2 , ~ 3then theorem, we have

which yields (12) by Nevanlinns's first fundamental theorem. If all r k are equal to each other, then we have P = a,(f - r k / h ) n , which contradicts a1 = 0 and a0 = 1. If two and only two of rk are different, then P assumes the form (14). Now we consider the case: Q # 0. From (34), we can see that a is a rational function in f with coefficients being small functions of f . If this rational function can not be reduced t o a small function of f , then by Lemma 3, we have

T ( r ,f

) I T ( r ,a ) + S ( r ,f ) I

(T,

1 / P )+ S ( r ,f ) ,

which leads to (12). If this rational function can be reduced to a small function of f , then T ( r ,a ) 5 S ( r ,f ) .Thus from the second equation in (19) we get

" r , 1/P) I

w,

l/a) I S(T,f

)

In terms of Theorem A, F has one of the three forms described in Theorem A . Since a1 = 0 and a0 = 1, the first form cannot occur. The second form and third form yield (13) and (14), respectively. Case 2: H I F 0 From (25), (26) and (29), we can deduce that

f'

2a

= -n .f2

-

1

(-&-+ a ; ) f + 3a3a

By the arguments similar to that in Case 1, we can obtain

(39)

28

where n

k=3

2(n - k n

3ka3 + 1)Uk-1 + 2 4 ak

k+l -

~

(41)

a2

with = 0. By similar arguments used for Case 1, we still get the conclusion that Case 2 holds. The proof of Lemma 5 is thus completed. Lemma 6. Let f ( z ) be a transcendental meromorphic function, and P ( f ) E P defined by (1). Then either

or one of the three cases an Theorem A holds. Proof. Without loss of generality, we assume a,-1 sider the transformation f = f - an-l/nan.Thus I

P ( f )= a n f n

= 0, otherwise, we can con-

+ an-2fn-' + ... + a l f +

(43)

a0

If all ao, ..., a,-2 vanish identically, then P ( f ) = a, f ,. This means that P ( f ) satisfies the first case of Theorem A. Suppose that one of ao, ..., and an-2 do not vanish identically. If an-z z 0, then by Lemma 2, we have

which implies (42). In the sequel, we assume an-2 0. If a0 = 0, then we can assume a0 = ...u k - 1 = 0, ak # 0 , l k 5 n - 2. Therefore, P ( f ) = f k @ , where @ = a n f n P k an-2 fn-'-' + ... + a k . By Lemma 2 , we get

+

Since N ( r ,l / P ( f ) ) = N(r,l / f ) + N ( r ,l/@) + S(r,f), the above equation also implies (42). Suppose a0 # 0. From (43)) we have P ( f ) = a n f n q , where 1 P = bohn blh"-l + ... + b,-2h 2 + 1, h = -, bi = ai/an, i = O , l , ..., n - 2.

+

f

+

Therefore, N ( r ,l / P ( f ) ) = N ( r , l / @ ) S(r, h ) . By Lemma 5, we have

-( N ;) I -( N i)+ S ( r , h ) r,-

r,-

which is equivalent to (42), or the following two cases hold: ( i ) There exists a small function GO 2 0 of h and positive integer p such that n = 2p, and

+ Go)'

Q = bo(h2

Hence P ( f )= a,b,(l

+ Gof2)'

= a,(f2

fall)',

a0

=

1 T

QO

29

which means that F satisfies the second case in Theorem A . (ii) There - exists a small function 61 2 - 0 of -h and positive integers pl,p2, constants Xland A2 such that n = p1 p2, Alp2 X2p1 = 0, and

-

+

@ = b,(h

+ X,&)"'(h

+ +

X 2 6 p

Hence

+x & f ) y l +

P ( f ) = a,b,(l = a,(f

--

- XlQl)fi'(f

-

X26lf)fiZ

- X2Crl)fiZ

where a1 = - l / ( X l A 2 & ) , A1 = X2, A 2 = X I . This means that P ( f ) satisfies the third case in Theorem A . This also completes the proof of Lemma 6. 3. Proof of Theorem 1

Let g = f - c. Then P ( f ) can be expressed as

P ( f )= bogn

+ bign-' + ... + b,-ig + b,

(44)

where b k ( k = 0, 1,..., n ) is a polynomial in ao,..., a,, and bo = a,, b, = U,C

If b,

n

+ an-lcn-' + ... + alc + uo

(45)

0, then from (44) we get

i.e., ( 2 ) holds. Suppose b, g 0. We have P ( f ) = gn@, where @ := b,h"

+ b,-lhn-l + ... + blh + bo

and h = l/g. It is obvious that

By Lemma 6, we have

which implies

( a ) , or one of the following three cases holds:

(A)

where

a0

g 0 is a small function of h, and p = n / 2 is a. positive integer.

(47)

30

where a1 g 0 is a small function of h, p1, p2 are positive integers, X I , Xa E C, A1 Z &,and p1+ p2 = n,Alp1 X 2 p 2 = 0. If needed, by a suitable transformation, we can see that P ( f )must satisfy one of the three cases in Theorem A. This also completes the proof of Theorem 1.

+

4. Proof of Theorem 2

If P ( f ) does not assume any form in (i), (ii), or (iii) of Theorem A, then by Lemma 1, we see that (2) holds for any c E @. For any positive number E , we can select a positive integer k such that k > 2 / ~ Let . e l , c2, ...,c,+ be distinct values in C.By Nevanlinna’s second fundamental theorem, we have

which yields (3). This completes the proof of Theorem 2 5. Proof of Corollary 1

Suppose that f ,g, P ( f )and Q(g) are as defined with (5) being satisfied.If P(f)does not satisfy any form described in Theorem A, then by Theorem 2 we have T(T,

f ) I l?

(T,

+)+

ET(T,

f) + S(T,f).

On the other hand, by (5) and (4), we have T ( r ,Q ( g ) ) = n T ( r ,f)

+ S(T,f ) ,and

Therefore,

which is impossible for E described in Theorem A.

< 1 - n / k . Hence P ( f ) must satisfy one of the form

6. Proof of Corollary 2

Since g k = P ( f ) we , have kT(r,g) = nT(r,f ) + S ( r ,f ) .If T ( r ,f ) 5 :N S ( r ,f ) ,then we have

(T,

l/P(f))+

31

+

Therefore, the above equation leads to T ( r , f ) 5 (3n/2k)T(r,f) S ( r , f ) , i . e . , T ( r , f )= S ( r , f ) ,a contradiction. If P ( f ) satisfies (ii) in Theorem A, then we have g k = a, (f2 + ao)’, where a0 2 0 is small function of f , and n = 2p.Therefore, the multiplicities of poles of f are at least k / n , and the multiplicities of zeros of f 2 a0 are least k / p . By Nevanlinna’s second fundamental theorem, we have

+

5

;w, f) + Lk w ,f2)+ T ( r ,f) + S(T,f),

which implies T ( Tf) , 5 (2n/lc)T(r, f) + S(T,f).This contradicts k > 2n. Hence F can not satisfy (ii) in Theorem A. By a similar argument, we can show that P ( f ) can not satisfy (iii) in Theorem A. Hence by Theorem 2, P ( f ) must satisfy (i) in Theorem A. This also completes the proof of Corollary 2.

7. Proof of Corollary 3 Suppose that ( f , g ) is a pair of admissible solutions of (5). Since a,-l aoa,#O, we see immediately that F can not satisfy (i) in Theorem A. If P ( f ) satisfies (iii) in Theorem A, then we have Sk = a,

0 and

(f - Xlal)pl (f - X z a l ) p z ,

(50)

where a1 2 0 is a small function of f, p1, 112 are positive integers, X I , XZ E C,A1 # X2,and p1 112 = n, A l p 1 A2112 = 0. Let d j be the greatest common divisor of k and p j ( j = 1,2). From (50), we see that the multiplicity of any zero of f - Xjal is at least k / d j . Since k and n are maturely prime, the multiplicity of any pole of f is at least k . By Nevanlinna’s second fundamental theorem, we have

+

+

1

1 1

+ +

1

+

which implies k 5 d l d2 1. Since p1 p2 = n < k and d j 5 pj , we have n < k 5 n + l . Therefore, k = n + l . If d l < p1 or d2 < p2, then d1+d2 < p1+p2 = n, which yields k 5 n, a contradiction t o the assumption. Hence d i = p j , j = 1,2. Without loss of generality, we assume d l 5 d2. Therefore, k 5 2d2 1. Note that n > 3 and k / d a is an integer. We deduce that k = 2d2. And thus d2 = d l 1, which shows that d l and d2 are relatively prime. Note that d l can divide k . Thus, there

+

+

32

are two possibilities: dl = 1 or d l = 2. Therefore, k = 4 or k = 6, which contradicts the assumption. Hence F can not satisfy (iii) in Theorem A. A similar arguments shows that P ( f ) can not satisfy (ii) in Theorem A as well. By Theorem 2, we see that ( 3 ) holds for any positive number E . Therefore,

T ( r ,f ) F

(r,

t) +

ET(T,

f ) + S ( r ,f ) F

T ( r , g )+ m r , f

) + S ( r ,f ) .

+

From (5), we get T ( r , g )= : T ( r , f ) S(r,f).It follows from the above inequality that (I - 2) T ( r ,f ) 5 ET(T, f ) + S( r ,f ) , which is impossible for E < 1 - n / k . This also completes the proof of Corollary 3 . References 1. Clunie, J., On integral and meromorphic functions, J. London Math. SOC.,37(1962), 17-27. 2. Hayman, W., Meromorphic Functions, Oxford, Clarendon Press, 1964. 3. Li, P. and Yang, C.-C., Admissible Solutions of Functional Equations of Diophantine Type, preprint. 4. Mohon’ko, A,, The Nevanlinna characteristics of certain meromorphic functions, Teor. Funktsi? Funktsional. Anal. i Prilozhen., 14(1971), 83-87. 5. Mues, E. and Steinmetz, N., The theorem of Tumura-Clunie for meromorphic function, J. London Math. SOC.,23(1981), no. 2, 113-122. 6. Yamanoi, K., The second main theorem for small functions and related problems, Acta Math. 192(2004), no. 2, 225-294. 7. Yamanoi, K., The second main theorem for small functions and related problems, Acta Math. 192(2004), no. 2, 225-294. 8. Yang, C.-C. and Yi, H.-X., Quantitative estimations on the zeros of differential polynmials, Science in China, Series A , 1(1993), no. 23, 9-20.

33

SPEED OF APPROXIMATION TO DEGENERATE QUASICONFORMAL MAPPINGS AND STABILITY PROBLEMS V.M. MIKLYUKOV Volgogrud St. University, Russia

There are many results concerning existence and uniqueness for degenerate Beltrami systems (see Pesin [1969], Lehto [1970], Miklyukov and Suvorov [1972], Kruglikov [1973], Belinskii [1974], David [1988], Tukia [1991], Brakalova, Jenkins [1998], Gutlyanskii, Martio, Sugawa and Vuorinen [2001], Martio and Miklyukov [2002] at al.) The proofs are based on an existence and uniqueness theorem for nondegenerate quasiconformal mappings and on approximations of the degenerate Beltrami system

wF= p ( z ) w,, esssup Ip(z)I = 1 , by nondegenerate systems

w,,?= p n ( z )w,,,, esssup Ipn(z)l < 1 , n = 1 , 2 , .. . . A principal part of the proofs consists of convergence of {wn}to a homeomorphic mapping w = w p(2). The different methods providing convergence imply different existence theorems. We study the speed of convergence

w,-+wp as n + m . This problem is connected with the problem of stability for conformal maps in the class of quasiconformal maps.

1. Isothermal coordinates

Let D c R2 be a domain, a,nd let R be a two-dimensional surface in Rm, m _> 3, defined by a monotone W,:: vector function

E

= f(ZlrZ2) = (fi(Z), . . . , f m ( X ) ) : D

-+

Iw”

(1)

which realizes a homeomorphic map of D onto a set f ( D ) whose metric (and topology) is induced from Rm .

34

The vector function f is differentiable a.e. on D . Below we assume that a.e. on D .

rank(df) = 2

(2)

At each point x E D , where f is differentiable, the condition ( 2 ) is equivalent to 9 = g11g22

-

91"2 > 0 ,

where

are the coefficients of the first quadratic form 0. If g11(x) = g22(x),

then

(XI,

g12(x) = 0

a.e. on

D,

(3)

x2) are the isothermal coordinates on 0. In this case, we have

+

d s i = X'(X) ( d ~ : d ~ z ) , X2(x) = gll(z) = g 2 2 ( ~ ) . The assumption (3) means that f : D + R preserves a.e. angles between the curves and that f is conformal a.e. in the usual sense. Let D be a simply connected domain in R2 and R be a surface given over D by a vector function (1) satisfying ( 2 ) . Let x = ( x I , x ~ E) D be a point, at which f is differentiable and (2) holds. The condition ( 2 ) implies that at this point df # 0, the metric d s does ~ not degenerate, and every infinitesimal circle with respect to dsn centered at x is an infinitesimal ellipse with respect to the Euclidean metric. Denote by p(x) 2 1 and 0 5 0(x) < 7r the characteristics of such ellipse, i.e., the ratio p of the lengths of its great and small axes and the angle 0 between the great -+ axis and the direction 0x1. A calculation gives

The characteristic 0(x) is well defined at each point where p(x)

> 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

<

w,(ao) = (0,O)

and

wn(al)= (1,O)

( n = 1 , 2 , . . .).

(5)

We call a sequence of maps F, = f o w;' : B, -+ Rm with w, : D -+ B, satisfying (5) the canonical sequence of representations of the surface R corresponding to {Q,} with (4). Fix a sequence of canonical representations F, : B, -+ Rm of a surface R. Definition 1.2. W e s a y t h a t a sequence {F,} converges locally u n i f o r m l y t o a canonical h o m e o m o r p h i s m F : B(0,R ) -+ Rm if:

( i ) t h e d o m a i n s B, converge as n -+ 00 t o a d o m a i n B = B ( 0 ,R) as their kernel with respect t o t h e p o i n t = 0 ( i n other words, there exists R = lim R,,

<

15 R

,-00

5 00);

(ii) t h e sequence {F,} converges as n -+ co t o t h e m a p F u n i f o r m l y in each subdomain U CC B(0,R ) ;

:

B ( 0 , R )-+ Rm

(iii) t h e sequence of inverse m a p s F;' : R -+ R2 converges as n + 00 t o t h e m a p Fpl : R -+ B ( 0 ,R ) u n i f o r m l y o n each s u b d o m a i n 0 1 cc 0. Definition 1.3. Let D c R2 be a d o m a i n . W e s a y t h a t a f u n c t i o n P : D -+ JR i s W'>'-majorized in D if there exists a f u n c t i o n K E W ' > ' ( D )s u c h t h a t

P ( x ) 5 K ( z ) a.e. o n D .

(6)

A f u n c t i o n P i s called locally W 1 , 2 - m a j o r i z e d in D , if it i s W 1 , ' ( D ' ) - m a j o r i z e d o n each s u b d o m a i n D' cc D . The bounded functions provide simplest examples of W1>' majorized functions. Let P : D -+ R be a bounded function defined on a domain D , X 2 ( D ) < co. Here

36

we can choose K = supzEDP ( x ). It is clear, that K E W 1 , 2 ( D )and (6) holds everywhere on D . It is not known, how reach is the class of W1,2 majorized functions in the general case. We will discuss this question later. The following theorem is proved in my paper [2004].

Theorem 1.1 L e t R be a two-dimensional surface defined by a vector f u n c t i o n (1) over a s i m p l y connected bounded d o m a i n D c R2 a n d satisfying (2). L e t {Q,} be a sequence satisfying (4). Suppose t h a t for a.e. x E D ,

T h e n there exists a n isothermal coordinate s y s t e m = (<1,<2) o n R defined by a canonical h o m e o m o r p h i s m y = F ( < ) of a disc B(0,R ) C R2 o n t o R t h a t i s a l i m i t of t h e canonical sequence of representations F,(<) 1 B(0,R,) + R of R corresponding to {Qn}. Moreover, there exists a l i m i t limn+m R, = R, 1 < R < 00, and t h e sequence {F,} converges t o F ( < ) locally u n i f o r m l y in t h e disc B = B(0,R ) . T h e canonical h o m e o m o r p h i s m F : B

+R

as unique.

We have obtained also some explicit two-sided estimates for the radius R in terms of the distance la1 - sol, integral

and the infimum of length arcs separating x1 from

a0

with the ends on a D .

The differential properties of the canonical homeomorphism

F : B(0,R)

+ IW"

and of its inverse F-' are described by certain properties of f and f - l . We will study them separately.

2. Canonical homeomorphisms Let R be a surface of the form (1).Since the metric on f ( D ) is induced from Rm, 2-dimensional Hausdorff measure 2' ( E ) ,E c f ( D ) , is defined on R. If R1 C Rn and 0 2 c Iw" are two-dimensional surfaces and h : R1 + 02 is a homeomorphism, then we say that a map h has the Lusin N-property, if for every subset E c 01, Y2 ( E ) = 0, we have 3t2 ( h ( E ) )= 0.

37

Let R be a surface of the form (1) satisfying (2). Suppose, in addition, that the vector function f E W,?;(D). Since this map f is homeomorphic, then we have for every subset E c D the equality x2

(f(E)= )

1d x d x l d x 2 .

(8)

E

It follows from (2) and (8) that

'H2 ( f ( E ) )= 0 if and only if 'N2 ( E ) = 0 and that the maps f , f - '

(9)

have the Lusin N-property.

Since f is differentiable, by (9) a.e. on R , there exists a tangent plane Ty(R). Thus, we may define a.e. on R the gradient (connection) V Q . Indeed, let 'p : R + R be a function and y E R be a fixed point, in which there exists a tangent plane Ty(R). By the standard scheme of definition of connection on surfaces, we extend 'p onto a some neighborhood of the point y in Rm. By the symbol (p we denote a new function. P u t

where 0= Va- is the standard differentiation in Rm and uT means the orthogonal projection of a vector 'LL onto the tangent plane Ty(R). Fore an open bounded set U @ R, we denote U @ R by W ' J ' ( U ) , p 2 1, the collection of functions 'p : U + Iw with the following property: for every point x E U there exists a neighborhood V C Rm and a sequence of C1-functions 'pk : V t 1w such that the contraction (Pk = ' p k l v n u converges t o cp uniformly and, in addition,

for some q = const

< co.By W,',.,"(R) we denote the set of the functions 'p : R + R

belonging to W 1 > P ( Ufor ) every open subset U c R. By Theorem 1.1, a canonical homeomorphism F : B(0,R) + R is differentiable a.e. in the disc B ( 0 , R ) .If f E C k ( D ) ,k 1, then F defines the isothermal coordinates of the class C k on R, and F is a conformal mapping from D onto R in the standard sense.

>

On the other hand, conformal mappings of nonregular surfaces were studied only in very special cases. H.A. Schwarz [1869] proved that every tetrahedron and cube may be mapped onto the unit sphere (see W. Gresky [Inaugural Dissertation, Leipzig 19281 and C. Caratheodory [Conformal representation, 19321). On conformal maps of general polyhedral surfaces (see R. Kuhnau [1970],D. Ells, B. Fuglede [Harmonic Maps between Riemannian Polyhedra, 20011) and of manifolds with ~

38

bounded curvature (see Yu.G. Reshetnyak [1989] and also T. Tor0 [1994], S. Muller and V.Sverak [1995]).A generalized Christoffel Schwarz formula for piecewise conformal homeomorphisms and the related questions were considered by I.M. Grudskii [1986], [1989]. ~

In the case of W11,':-surfaces we can consider the conformality at the points where f is differentiable, i.e. the conformality of f : D + R almost everywhere in the domain D . The notion of conformality is not defined in general, since one encounters substantial difficulties while trying to make sense of conformal maps of non-regular (singular) surfaces. This issue is similar, for example, to the well-known problem of removability of singular sets in the flat case. As in the planar case, we can study different classes of mappings conformal outside of the singular set and having different properties 'in the large' (see, for example, D.E. Menchoff [1936], Yu.Yu. Trohimchuk [1963], E.P. Dolzhenko [1963], S.Ya. Khavinson [1963], G. David and P. Mattila [2000], et al.) The following theorem connects the properties of a map f : D + Rm with the properties of its canonical homeomorphism F . Theorem 2.1 L e t R be a surface given by a vector f u n c t i o n (1) over a simply connected bounded d o m a i n D c R2 and satisfying (2). Suppose t h a t a f u n c t i o n P defined by (7) i s W 1 > 2 - m a j o r i z e do n D . T h e n we have:

(i) t h e canonical h o m e o m o r p h i s m F : B(0,R ) + R and t h e inverse homeom o r p h i s m F-' : R + B(0,R ) ) have t h e L u s i n N - p r o p e r t y if a n d only i f f : D + R (respectively, f-' : R + D ) have this property; ( i i ) if f E W;:(D),

(iii) i f f - ' DIED:

t h e n F E Wly:(B(O,R));

E Wlkr(fl) f o r a n u m b e r

J

4 (911922

-g:2)

Q,

25

Q

5

m, and for every subdomain

Q

dx1dx2 < 00, P = (2-2'

(10)

Di

t h e n t h e inverse m a p F-l : R

+ B(0,R ) belongs to W,1,':(R).

3. Main Theorem

In addition t o Theorem 1.1,we shall establish the estimate for speed approximations of the canonical homeomorphism F : B(0,R) + R" by maps F,. For this purpose, we use the well-known concept of stability of conformal mapping in the class of mappings with bounded Dirichlkt integral and show that F, and F are close and can be obtained one from another by a small deformation. There are many results describing such closeness. Below we formulate the simplest statement. Assume that the surface 0 c R" has the conformal hyperbolic

39

type and the vector function

f in (2)

is chosen so that f : B(0,R) + Rm for some

O
I,

p,(z) = 1 for

5

, n = 1 , 2 , .. . ,

E I,,

Assume also that F ( z ) = f o w ( z ) ,F,(z) = fow,(z) and W ( X ) : B ( 0 , R )-+ B(0,R) have Lavrentiev’s characteristics ( p ,19)and (p,, 6) respectively. Assuming that the mappings w ( [ ) and w,(<) are automorphisms of disk B(0,R ) , R > I, normalized by conditions 5, we have the following theorem. Theorem 3.1. Suppose that the functions ( p 2 ( z ) - 1f o r

5

E I,,

have majorants K,(Z) of the class W,’>’(B(O, R ) ) ,n = 1 , 2 , .. .. Let

s,

=

61

1

(VK:/’(5)12p ( 5 ) d z l d z z

lI2 > 0 such that f o r all n 2 N and 0

(023

There are N >_ 1 and

EO

(11)

Ih(z) - 21 5 C’ A(RJZ6,), and f o r every measurable set E

C

h = W,

0

I.(

w-l

,

E B(O,R),

(12)

B(0,R ) the following estimate holds

I’H2(h(E)) - ‘Hz(E)l5 C”A,(RJZS,).

Here

A, ( E )

< 6, < E~

G

+

C A(&) &RE,

(13)

40

M = const 5 8 2 R .

The proof of this theorem is based on stability theorems of conformal maps in the class of maps with bounded Dirichlkt integral. The stability problem goes back t o M.A. Lavrentiev [1935], where the first qualitative solution was given for (1 E)-quasiconformal mappings of a disk onto a disk. The quantitative aspects of the Lavrentiev result were revised by P.P. Belinskii [1953] [1960]. The further development of this problems is connected with papers of S.L. Krushkal [1967], G.D. Suvorov [1968], J. Lawrynowicz [1969], V.I. Kruglikov, V.M. Miklyukov [1972], LA. Volynec [1977] ( n = 2) and with papers Yu.G. Reshetnyak [1961], [1967], V.M. Miklyukov [1969], P.P. Belinski [1973], A.P. Kopylov [2002] and others (n > 2). In the proof of Theorem 3.1, we used the corresponding results of V.I. Kruglikov, V.M. Miklyukov [1972] and I.A. Volynec [1977].

+

~

4. Remarks on W17’ majorized functions

Until now, no complete description of the W1i2majorized functions has been obtained. We notice here only some their limit properties, arising from the theory of functions with generalized derivatives.

A. On the set P,. It follows from the definition of W1i2 functions that the restrictions of the W 1 , 2majorized functions P onto almost all horizontal and vertical intervals are locally bounded. Assume that a domain D is the disk and the function P ( z ) defined by (7) is a W1>’majorized in D . In other words, there exists a function K ( z ) E W’,2(D),for which

P ( Z )5 K ( z ) for a.e.

IC

E D.

Denote by Lf(R’) the set of the functions cp(z) which are as the Bessel potentials cp(X)

=

1

Gl(lZ - Y O 4 Y ) dYldY2,

w2

of the functions u E L2(R2)with the Bessel kernel of the order 1 (see, e.g., Adams and H e d b e ~ - g ~Since ~ ) . D is a disk, there exists by Theorem 1.2.3 of34 for

41

every function p(x) E W’>’(D) a function cp*(x) E L:(IR2) such that a.e. cp*(x)= p(x). Thus, since K E W112(D),there is a function u E L 2 ( R 2 )such that

K ( x )=

J

G1(1x - y I ) u ( y ) d l ~ l d y z a.e. on

D.

w2

By Theorem 6.2.1,34the set

{x

E

D : lim K ( < )= co} C-tx

has zero conformal capacity, and therefore the following statement holds. Theorem 4.1. If P i s W1>’-majorized,the set

P, = {z E D

:

lim T+O

-

r2

J

P(y)dxn=ca}

B(x,r)

is of zero capacity. Problem I. Describe the class of functions p ( z ) : D majorized in D .

+ [ l , ~that ) are W1>’-

B. The class is not empty. We privide the conditions for characteristic p ( z ) which ensure that the quantities 6, --+ 0 and, consequently, the class of mappings described by Theorem 3.1 is not empty. Namely, the following statement holds. Theorem 4.2. Let p(x) : D + [l,co]be a continuous function of the class W’i2(D). Then the quantities S, defined b y

satisfy the condition lim S, = 0 ,--to3

Sketch of the proof. Let ( p ( z ) ,O(x))be the Lavrentiev’s characteristics in D, and let p : D -+ [l,fco] be continuous. For a given sequence Q , --+ 00, we find the sets I , c D and the functions P, involved in Theorem 3.1. Since p is continuous, the sets I , are closed. Suppose that for some Q , > 1, the set I , G D . Without loss of generality, we may assume that D \ I , is a domain.

42

For every function K, we have

6: 5

J’ lVI(:,/2(z)I2 p ( z )dz1dz2 D

Let cp : D \ I , be a function which is extremal for the condenser capacity (I,,aD; D ) , that is, cp = 0 on a D , cp = 1 on In and

J’

~ ~ c p ( zdzldzZ ) l ~ = cap (I,, a D ; D )

D\In

We choose K, of the form

This function has the desired properties and by the proved above,

Thus, if lim

n+m

Qb cap ( I n ,a D ; D ) = 0 ,

then 6, + 0. We show that the assumption p ( z )

An,,, =

J’

W ’,2 (D) implies (14).Indeed, let

1 ~ p ( z ) 1dzIdz2 2 <m.

In0 \In

Suppose that the set P, is not empty. We choose Qno > 1 so that the set In, lies strongly interior to D . Fix Q n > Qno. Consider the condenser ( D \ I n o , I,; D ) . The function

43

is admissible in the variational problem for the capacity of the condenser. Therefore,

This yields

n

we get that Setting Qn, = $Qn, + 00, and the theorem is proved.

the condition (14) holds. Thus, 6,

Problem 11. Describe the class of continuous functions p ( z ) : D which

+0

as 0

+ [l,co) for

liminf 6, = 0 . n+oo

References 1. I.N. PESIN:Mappings which are quasiconformal in the mean. - Dokl. Akad. Nauk SSSR 187, 740-742, 1969 (in Russian). 2. 0. LEHTO:Homeomorphisms with a given dilatation. - Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968), Springer-Verlag, 55-73, 1970. 3. 0. LEHTO:Remarks o n generalized Beltrami equations and conformal mappings. Proceedings of the Romanian-Finnish Seminar on Teichmuller Spaces and Quasiconformal Mappings (Bragov, 1969), Pub1 House of the Acad. of the Socialist Republic of Romania, Bucharest, 203-214, 1971. and G.D. SUVOROV: O n the existence and uniqueness of quasi4. V.M. MIKLYUKOV conformal mappings with unbounded characteristics (in Russian). - Investigations in the theory of functions of a complex variable and its applications, Inst. Mat. Akad. Nauk Ukr., Kiev, 45-53, 1972. Existence and Uniqueness of Quasiconformal “in Mean” Mappings 5. V. I. KRUGLIKOV: (in Russian). - Metric questions of functions theory and mappings, n. 4, 123-147, 1973. General properties of quasiconformal mappings (in Russian). 6. P. P. BELINSKII: Novosibirsk, Nauka, Sibirskoe otdelenie, 1974. 7. G. DAVID:Solutions de l’e‘quation de Beltrami avec Ilplloo = 1. - Ann. Acad. Sci. Fenn. Ser. A I Math. 13, 25-70, 1988. Compactness properties of p-homeomorphisms. - Ann. Acad. Sci. Fenn. 8. P . TUKIA: Ser. A I Math. 16, 47-69, 1991. and J.A. JENKINS: O n solutions of the Beltrami equation. - J . Anal. 9. M.A. BRAKALOVA Math. 76, 67-92, 1998. 0 . MARTIO,T. SUGAWA and M. VUORINEN: O n the Degenerate 10. V. GUTLYANSKII, Beltrami Equation. Trans. Amer. Math. SOC.,v. 357, n. 3, 2005, 875-900. 11. 0. MARTIO,V.M. MIKLYUKOV: O n existence and uniqueness of degenerate Beltrami equation. - Complex Var. Theory Appl. 49, n. 7-9, 2004, 647-656., 12. V.M. MIKLYUKOV: Isothermac coordinates o n singular surfaces. - Mat. Sb. 195,n. 1, 2004, 69-88. Conformal representation. - Ca.mhridge at the University Press, 13. C . CARATH~ODORY: 1932.

44

14. J. EELLSand B. FUGLEDE: Harmonic m a p s between R i e m a n n i a n polyhedra. - Cambridge Tracts in Mathematics 142, Cambridge Univ. Press, UK, 2001. Two-dimensional manifolds of bounded curvature, - Modern 15. Yu.G. RESHETNYAK: problems of math. Fundamental directions, v. 70, Itogi nauki i thechniki, VINITI, Moscow: 1989, 8-189. 16. T. TORO:Surfaces with generalized second fundamental f o r m in L2 are Lipschitz manifoZds. - J. Differential Geometry, v. 39, 1994, 65-101. 17. S. MULLER,V. S V E R ~ K On : surfaces of finite total curvature. - J. Differential Geometry, v. 42, 1995, 229-258. Construction of i n n e r coordinates in composite R i e m a n n i a n surfaces. 18. I.M. GRUDSKII: ”Differential, integral equations and complex analysis”, Elista, 1986, 30-45. 19. I.M. GRUDSKII: Christoffel - Schwarz f o r m u l a f o r polyhedral surfaces. - DAN SSSR, v. 307, n. 1, 1989, 15-17. Les conditions de monoge‘ne‘ite‘. - Act. sci. et ind. v. 329, 1936, 1-52. 20. D . E . MENCHOFF: 21. Yu.Yu. TROHIMCHUK: Continuous mappings and conditions of monogeneity. GIFML, Moskow, 1963. 22. E.P. DOLZHENKO: O n ”erasure” of singularities of analytic functions. - Uspechi math. nauk, v. 18, n. 4, 1963, 135-142. On erasure of singularities. - Litov. math. sb., 111, 1, 1963, 27123. S.YA. KHAVINSON: 287. 24. G . DAVIDand P. MATTILA:Removable sets f o r Lipschitz harmonic functions in the plane. - Revista Matem. Iberoamericana, v. 16, n. 1, 2000, 137-215. 25. M.A. LAVRENTIEV: S u r u n e classe de representations continue. - Math. sb., v. 42, n. 4, 1935, 407-424. On distortion under quasiconformal mappings. - DAN SSSR, v. 91, 26. P.P. BELINSKII: n. 5, 1953, 997-998. 27. P.P. BELINSKII: On measure area under quasiconformal mappings . - DAN SSSR, v. 121, n. 1, 1958, 16-17. Solution of extremal problems of the quasiconformal mappings theory 28. P . P . BELINSKII: with the variational method. - Sib. math. j., v. 1, n. 3, 1960, 303-330. 29. S.L. KRUSHKAL’: O n mappings E-quasiconformal in ”means”. - Sib. math. j., v. 8, n. 4, 1967, 798-806. Equistability of canformal mappings of closed domains. - Ukr. math. 30. G . D . SUVOROV: j., v. 20, n. 1, 1968, 78-84. On a class of quasiconformal mappings with invariant boundary 31. J. LAWRYNOWICZ: points, II, Applications and generalizations. - Ann. polon. math., v. 21, n. 3, 1969. 32. V.I. KRUGLIKOV, V.M. MIKLJUKOV: Stability theorems f o r B L - m a p p i n g s . - Dopov. AN USSR. ser. A, 1972, n. 5, 421-423 (in Ukr6inian); In sb. ” Metr. vopr. teor. funct. i otobr.” , vyp. 111, ”Naukova Dumka” , Kiev, 1971, 55-70 (in Russian). 33. I.A. VOLYNEC: On distortion u n d e r B L - m a p p i n g s . - Sib. math. j., v. 18, n. 6, 1977, 1259-1270 (in Russian). 34. D.R. ADAMSand L.I. HEDBERG: f i n c t i o n Spaces and Potential Theory. - SpringerVerlag, Berlin - Heidelberg - New York etc., 1996. 35. V.M. MIKLYUKOV: Conformal mapping of nonregular surface and its application. Volgograd State University Press, Volgograd, 2005 (in Russian). 36. H.A. SCHWARZ:Conforme Abbildung der Oberflache eines Tetraeders auf die OberfEache einer Kugel. - J reine angew. Math., v. 70, 1869, 121-136. 37. REINERKUHNAU:Trianguliere Riemannsche Mannigfaltigkeiten mit ganz-linear Bezugssubstitutionen und quasikonforme Abbildungen mat stuckweise kpnstanter k o m plexer Dilatation. - Mathematische Nachrichten, Band 46, Heft 1-6, 1970, 243 -261.

45

38. WALTER GRESKY: Konforme Abbildung der Oberflache eines rektangularen Hexaeders auf die Kugeloberflache. - Inaugural - Dissertation zur Erlangung der Doktorwurde der Hohen Philosophischen Fakultat der Universitat Leipzig, Weida, i. Thur. 1928, 1-74.

46

GRUNSKY INEQUALITIES, FREDHOLM EIGENVALUES, REFLECTION COEFFICIENTS REINER KUHNAU

F B Mathematik und Informatik Martin-Luther- Universitat Halle- Wittenberg 0-06099 Halle-Saale, Germany [email protected] We give the exact domain of variability of a fixed Grunsky functional in the class of all hydrodynamically normalized schlicht conformal mappings of the exterior of the unit disk for which the unit circle transforms onto a quasicircle with a given Fredholm eigenvalue.

Keywords: Univalent functions, Grunsky coefficients, quasiconformal map, Fredholm eigenvalues.

1. Introduction

Today a central role in Geometric Function Theory plays the Grunsky functional. We will restrict ourself here to the simply-connected case. Then we consider in the complex plane the class of all schlicht conformal mappings w = f ( z ) of IzI > 1 with the hydrodynamical normalization

The corresponding Grunsky coefficients a k l are defined by the development

and with a given fixed system of complex numbers the Grunsky functional Gf is defined as

Gs =

C", k I-1

-

11.

~ 1 ,. ..

aklxkxl 1Zkl2

ck=l7

.

, xn ( n 2

1,

1xk

l2 > 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

<

cp'(z) = z1 has its zero zo =

+

222

-2 in the unit disk IzI 5 1. Because the left-hand side of ( l a ) , 1 q w )= q w + - 22(w2 2 b l ) , 2 -

-2= zo, the equality qwo) = cpo

has its zero of the derivative at wo =

cp(Z0) -

yields

-

2bl = -KX:.

Therefore the equation (9) becomes

22(w

-@

=

{

z,(z

-~

0 - KE ) ~

2

($ - z)for IzI 2 I, (16)

z2(z - ~

0 - )KQ ~ ( 2

2

- zo)

-2

for IzI 5 I .

With (16) and zo = the parameter problem, mentioned at the very beginning of this section, is completely solved in the case n = 2, lzol 1. Particularly, we obtain for IzI = 1 the corresponding image C with = qc = K by the chain of elementary mappings

&

3 = .2(z

- 20)

2

, tu = 3 - K j , w = zo +

<

a.

(17)

"Especially, this means that the derivative of q ( z ) and of q ( z ) - n q ( l / F ) must have the same number of zeroes for 1z/2 1. This also follows by RouchB's theorem.

52

A s a result, w e obtain a f a m i l y of quasicircles C , depending o n t h e p a r a m e t e r a n d w i t h explicitly k n o w n values = qc. In the special case lzol = 1 we have a cardioid in the 3-plane. We repeat that the final quasicircle C is always analytic. x1,x2,

Finally, we leave the friendly reader with the following general inverse problem.

&

Problem. Is it possible t o produce a n y quasicircle C with = qc as t h e i m a g e of t h e unit circle by t h e special m a p p i n g s (9), generalized by a suitable infinite case, perhaps also w i t h a n additional approximation pracedure ? Here one has to bear in mind also the cases of quasicircles C without a corresponding quadratic differential for the extremal quasiconformal reflection; cf., for example, the case of a square C [Ku6]; further examples e.g. in [Kr3,4], [ K u ~ ][W]. , 4. Analogue: Golusin's inequalities

Completely similar to Theorem 2 there holds (always with the usual branch of the logarithm) Theorem 3. For every fixed s y s t e m of complex n u m b e r s 7 1 , . . . ,yn ( n 2 ~ > ~ 0 ) and 1 ~every fixed a n d distinct p o i n t s 2 1 , . . . , z n w i t h l z k l > 1, in 1, 1 t h e class C ( I C )t h e d o m a i n of variability of Golusin's f u n c t i o n a l

i s t h e closed disk defined by

lGjl

5 IC .

(19)

Equality holds for t h e m a p p i n g which i s uniquely a n d defined implicitly by

and i t s modifications by replacing all Y k by e i e y k . T h e i m a g e of 1.z = 1 u n d e r these m a p p i n g s is always a closed analytic J o r d a n curve C w i t h XC = qc = IC. Again it remains as an open question: Are there further extremal mappings ? Proof. We can write (20) again in form of (12), now with n k=l

53

and a corresponding @(w).Therefore we can use for the proof of Theorem 3 the same ideas as in Section 3. The great similarity of Theorem 2 and Theorem 3 is not a marvel because of the equivalence of the analogous theorems for the class E ( K ) ;cf. [Kul], [KKl], [PI.

References Gr. H. Grunsky, Koefizientenbedingungen fur schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), 29 - 61. Krl. S. L. Krushkal’, Quasiconformal mappings and R i e m a n n surfaces, V. H. Winston & Sons, Washington, D. C./ John Wiley & Sons, New York etc. 1979 (Russian original: Izdat. “Nauka” , Sibirsk. Otd., Novosibirsk 1975). Kr2. S. L. Krushkal, O n the Grunsky coeficient conditions, Siberian Math. J. 28 (1987), 104 - 110. Kr3. S. L. Krushkal, Variational principles in the theory of quasiconformal maps, Handbook of Complex Analysis: Geometric Function Theory, V01.2 (ed. R. Kuhnau), Elsevier, Amsterdam etc. 2005, 31 - 98. Kr4. S. L. Krushkal, Univalent holomorphic functions with quasiconformal extensions (variational approach), Handbook of Complex Analysis: Geometric Function Theory, V01.2 (ed. R. Kuhnau), Elsevier, Amsterdam etc. 2005, 165 - 241. KK1. S. L. Kruschkal und R. Kuhnau, Quasikonforme Abbildungen - neue Methoden und Anwendungen, B. G. Teubner Verlagsgesellschaft, Leipzig 1983 (Russian edition: Izdat. “Nauka”, Sibirsk. Otd., Novosibirsk 1984). KK2. S. Krushkal and R. Kuhnau, Grunsky inequalities and quasiconformal extension, Israel J. of Math. 152,49 - 59 (2006). Kul. R. Kuhnau, Verzerrungssatze und Koefizientenbedingungen v o m GRUNSKYschen Typ fur quasikonforme Abbildungen, Math. Nachr. 48 (1971), 77 - 105. Ku2. R. Kuhnau, Z u den Grunskyschen Coefizientenbedingungen, Annales Academiae Scientiarum Fennicae, Ser.A.1. Math. 6 (1981), 125 - 130. Ku3. R. Kuhnau, Quasikonforme Fortsetzbarkeit, Fredholmsche Eigenwerte und Grunskysche Koefizientenbedingungen, Annales Academiae Scientiarum Fennicae, Ser.A.1. Math. 7 (1982), 383 - 391. Ku4. R. Kuhnau, Z u r Berechnung der Fredholmschen Eigenwerte ebener Kurven, Zeitschr. angew. Math. Mech. (ZAMM) 66 (1986), 193 -200. Ku5. R. Kuhnau, W a n n sind die Grunskyschen Koefizientenbedingungen hinreichend fur Q-quasikonforme Fortsetzbarkeit 2, Comment. Math. Helvetici 61 (1986), 290 - 307. Ku6. R. Kuhnau, Moglichst konforme Spiegelung a n einer Jordankurve, Jahresber. d. Deutsch. Math.-Verein. 90 (1988), 90 - 109. Ku7. R. Kuhnau, Quasiconformal reflection coeficient and Fredholm eigenvalue of a n ellipse of hyperbolic geometry, Publ. de 1’Inst. Math, Nouv. S6r. (Beograd), 75 (89) (2004), 77 - 86. P. Ch. Pommerenke, Univalent functions, with a chapter o n quadratic differentials by Gerd Jensen, Vandenhoeck & Ruprecht, Gottingen 1975. S. M. Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149 - 164. W. St. Werner, Spiegelungskoefizient und Fredholmscher Eigenwert fur gewisse Polygone, Annales Academiae Scientiarum Fennicae, Math. 22 (1997), 165 - 186.

54

SUMS OF RECIPROCAL EIGENVALUES B O D 0 DITTMAR Institut fur Mathematik Martin-Luther- Universitat Halle- Wittenberg bodo. [email protected]. de

Contents 1. 2. 2.1 2.2 3. 3.1 3.2

Introduction Membrane problems Fixed membrane Free membrane Stekloff eigenvalue problems Stekloff problem Mixed Stekloff problem

1. Introduction

Eigenvalues and extremal problems for eigenvalues of partial differential equations with various boundary conditions are on the one hand important for people interested in applications such as acoustics, theoretical physics, quantum mechanics, solid mechanics and on the other hand this field is one of the most fascinating of mathematical analysis. The purpose of the lecture are formulas and isoperimetric inequalities for sums of reciprocal eigenvalues of the Laplacian and related eigenvalue problems. For the eigenvalues of the fixed membrane problem it was proven by P d y a and Schiffer 195325

where X j are the eigenvalues of a simply connected and bounded domain in the plane with the maximum of the conformal radius i. and XI"' are the eigenvalues of the unit disk. This result has been generalized and extended by many authors. For a fuller treatment we refer the reader to.1>7)12 The eigenvalue problems which we consider here are eigenvalue problems of Green's functions and a simply consequence of the well-known Expansion Theorem of the classical theory of integral

55

equation are formulas for the sum over the squares of all reciprocals. After conformal transplantation into the unit disk this formulas are given in terms of the coefficients of the Taylor series for the conformal mapping. This formulas allow on the one hand to calculate the exact value of the C r A j 2 in some cases and on the other hand a variational characterization for the sum of the first n reciprocal eigenvalues has been obtained. Basing on this variational characterizations we are able to prove isoperimetric results for sums of reciprocals eigenvalues. We consider the fixed and free membrane eigenvalue problem and the Stekloff and mixed Stekloff eigenvalue problem. Among others the above inequality is proven for the free membrane eigenvalues. 2. Membrane problems

Let D be a simply connected domain in the plane, which is bounded. We consider the following eigenvalue problems = 0 in D ,

&+Xu

u = 0 on d D ,

(1)

and

Av + pu = 0 in D , dV

= 0 on d D dn n stands for the normal to d D , X and p for the eigenvalue parameters. It is wellknown that there exists an infinity of eigenvalues with finite multiplicity -

The eigenvalues are the stationary values of the Rayleigh quotient

where Lj ranges over all j-dimensional subspaces of the Sobolev space H i ( D ) in problem (1) and H 1 ( D )in problem ( 2 ) . 2.1. Fixed membrane

The basic isoperimetric inequality on vibrating membranes with fixed boundary is that of Rayleigh, Faber and Krahn1~12~19~24 Theorem l.(Faber, Krahn) Among all membranes of given area, the circle hast the lowest first eigenvalue.

56

Proofs were given independently by Faber and Krahn in the early 1920s.. A counterpart of the Faber-Krahn inequality was derived using conformal mappings by P6lya and S ~ e g .o ~ ~ Theorem 2.(P61ya and Szego 1951) For all simply connected domains in the plane holds .2

<

.2

lr - 3 ,

(6)

where j denotes the first positive root of the Bessel function Jo, and 7: means the maximum of the inner conformal radius of D . Equality holds if and only if D is a disk. Of course, the products Xi+ and p i + depend only on the shape of the domain

D , not on its size. For higher values of n the products Xt7: and p i + do not attain the maximum for the disk and so do A, and p n for domains with the same i . But for domains with a sufficiently high symmetry the circle yields the maximum of A i r and p i + , more precisely22 . Theorem 3.(P61ya 1955) If the domain D is convex and possesses a symmetry of the order not lower than 2n 1, then

+

where Z(n) and m(n) are strictly increasing functions of n, will be inferior to the corresponding quantity belonging to a circle. On the other hand, from Weyl's well-known asymptotic result28

where A = ID1 is the area of D , it follows that for sufficiently large n it holds X,(D) < Xn(U1) and p L , ( D ) < pn(U1) if D is the conformal image of the unit disk U1 under a normalized conformal map such that i is the same because then ID1 > lull. P6lya and Schiffer derived an interesting result. They use conformal transplantation and convexity arguments in order to prove25 Theorem 4.(P61ya and Schiffer 1953) For any n

where Xio) denotes the eigenvalues for the unit disk. Equality holds if and only if D is the unit disk. There are generalizations given by C. Bandle and closely related papers by J. H e r ~ c h ' >. ~ An easy consequence of the Expansion Theorem is the following formula for the sum of all reciprocals3

57

Theorem 5.(Dittmar 2002) Let f be a conformal mapping from the unit disk U1 onto the domain D with the area A, then it holds

where G ( z ,() denotes Green's function of the unit disk. From this theorem it follows a formula'l in terms of the coefficients of the Taylor series for the conformal mapping which is similar to a formula for the torsional rigidity given in [25, p.3301 Theorem 6.(Hantke 2006) For the eigenvalues of the fixed membrane problem holds 0 0 .

0 3 0 3

k=l m=l 1=1

k=2

m=l l=1

where the coefficients A, B, C , D , E are known and 00

0 3 0 3

n=O

m=l n=l

It is worth pointing out that in some cases it is possible to calculate (10) exactly.'' Examples 1. Disk Let D1 be the unit disk then holds 4

C;l*=,,

8

03

_ _ _5

- 7r2

+ c n = 1 4(2n+4)(4n+4)(2n+2) For the unit disk see also [3, p.511.

48

32

'

2. Cardioid

f ( z ) = z + z 2 / 2 maps the unit disk onto a cardioid. It follows from the formula above C;, = &x2 - &. Similar results are given in" for the image of the

$

+

unit disk by f n ( z ) = z ;zn,n = 3,4, ... . For the cardioid follows in the same matter also the value ll7r/48 for the torsional rigidity" . Starting with (10) it follows3 Corollary 1. Let f be the conformal mapping from the unit disk U1 onto the domain D with the area A = If'(z)12dA< m, then

s,

58

with (15) where uy' is the j-th eigenfunction of U1. The following lemmas are essential tools in order t o give isoperimetric r e ~ u l t s ~ , ~ Lemma 1. Let f ( z ) = a l z + a2z2 + ... be an univalent conformal map of the unit disk onto U I . We denote by ui")and Xi"' the j-th eigenfunction and eigenvalues of the unit disk. If uy' is radial, then it holds

If up' is not radial, then let u?' and u E l be the eigenfunctions whose sum of the squares is radial and both of them have the same eigenvalue Xi"', then it holds

Equality occurs in both inequalities if and only if f ( z ) = a l z . Lemma 2. For the eigenvalues of the fixed membrane problem holds

s,

lf'(z)l'~.i(z).j(z)dA, = & , j , i , j = 1,2, ..., n, uj E L 2 ( U ) . where Using this variational characterization we obtain following5 the P6lya-Schiffer result (9) and with a result from Hardy, Littlewood and P61ya5 follows18 Theorem 7.(Laugesen, Morpurgo 1998) Let @ ( a )be convex and increasing for a >_ 0. Then

where n is either a positive integer or +m, and D is the conformal image of the unit disk U1 under the conformal mapping f with If'(0)l = 1, with strict inequality unless D equals the unit disk. 2 . 2 . Free membrane

In the free membrane case the situation is more complicated. Encouraged by a conjecture of Kornhauser and Stakgold it was proven by Szego 1954 that of all simply connected domains of a given area, the circle yields the maximum value of p2. Szego and Weinberger noticed that the same proof causes the disk to minimize

59

-1+ P2

1 P3

of all simply connected domains of a given area. Generalizations are given by C. Bandle 1972, T. Gasser and J. Hersch 19681’7 and N. Nadirashvili 1997.” In order t o derive a formula similar to (10) we introduce a special symmetric function which will be the kernel in an integral equation for the eigenfunctions of Problem (2).5 Lemma 3. Let N f ( z , C ) be the following symmetric function depending on an univalent conformal map f

where

and A =

sulIf‘(z)12dAZ< co.Then 1

Hf(4= ,If(.)12 + M z ) , where h ( z ) is a harmonic function in U1 with = A/(27r) - 1/4% sufficiently smooth f ( z ) and n is the outward pointing normal. In particular 1

2

HfEZ(Z) = -121 .

4

(25) on dU1 for a

(26)

Now we are able t o give an integral equation for the eigenfunctions vjlf’l with the eigenvalues pj/A . Theorem 8. For the eigenfunctions of the free membrane Problem (2) it holds, if D is sufficiently smoothly bounded ~ j ( 5= )

sul

A

ll

Nf(z,5)~~j(z)lf’(z)I~dA,,j = 2,3, ... ,

where A = lf’(z)I2dA < 03 and N f referring to Lemma 2.2. The kernel N f ( z ,< ) l f ’ ( z ) l l f ’ ( ~ ) \has only the eigenfunctions vj(z)lf’(z)l with the eigenvalues pj/A in the space V f = L2(U1) n {u : uIf’(z)IdA = O}. A consequence of Theorem 2.2 is the following Theorem S.(Dittmar 2002) If D is a simply connected sufficiently smoothly

,s,

60

bounded domain with the area A = of the free membrane it holds

,s,

If’(z)12dAZ< m. Then for the eigenvalues

--&

Ju J,N(z, C)If’(z)121f’(C)12dA,dA~. where C = The following lemmas are helpful in order to get isoperimetric results3 . Corollary 2. If D is a simply connected sufficiently smoothly bounded domain with the area A = If’(z)12dA, < m. Then

sul

with

Lemma 4. For a radial eigenfunction u k ’ ( z ) = aoJo(k,,,r), conformal map f ( z ) = a l z u2z2 ...

+

+

we have for every

Let

u$’(z) = a,J,(km,,r)cosmcp, V,+~(Z) (0)

(32)

= am~m(km,nr)sinmcp,z = rezv,

be eigenfunctions belonging to the same eigenvalue &I. f ( z ) = a1z + a 2 2 2 + ...

Then it holds for every

Equality occurs in both inequalities if and only if f ( z ) = u p . The preceding lemmas lead to the following isoperimetric result. Theorem lO.(Dittmar 2002) For all simply connected domains with the maximal conformal radius 1, the unit disk and only the unit disk yields the minimum of

61

where A is the area of the domain. Remark 1. 1. We can also calculate the minimum of the preceding theorem. It follows 25 5 N 2 ( z ,C)dA,dAi = --7r2 (35) 32 48

l

+

2. A still open conjecture of P6lya is p, < 47rnA-' < A., It was proven by Kroger in 1992 that p n < 87rnA-l. Using Krogers result it is easy to check that a minimizer of A2 C pLj2 must have an area less than 2.163..7r For the finite case a variational characterization is also given.5 Theorem 11.

s,

with hjIf'(z)12dA, = O,S,hihjIf'(~)1~dA, = Sij. We take the functions uj = v j( 0 ) + c j , j = 2, ...,n,with cjA = - ~ , v ~ o ) I f ' ( z ) 1 2 d A such r , that S , u j ( f ' ( ~ ) ( ~ = dA

0.

A consequence is the following isoperimetric result Theorem 12.(Dittmar 2005) Let D be a domain with the area A < n 2 2, we have

03,

for any

( where i- denotes the maximal value for the inner conformal radius of D and pj()) denote the eigenvalues of the unit disk. Equality occurs if and only if D = U . It is worth pointing out that a slight change in the proof in5 gives more, namely the following6 Theorem 13.(Dittmar 2006) Let D be a domain with the area A < 03, for any n 2 2, we have

where r denotes the maximal value for the inner conformal radius of D and p (j o ) denote the eigenvalues of the unit disk. Equality occurs if and only if D = U .

3. Stekloff eigenvalue problems We consider now the following classical eigenvalue problem

Au = 0 in D , dU

- = uu on d D ,

dn An infinity of eigenvalues also exists with finite multiplicity 0 = v1

< u2 5 v3 5 ... .

(36)

(37)

62

It is usual t o call this eigenvalue problem Stekloff problem. We consider also the mixed Stekloff problem, which is the following one

Au = 0 in D ,

au = gu on CI,

-

dn u = 0 on

C2,

where d D = C = CI U C2. There is also an infinity of eigenvalues with finite multiplicity

3.1. Stekloffproblem

Transplanting the eigenvalue problem (36) conformally in the unit disk and using the Neumann function of the unit disk and the well-known fact that the Stekloff eigenfunctions are eigenfunctions of the Neumann function on the boundary, it follows a formula for the sum of all squares of reciprocals. Theorem 14.(Dittmar 1998) I f f is the univalent conformal mapping of the interior of the unit disk onto the domain D with f(0) = f'(0)-1 = 0, then for the eigenvalues of problem (36) with p 1 we get

=

where N o is Neumann's function of the domain D . It follows once more4 Corollary 3. If D is a simply connected sufficiently smoothly bounded domain. Then 7

with a harmonic function bj in the unit disk U which satisfied

where us"'and u;"' are the eigenvalues of the unit disk. It holds also here a lemma analogously to Lemma 1 and Lemma 4. Lemma 5. If D is a simply connected smoothly bounded domain with L = ldDl then it holds

63

In the same way follows4 Theorem 15. Let D be a sufficiently smoothly bounded domain with the maximal conformal radius 1, then holds

It is also possible to obtain isoperimetric results for the finite case. Using these and the Hardy, Littlewood and P6lya result it is easy to see that the inequalities are weaker than the result given by Hersch, Payne and Schiffer 1976.13 3.2. Mixed Stelcloff problem

The eigenfunctions of the mixed Stekloff problem are the eigenfunctions of the Robin function. This is Green's function for the mixed problem. There are a lot of interesting results for the Robin function given by Duren and Schiffer." The situation in the mixed Stekloff problem is easier because the Robin function is invariant under conformal maps. We restrict ourselves to the case of the ring domain where Cz is one of the components of the boundary. Hersch and Payne found out that in this case the conformal modulus of the domain D can be characterized by the mixed Stekloff eigenvalues. It holds the following Theorem 16.(Dittmar 1998) Let f m a p the annulus A ( l , R ) onto the piece-wise analytically bounded ring domain D such that the circle ( 1 . ~ 1 = l} corresponds to the boundary component C1, then it holds

The Robin function of the annulus A ( l , R ) was given by Duren and Schiffer in 1991: In R

R(z'

=

c O0

+

n=l

(Rn-Kn) Tn(Rn+ R-n) cosn(19 - cp),z = Reip,<= Rei8.

(45)

A simple calculation yields

with

and for k = 1 , 2 , ...

Ba =

In R ( R k- R - k ) x2k(Rk R - k ) " ' .

+

(48)

64

For this mixed Stekloff problem the following result holds Corollary 4.(Dittmar 1998) Of all doubly connected conformally equivalent domains with a piece-wise analytic Jordan curve C1 without outward-pointing cusps and a given length L of the boundary curve C2 the annulus has the lowest amount of C gj-’). In other words L2

- > -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 <

+ lv(z)\ < 1) Z;rf all the map-

JWFJI I ) .

Actually the proof in [27] works for linear families of sense preserving mappings. Our Theorem 3.1 below (see also [19]) states that any uniformly elliptic general Beltrami equation (0.2) admits a primary pair of solutions. Theorem 3.1 gives a positive answer to Conjecture 1 in [27]. Is this true for not necessary uniformly elliptic systems is an interesting delicate question, presumably not easy. I t is to be expected that no analogues of linear k-dimensional families of orientation preserving QC-mappings exist in dimensions n 2 3 , k > 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
= 1 and

forIp-21<&

(2.7)

for some sufficiently small positive E , depending on the uniform ellipticity constant in (0.4). Then the system (0.2) for ~ ( zreduces ) to the system of linear singular equations for the density w -

w - pSw

-

uSw = h, h = up + ?iu

(2.8)

in the space LP(@)for p satisfying (2.7). The equation (2.8) acts as an invertible operator in LP(@)for the range of p defined by (2.7). Most important is the case p > 2 . Then the function W ( Z ) in (2.3) realizes a continuous, in view of the Sobolev imbedding theorems, even Holder continuous (with exponent a = 1- > 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 .

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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.

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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

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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).

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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.

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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 .

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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

<$

or equivalently

K<3

(4.1)

when formulated in terms of associated K-QC-mappings. An interesting special class of Beltrami equations (0.2) arises when u+p=O,

u=-p

and (0.2) takes the form wz

-

-

p(wz - w,) = 0.

(4.2)

The ellipticity condition (0.3) then reads Ipl < $ < $. A characteristic feature of (4.2) is that the identity mapping w(z) = z is a principal homeomorphism of (4.2) for arbitrary dilatation p. Then the condition (1.2) for a primary pair ( z , $ ( z ) ) takes the form Im($,)

#0

a.e.

(4.3)

Theorem 4. T h e B e l t r a m i equation -

wz - q(z)(wz - w,) = 0

w i t h t h e compactly supported measurable dilatation ( q = 0 in t h e neighborhood of 00) satisfying t h e ellipticity condition (0.4) a d m i t s a global solution $ : C + C satisf y i n g condition (4.3). T h e n t h e pair ( z , $ ( z ) ) i s a p r i m a r y pair f o r t h e B e l t r a m i equation (4.2).

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The main idea of the proof is the “factorization” of solutions of (4.2) through the solutions of a Beltrami equation

and the “conjugate” or inverse Beltrami equation -

dW

(4.5) with a special choice of the auxiliary dilatations q l ( z ) and F({) in (4.4) and (4.5). The function yl(z) in (4.4) is determined by q ( z ) in (4.2) by the formula

The uniform ellipticity condition (0.4) ensures that (4.4) is uniformly elliptic also. The choice of q l ( z ) is “canonical”: it depends on q ( z ) in (4.2) only. The choice of F({) for the “conjugate” Beltrami equation depends on the selected global homeomorphic solution of (4.4). If w~ is a normalised-“principal”homeomorphism of (4.4) (i.e. it is determined by putting a = 1 in ( 2 . 3 ) ) then any other global homeomorphism { ( z ) is determined by the formula ((2)

= L(.Wo(z))

(4.6)

+

for some linear L(w0)= bwO c. Assuming a ( ( z ) fixed we determine

F(<) = 41 (z-%))

F({) in the equation or

F(<(2))

(4.5) by

= 41(z).

(4.7)

For further details of the proof see [19]. Since the primary pair (4,$) for (0.2) allows to recover the coefficients p and u of the general Beltrami equation (0.2),

it plays such a fundamental role in describing the totality (infinite dimensional) of all solutions of (0.2), in particular in discussing the G-convergence phenomenon and the G-closed classes of elliptic operators in the complex plane (see [27)]. As is well known (see e.g. Vekua’s monograph [44]), the auxiliary equations (4.4) and (4.5) admit global solutions without necessarily assuming that lpl Iu/is compactly supported. The factorization procedure used in the proof of our theorem then can be extended to the general Beltrami equation (0.1) in the whole complex plane C. There are also many other interesting analytical and geometrical problems related with the important class of primary solutions of Beltrami equations (0.2). Some of these will be hopefully discussed in a subsequent publication 1251.

+

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5. Lavrentiev fields and quasiconformal mappings The infinitesimal geometric meaning of a differentiable transformation w a point z0 is defined by the linear tangent map

= w ( z ) at

It transforms invariantly ellipses in the tangent plane at z0 into ellipses in the tangent plane at the image point w ( z 0 ) . Ellipses centered at z are defined up to a similarity transformation by the ratio p 2 1 of their semiaxes and, if p > 1, the angle Q mod T between majoraxis and the positive z-axis, and denoted by & ( p , 8 ; z ) or &h(p,8 ; 2 ) where h is the length of the same minoraxis. The pair ( p , Q) is called the characteristic of the infinitesimal ellipse, and the family Eh(p, 8 ; z ) , h > 0 , z E G, is a field of infinitesimal ellipses (Lavrentiev field). A homeomorphism w = w ( z ) is said to map the infinitesimal ellipse E ( p , 8 ; z ) onto the infinitesimal ellipse € ( P I , 81; w ( z ) ) if the tangent map D w ( z ) transforms € ( p , 8 ; z ) onto I ( p 1 , Q I ;~ ( z ) In ) . the Euclidean case the infinitesimal ellipse E ( p , 8 ; z ) at z is identified with geometric curve-ellipse-in the mapped domain G, for h sufficiently small and then it is meaningful to consider the image W(&(P, Q; z ) )

~h

as a geometric curve ~h in the w-plane. If the image curve ~h lies in the ring domain ) & l ( p l , & ; w ( z ) ) k, 5 1 , such that between the curves E k ( p 1 , Q l ;~ ( z )and

.

I k

lim -

h-0

=1

(5.2)

then the homeomorphism w = w ( z ) is said to map the infinitesimal ellipse E ( p , 8 ; z ) at z onto the infinitesimal ellipse & ( p l , Q 1 ; w ( z ) at ) w = ~ ( z )The . idea of Lavrentiev was to consider the infinitesimal fields of ellipses as conformal structures and the quasiconformal mappings as mappings compatible with the assigned conformal structures.

Proposition 5. The general Beltrami equation WE -

p(z)w,

-

v ( z ) G z= 0

in a domain G of complex plane determines two conformal structures & ( p ,0; z ) and E ( p 1 , Q l ;z)’ on G such that the QC-mapping determined by the solution w = w ( z ) of the equation maps the infinitesimal field &(p, 8; z ) onto & ( P I , 81; z ) . 2We freely follow here, strictly speaking, the uncorrect tradition of identifying the notation &(PI,81;z ) with the notation & ( p l ( w ( z ) Q ) ,l ( w ( z ) )w ; ( z ) ) ; it is legitimate only when the “source” z and the image w = w ( z ) are described in the same Euclidean coordinate system. Correctness would require to call & ( P I , 81,~) the pull-back by the map w = w(z) of a Lavrentiev field on the image.

80

The fields are determined by the coefficients or complex dilatations of the Beltrami equations (0.2) by the (invertible) formulas

CLk)=

-

P - P-l P + P - l + P l +P,l

,%i@,

().

=

PT1 p-tp-1 + P I +PT1 P1-

e 2 i 6 ~.

(5.3)

For details of the corresponding calculations see [41] or [45]. The conformal structure in the sense of Lavrentiev on the complex planeone-dimensional complex manifold--corresponds to Lavrentiev fields with characteristics p l = 1, p-uniformly bounded. They correspond to the complex (proper) Beltrami equations of the type (0.1). We should stress that the clu of the Lavrentiev idea is that the %ource” characteristic ( p , Q), is mapped into the “target” characteristic (pI,&) independently of the considered particular solution of the general Beltrami equation (0.2) as long a s the relation source z-target w = w(z) is preserved. The formulae (5.3) describe then a pair of “distinguished” or canonical Lavrentiev fields intrinsic for the Bel. trami system considered and a selected, pointwise correspondence w = ~ ( 2 ) Of course any chosen solution w = w(z) at every differentiability point transforms an arbitrary Lavrentiev field of infinitesimal ellipses into “some” Lavrentiev field whose characteristics p l , 81 at the image point w(z) depend on the behavior of the map at neighboring points, i.e., on the values of the derivatives w,, w, at z . Lavrentiev in his seminal paper of 1935 [32] defined QC-mappings as homeomorphic mappings of the unit disc D onto itself such that at every point z E D the infinitesimal ellipse € ( p ( z ) Q , ( z )2; ) is mapped onto an infinitesimal circle in the sense defined above. He also proved the existence theorem for such mappings by a direct geometric construction without referring to any classical solutions of boundary value problems, e.g. in the L. Lichtenstein paper [36]. Thus Lavrentiev is probably to be credited for the first direct self-contained proof of the global Riemann mapping theorem for a rather general class of complex Beltrami equations with continuous coefficients &). In the following years the Lavrentiev ideas, unfortunately, seem to have been considered and developed mostly in the Russian language mathematical literature. The only paper in English is probably the one of M. Naatanen [40]. P. P. Belinski’i and I. N. Pesin in a series of papers starting with [ll]made substantial progress in the Lavrentiev theory, introducing the class of generalized QC-maps and discussing their various subtle properties. Later P. Belinskii in [lo] has shown that Lavrentiev’s proof from 1935 [32] may be extended t o measurable characteristics p thus giving another full direct proof of the measurable Riemann mapping theorem. The ideas of Belinskii and Pesin have many geometric features relating this theory with the deep investigations of D. E. Menshov [37], [38]on the monogeneity and the pointwise theory of solutions of Cauchy-Riemann equations in the setsting of classical real analysis. From the point of view of partial differential equations the state of the art at

81

this moment is described by the following theorem obtained by the parallel efforts of P. P. Belinskii, B. Bojarski, I. N. Pesin, see also [40], [16].

Theorem 5. Orientation preserving Lavrentiev maps are the generalized homeomorphic solutions of the complex Beltrami equations with continuous coefficients. The generalized QC-mappings in the sense of I. N . Pesin [ll],[lo]are Sobolev space W12F(C) ( p > 2) solutions of the general, uniformly elliptic, Beltrami equations with measurable coeficients p , u . In this area there are interesting deep problems to be studied, in particular in the spirit of D. E. Menshov’s geometric theory of Cauchy-Riemann equations discussed in the language of advanced subtle methods of Real Analysis in the early decades of the last century.

6. Uniqueness in the general measurable Riemann mapping theorem Here fundamental is the following proposition.

Proposition 6. Let f

=

f ( z ) be the generalized solution of the equation

mapping the disc IzI 5 1 onto the disc I f I 5 1. T h e n f holds :

(2)

=z

i f one of the following

1) f ( z ) E z in three different points of boundary IzI = 1, 2) f (z)e z in one interior point of the disc and one boundary point, 3) f ( z ) = z in two different points of the open disc. The equations (6.1) and (4.2) are obviously equivalent. This reflects the fact that the reduction of the uniqueness question for real Beltrami systems (0.2) to the consideration of (6.1) is analogous to the reduction of the search of primary pair for (0.2) to the problem (4.2). The proof of Proposition 6 is analytically rather complicated; in particular it requires to appeal to the properties of the general homogeneous elliptic systems of the form (0.10) (with C = 0). In infinitesimal geometric interpretation in terms of Lavrentiev fields the equation (6.1) means that the solutions of (6.1) just realize the parallel shift (or translation) .( ? p ) of the corresponding Lavrentiev ellipses. If, additionally, the global curve: the boundary IzI = 1 is preserved and has three fixed points then the mapping does not move any point at all. Proposition 6 should be compared with the famous D. E. Menshov [37] theorem on monogeneity conditions expressed in terms of the geometric requirement to preserve almost all infinitesimal circles.

82

References 1. L. Ahlfors, Zur Theorie der UberlagerungsfEiichen, Acta Math. 65 (1935). 2. L. Ahlfors, Conformality with respect t o R i e m a n n i a n metrics, Ann. Acad. Sci. Fenn. Ser. A. I, 1955, no. 206, 22 pp. 3 . L. Ahlfors, Lectures o n Quasiconformal Mappings, Van Nostrand, Princeton 1966. 4. L. Ahlfors, L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385-404. 5. C. Andreian Cazacu, Foundatzons of quasiconformal mappings, in: Handbook of Complex Analysis: Geometric Function Theory, vol. 2, Ed. R. Kuhnau, Elsevier, 2005. 6. K. Astala, Analytic aspects of quasiconformality, in: Proceedings of the International Congress of Mathematicians (Berlin 1998), vol. 11. 7. K. Astala, T. Iwaniec, G. Martin, Elliptic equations and quasiconformal mappings in the plane, t o appear. 8. K. Astala, T. Iwaniec, E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), 27-56. 9. K. Astala, V. Nesi, Composites and quasiconformal mappings: n e w optimal bounds in two dimensions, Calc. Var. Partial Differential Equations 18 (2003), 335-355. 10. P. P. Belinski?, General Properties of Quasiconformal Mappings, Nauka, Novosibirsk 1974 (in Russian). 11. P. P. Belinskii, I. N. Pesin, O n the closure of the class of differentiable quasiconformal mappings, Dokl. Akad. Nauk SSSR 102 (1955), 865-866 (in Russian). 12. L. Bers, L. Nirenberg, O n a representation theorem for linear elliptic systems with discontinuous coeficients and its applications, in: Atti del Convegno Internaz. sulle Equazioni alle Derivate Parziali, Trieste 1954, Ed. Cremonese, Roma 1955, 111-140. 13. B. Bojarski, Homeomorphic solutions of the Beltrami systems, Doklady Akad. Nauk SSSR (N.S.) 102 (1955), 661-664 (in Russian). 14. B. Bojarski, O n solutions of elliptic systems in the plane, Doklady Akad. Nauk SSSR (N.S.) 102 (1955), 871-874 (in Russian). 15. B. Bojarski, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coeficients, Mat. Sb. N.S. 43(85) (1957), 451503 (in Russian). 16. B. Bojarski, General properties of solutions of elliptic systems in the plane, in: Issled. sovr. probl. teor. funktsii kompl. peremen., ed. I. A. Markushevich, Gosfizmatizdat, Moscow 1960, 461-483 (in Russian). Transl. in: Fonctions d'une variable complexe, Probl. contemporaires, Paris 1962, 254-268. 17. B. Bojarski, Quasiconformal mappings and general structural properties of systems of non-linear equations elliptic in the sense of Lavrentiev, in: Symposia Mathernatica XVIII (Convegno sulle Trasformazioni Quasiconformi e Questioni Connesse, INDAM, Rome 1974), Academic Press, London 1976, 485-499. 18. B. Bojarski, Old and n e w o n Beltrami equation, in: Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations (Trieste 1988), World Scientific, River Edge 1990. 19. B. Bojarski, P r i m a r y solutions of general Beltrami equations, t o appear. 20. B. Bojarski, L. D'Onofrio, T. Iwaniec, C. Sbordone, G-closed classes of elliptic operators in the complex plane, to appear. 21. B. Bojarski, V. Ya. Gutlyanskii, O n the Beltrami equation, in: Conference Proceedings and Lecture Notes on Analysis I (Tianjin 1992), ed. Zhong Li, International Press, Cambridge, MA 1994, 8-33. 22. B. Bojarski, T. Iwaniec, Quasiconformal mappings and non-linear elliptic equations I, Bull. Pol. Acad. Sci. 27 (1974), 473-478.

83

23. B. Bojarski, T. Iwaniec, Quasiconformal mappings and non-linear elliptic equations 11,

Bull. Pol. Acad. Sci. 27 (1974), 479-484. 24. B. Bojarski, T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in Rn,Ann. Acad. Sci. Fenn. Ser. A. I Math. 8 (1983), 257-324. 25. B. Bojarski, T. Iwaniec, Primary solutions of planar elliptic systems, in preparation. 26. F. W. Gehring, Definitions f o r a class of plane quasiconformal mappings, Nagoya Math. J. 29 (1967), 175-184. 27. F. Giannetti, T. Iwaniec, L. Kovalev, G. Moscariello, C. Sbordone, O n g-compactness of the Beltrami operators, in: Nonlinear Homogenization and its Applications to Composites, Polycristals and Smart Materials, Kluwer, Dordrecht 2004, 107-138. 28. H. Grotzsch, Uber einige Extremalprobleme der konformerL Abbildung I, 11, Ber. Verh. Sachs. Akad. Wiss. Math.-Nat. K1. Leipzig 80 (1928), 367-376, 497-502. 29. H. Grotzsch, Uber die Verzerrung bei schlichten nichtkonformen Abbildungen und fiber eine darruit zusammenhangende Erweiterung des Picardschen Satzes, Ber. Verh. Sachs. Akad. Wiss. Math.-Nat. K1. Leipzig 80 (1928), 503-507. 30. V. V. Jikov, S. M. Kozlov, 0 . A. Oleinik, Homogenization of differential operators and integral functionals, translated from the Russian by G. A. Yosifian, Springer, Berlin 1994. 31. R. Kiihnau, Herbert Grotzsch z u m Gedachtnis, Jahresber. Deutsch. Math.-Verein. 94 (1992), 141-169. 32. M. A. Lavrentiev, S u r une classe de repre'sentations continues, Rec. Math. Moscou 42 (1935). 33. M. A. Lavrentiev, A general problem of the theory of quasiconformal representation of plane regions, Mat. Sb. N.S. 21 (1947) (in Russian). 34. M. A. Lavrentiev, A fundamental theorem of the theory of quasiconformal mapping of plane regions, Izv. Akad. Nauk SSSR 12 (1948) (in Russian). 35. 0 . Lehto, K. I. Virtanen, Quasikonforme Abbildungen, Grundlehren Math. Wiss. 126, Springer, Berlin 1965; 2nd ed.: Quasiconformal Mappings in the Plane, Springer, Berlin 1973. 36. L. Lichtenstein, Z u r Theorie der konformen Abbildung: Konforme Abbildung nichtanalytischer, singularitatenfreier Flachenstiicke auf ebene Gebiete, Bull. Acad. Sci. Cracovie, 1916, 192-217. 37. D. E. Menshov, Les conditions de monogknkite, Act. Sci. et Ind. 1936, vol. 329, 1-52. 38. D. E. Menshov, O n the asymptotic monogeneity, Mat. Sb. 43 (1936), no. 2 (in Russian). 39. C. B. Morrey, Jr., O n the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. 40. M. Naatanen, Maps with continuous characteristic as a subclass of quasiconformal m a p s , Ann. Acad. Sci. Fenn. Ser. A. I, 1967, no. 410, 28 pp. 41. H. Renelt, Quasikonforme Abbildungen und elliptische Systeme, Teubner Texte z. Mathematik 46, Leipzig 1982. 42. S. Stoi'low, LeCons sur les principes topologiques de la the'orie des fonctions analytiques, Collection Borel, Gauthier-Villars, Paris 1938; 2nd ed. 1956; Russian ed.: Nauka, Moscow 1964. 43. I. N. Vekua, The problem of reducing differential forms of elliptic type to canonical f o r m and the generalized Cauchy-Riemann system, Doklady Akad. Nauk SSSR 100 (1955), 197-200 (in Russian). 44. I. N. Vekua, Obobshchennye analiticheskie funktsii, Fizmatgiz, Moscow 1959. 45. L. I. Volkovysky, Quasiconformal mappings, Lvov 1954 (in Russian).

84

A PARTICULAR POLYHARMONIC DIRICHLET PROBLEM H. BEGEHR I. Math. Inst., F U Berlin, Arnimallee 3,0-14195, Germany; begehromath.fu-berlin.de A certain Dirichlet boundary value problem for the polyharmonic operator of arbitrary order n is explicitly solved in case of the unit disc of the complex plane. Bassisc tools are a related Cauchy-Pompeiu representation formula modified with a proper polyharmonic Green function. A similar but asymmetric Dirichlet problem is recently treated with the same method. Keywords: Polyharmonic operator, Dirichlet problem, unit disc, Cauchy-Pompeiu representation, Green function

1. Biharmonic boundary value problems

For the biharmonic differential equation

(d,dz)2w = f (1) in some regular plane domain D of the complex plane C two boundary conditions may be described determining the solution. Preferring the Dirichlet condition two kinds of boundary value problems are possible.

Dirichlet-Dirichlet problem Find a solution to (1) in D satisfying

w = y0,d,dzw = y2 on d D Dirichlet-Neumann problem Find a solution to (1) in D satisfying w = yo,dud,dzw = y3 on d D

(2)

(3)

Here denotes the outward normal derivative on D . Similarly with respect to the Neumann condition there are two more boundary value problems.

Neumann-Dirichlet problem Find a solution to (1) in D satisfying d,w = y~,d,d,w = 7 2 on d D

(4)

Neumann-Neumann problem Find a solution to (1) in D satisfying

d,w = yl,d,d,dZw = y3 on d D

(5)

85

With proper biharmonic Green and Neumann functions problems (2) and (5) respectively can be solved. Problems (3) and (4) are solvable using a hybrid GreenNeumann function, ~ e e .Of ~ ,course ~ wherever a Neumann condition is involved as well some solvability condition as a normalization condition appears. However there is another biharmonic boundary value problem different from the ones stated before.

Dirichlet problem Find a solution to (1) in D satisfying w = yO,duw = y1 on d D

(6)

While problems (2) to (5) are corresponding to the decomposition of equation (1) into the system of two Poisson equations

aza,-w= w,aza,-w= f and their solutions can be attained by iterating the solutions to these equations, problem (6) does not go along with this decomposition. Problem (6) requires another kind of biharmonic Green function.’ Moreover it requires more smoothness assumptions on the boundary data.4i5i14p16 A problem related t o (6) is w = yO,dzw = y1 on d D

(7)

w = yo,d,w = y1 on d D

(7’)

or equivalently

. Obviously the variety of boundary value problems for the polyharmonic operator grows with its order. All kinds of combinations of conditions from (2) to (7’) are possible and other conditions as e.g. of Robin type like

see14p16

w

+ aduw = y

on d D

can be added. In14-16 Dirichlet problems of type (7) are treated for the polyharmonic equation in the unit disc and the upper half plane, respectively. Here the Dirichlet problem of type (6) is solved for the ployharmonic equation of arbitrary order.

2. A representation formula In order to be explicit the domain is chosen to be the unit disc A of the complex plane @.

Dirichlet problem Find a solution to

(dzdr)”w= f in ID satisfying on dD

(d,d,)fi w

= yfi, 0

5 2p 5 n - 1,

du(dzdz)@w = Tfi,0 5 24.1 5 n - 2.

86

The solution will be described with the polyharmonic Green-Almansi function2

It has the properties 0

0 0

G n ( z ,C) is polyharmonic for z E ID\{(}

+

for

< E D,

G,(z, C) (n-1)!2 log - zI2 is polyharmonic for z E D, C E D, (&3,)PGn(z, C) = 0 for 0 5 2p I n - 1 for z E 8D, C E D, 1
au(i3Z&)PG,(z,C) = 0 for 0 5 2p 5 n - 2 for z E D,

G n ( z ,C) = Gn(C,z ) for z , C E D, 2 #

< E D,

C.

Basic for treating boundary value problems are integral representation formulas For the polyharmonic operator this is, see” , of Cauchy-Pompeiu

By combining this formula with its complex conjugate counterpart the representa-

87

tion becomes symmetric in z and 2 , namely

Denoting the sum of the kernel function in the area integral of (12) and G n ( z , < ) by A, then as in14

( n - 1)!2An=

-

z/2(n-1) log 11 -

-

p= 1

Lemma 1 For 2 5

R

-1 1

2

P

88

where for 1 5 v 5 n - 1

with, see2 ,

Proof Differentiating and summing up geometric series shows

-IC

-

which leads to (14) and

89

which sums up to (15). Theorem 1 Any w E C2"(D;C ) n C2n-1[D; C)is representable as

1

-1 n

Gn(2,c)(aca,)"w(C)dEd?l.

A

Proof Applying the Gauss theorem, see,' in the forms

and

shows

=-J' 1 4ni

A, 8, (a&),-

ann

d< "T

1

90

In the same manner induction gives for 0 5 k 5 n - 1

As

the last relation becomes

n-1

= p=l

/ D

a,(dCd<)lLwdEdq

91

Rewriting (12) as

+-71r D (An

-

Gn)(a
and substituting (19) proves (17). Theorem 2 Any w E C2”(D;C)n CZn-l(B;C)i s representable as

Proof The integral under the sum in (1) is by repeated application of the G a d theorems (18) and (18’) and using (16)

1/ ( - 1 ) V 7r

- IClZ)%1(.>

C)(a&)@w(odEdV

D

1 =

-7r

J’aca&(., D

C)(a&)”w(C)dEdv

92

On the basis of2 , Lemma 2 , see3 for 1 5

K

5p-1

1 ( K - I)! ( p - l)!p!(a,at)KGL+l ( z ,0= p ! ( p - K ) !

'

r=O

min{K-l,p-K-l}

-

- I)! p!(p- K)! 2(K

- K)!

min{K-1 , p - ~ }

+-

(K -

P!

and

r=O

(P

'T!(P-K-T-l)!

l)!

c

r=O

1 T ! ( p- K - T ) !

( 1 ~ -1 ~ I ) + - ~

93

-I -

2n-j1 p!(p-K-l)! 2(n-l)!

(p-tiX1+X)S2n-@+1-X(Z,

0

2n-p-1

+-

(6-l)!

Thus

p = l p=1

c

X=O

1 1 1 Y

Here the last sum becomes

(p-n;l+X) X=O

(2K - p

-

A)

94

Thus (20) is attained

3. A polyharmonic Dirichlet problem

As is known e.g. from the Cauchy formula not every integral representation formula provides a solution to a related boundary value problem. The problem may be overdetermined so that it is sovable only under some solvability conditions, see e.g.4,5 The representation formula (20) gives rise to the solution of a Dirichlet problem for the polyharmonic equation. Theorem 3 The unique solution to the Dirichlet problem

(az&)”w(z) = f ( z ) in D, (az&)pw(z) = y p ( z ) on 8ID f o r O 5 2p 5 n - 1, a!ag+’w(z) = 9 p ( z ) on ID for o 5 2p 5 n - 2, for f E &(D; C), yp E Cn-2fi(D;C), 9p E Cn-’-’p(D; C)in the distributional sense is given as

[41-1 -

n-1

p-2tc

ccC n=l p=Zn+l

x S X b , C)(1

X=l

p-K-1-A

(

1

l)!

2K!(K+

dC - /~IP)i”jrK(C)}

-

6-1

1

(n;l)

1

1 ; G n b ,C)f(S)dldrl. D

Proof The uniqueness of the solution follows from the representation formula (20). If there is a function satisfying the inhomogeneous polyharmonic equation

95

and the above Dirichlet conditions then it is representable via (21). Thus the related homogeneous problem is only trivially solvable. The existence of a solution can be proved by verifying (21) to be a solution. For this purpose the properties

(a,aE)pGn(z,<) = 0 on A for 0 5 2p 5 n - 1, a;ap+lG,(z,

<) = o on A for

o 5 2p 5 n

-

2,

of the polyharmonic Green function G n ( z ,<) are important, see,’ which together with

(aM{-1 7T J’Gn(z,
in the weak (distributional) sense shows that this area integral in (21) provides a solution to the inhomogeneous polyharmonic equation satisfying the above homogeneous Dirichlet data. As the g y ( z ,<) are harmonic in z (and in <) the boundary integral in (21) obviously is a polyharmonic function of order n. Hence it only remains to prove the proper boundary behaviour of this boundary integral. For this aim the respective derivatives have to be calculated. The following rule will be used. Lemma A For f , g E C’P(D;C!),1 5 p , D c C open, and g harmonic, i.e. a,azg = 0,in D

Verifying the boundary conditions will be done on the basis of the next two lemmas. Lemma B Let y E C(aQ C!) and 1 5 k 5 n, k , n E N. T h e n for any E dD

Proof Observing the property of the Poisson kernel function

the first part of the lemma follows from

The second follows similarly or by complex conjugation.

96

Lemma C Let y E Cm(dD;C), m. n E N. Then for a n y
Proof Arguing as before after m times integrating by parts shows

and

E dD.

tend to zero with z E D tending to (i) Differentiating

once with respect to gives (1 - Iz12)n-l (1 - ,<)"+1 '

AZ = n<

m-1

r-u

r

m-r

for 2 5 2m 5 n - 1 and m

T

r-u

m--7

97

m

r--6

m-r

(22'")

(ii) Similarly for

it is seen

(";

l)

(;)

r

') ;")

(m

-:'> -

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,<) [(l- ) Z y ) n - u - - r - - L

5X5n Lemma 5

for 0

-

- (1-

Iz12)n-U-T-L-1

1

2 ~ . K-1

(a,a,)"c,,= a,azc,1 + -y(dzdz)"'

[cKK-r(l - (z(2)n-2T]7

r=1

where a,azCKlis given in Corollary 1 and the terms (dzdz)K-T[C,K-T(l-(~12)n-2T] are expressible as in L e m m a 4 with (T = K . Lemma 5' K

dU,(d,dz)"C,,

=

a,,c,o + ~a",(azaz)~-r[c,-r(l - Iz12)n--2r], r=l

where aVzCK0is given in Corollary 1 and the last s u m is expressible according t o L e m m a 4' with CT = K . Lemma 6 For 1 5 (T,2 ( +~ (T) 5 n - 1 (dZdZ)K+UC,,

;+::

= (dZz)ucKo

(azaz)K+u-r

[c,-r(l - Iz12)n--2T].

102

Here 8,&Cn~i s given in Corollary I and (aZaz)"CK0 is linearly composed by t e r m s as

u-1

u

u-r

u-1-z/

u=o

r=O

x=o

G O

ccc c

=

(" ,yl-

(-l)V+X+yu

-

+

1) ( n - 2~ u - v ) ( u a-1-u

1)!2

("?;: ")

;

1) ( u

-

u) (u

-

;

7)

and another o n e like this with n f o r m a l l y replaced by n - 1. T h e other t e r m s are expressible as in L e m m a 4 w i t h K u instead of u . Lemma 6' For 1 5 o , 2 ( + ~ o) 5 n - 2

+

T h e first t e r m i s linearly expressible by t e r m s like

c cc

u-1

=

u--7

u-1-u

;=o

x=o

u=o

("

+

1=0

26 u - u ) u-1-v

-

- 1)!2

(-l)v+X+'(u

(" ;1) ("

-

;

- u)

n-2K-u

n-2K-1 u-1

( )( ("; ') u--7

)

and t h e s a m e expression with n f o r m a l l y replaced by n - 1. T h e remaining t e r m s can be expressed as in L e m m a 4' w i t h p u instead of u .

+

(iv) Let

103

with

Lemma 7 For 0 5 p 5 n - 2

where n-2n-1

p=l

p

p+n-A-p-1 K-p-1

(

1

X=l

n-p+l

and n-2~ Dnn-p-1

=

c

X=l

Lemma 7’ For

holds

with

1 (6 - p - l)!(K

- p)!

1

104

and

Lemma 8 For0 5 u 5

1

K -

where (8z&)u-r[ f i K n - T ( l- I certain linear combination of

(

z ~ ~ ) ~ is- ~ according ~ ]

) (;)

n-r -u

(A

u--7-u-p

-

1)

(0

I

r)

(0

-r -u- p L

xgx+p(z, <)(I-

Lemma 8' For 0 5 u 5

+;

to the definition in Lemma 7 a

IZ12)n-u-r--L

K -

1

105

is a certain linear combination of (&&)'-T

[gx(z, 5 ) ( 1 -

JZ12)n-2T]

for 1 5 X 5 n - 2 K Lemma 9

where D o , E o are defined an Lemma 7 ' and the terms (azdy)n-TDK,-T(l-(z(2)n-2T are expressible as in Lemma 8 with 0 = K . Lemma 9'

where the last sums are linear combinations of terms as in Lemma 8' with and &Dno is given i n (29).

0

=

K

106

Lemma 10

L

X=l

where the last s u m is a certain linear combination of terms as in Lemma 8 with O = K f l .

Lemma 11 For 1 5

+ u) 5 n

o , ~ ( K

-

1

r=l

Here the terms in the first two sums are

=

cC c

u-1

p

p=o

7=o

u-1-p

p=o

p-r

-

C(-l)p'p+y" L=o

1)!2

(Y?)

for either m = n - 2 or m = n - 1 and the last sum is a linear combination of terms as in Lemma 8 with 0 replaced b y p g . Here 1 5 A 5 n - 2 ~ .

-+

107

where the terms in the first two sums are expressible as allr

(&W-'[gx(z, C)(l

0-1

p

u-1-p

cc c

= p=o

7=0

( m+

-2(m

p=o

-

-

IZ12)m--2n]

p-r

-y(-l)"+'"+'(cT L=o

):I:( 1)!2

-

") ; (" -;

+ 1- 2 K

1)

(0

- L)gX+n(Z,

-

C) [(I

P)

(0

;

i ) (A

- /Z/2)m--2K-L

-

-

;;;

-p

(1 - Iz12)m--2n-L+l

I

m = n - 2 and m = n - 1 and the third s u m is a linear combination of terms as in Lemma 8' with IS replaced b y K + m. Here 1 5 A 5 n - 2.

for

The proof of some of these lemmas will be given in the Appendix. From (Z),the property of the Poisson kernel g l ( z , C), Lemma B and Lemma C lim w(z) = TO(<) for z--c

C E aID

follows. As yo E C"(dD; C)there is no contribution from A to the boundary values or the derivatives of w. From (23') and lemmata B and C similarly lim dv,w(z) = -i/o(C) for

C E dD

2Z-c

is seen and (23") together with (23'"), (23'") indicate that no contribution may be expected for the other higher order derivatives from this second summand because

qo € c"-l(dD; C).

108

For the third summand Corollary 1 and lemmata 2 to 6' for p = 1 show lim dzdzw(z)= TI(() for z+c

< E dD

and also that no contribution comes from this part to the boundary behaviour of the other higher order derivatives involved as y1 E Cn-2(dD;C). Proceeding inductively next the term C,,y, is considered for 2 5 2 6 5 n - 1. Rewriting (25) and (26) as

a,azc,,-, iteration gives for 1 5

B

= Cnn-p-l

+ C,&-,-l(l

-

Iz12)n--2P-2

5 K, - I

in particular

On the basis of Corollary 1 and lemmata 2 to 6' then lim ( d z d z ) n w= y, for z+c

< E dD

is seen and also that this term does not contribute to the boundary behaviour of any other higher order derivative involved because 7, E C"-2n(dD;C). At last the terms D,,Y,,2 5 2 ~5, n - 2, are considered. As for the C,, from (27) follows for 1 5 B 5 K, - 1

in particular

109

Hence, observing lemmata 7 to 11' lim d y z ( d , d z ) K ~ (= z )TK(C)for

C E A.

z-tc

Moreover, from (30) and Lemma 10 (&&)fi+lDKK= ( n - 2K: - 1)(n - 2 K )

is seen, showing that this term does not contribute to any other higher order derivative on the boundary. Here qK E Cn-2K-1(dD;C)is needed. That higher order terms vice versa do not contribute to lower order derivatives of w on the boundary follows from the size of powers of the factor (1- lz12) in relation to the powers in the denominators, see Lemma B. 4. Appendix

Here the proofs of some lemmas in Section 3 are provided.

110

Observing for 2 5

~i

and

shows

&&Cnnl

=

(2K

-

a)!

111

-

>, 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:

<

<

fine lim

CEG

Let G be a finely open set in h : G + let us denote:

hco,G,f

:==

{ {

h

SUP~+,, CtG

fine lim

:=

e

e,

e

ha,G,f

c).

e,

C and a

40 =: G G , ~ ( z ) , lh(<)l=: & G , ~ ( z ) . E

CG be a fixed point. For a function

fine lim SuPC+a, CEG loglfol liogl<-zll when

0

when

fine lim suP<+m, CEG loglcl log'f(C)l

0

a

E

dfG,

a $ dfG.

when cc E &G; when

00

f &G,

If the function f is finely meromorphic in G then we denote by Ic(f,w) the multiplicity of its value f(w)at the point w E G. Given functions p : X + and q : X + on a set X c and a finely limit point w for X , we use the following notations. If there exist a finite number 1 2 0 and a fine neighbourhood U of w for which Ip(z)I 5 1 /q(z)I b' z E X n U , then we write

e

e

p ( z ) = fine O ( q ( z ) ) ( z + w,zE X ) ,

and if for every E > 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(<) = fine o(l
(< + 00,

C E G);

(1)

(B0,co) 03 # b(CG), pco 2 0 and (1) is true. If z E @. is a fixed point, then we consider also the following conditions: (A,z) z E b(CG) and for every finely connected component T of G with z E b(T) there holds h,,T,f < + co; (B,z) z # b(CG), p0 < 03 and

+

h(<) = fine o(lC - zlmo-') (< t z ,

< E G);

(2)

( B l, z) z # b(CG), po 5 1 and ( 2 ) is true. We will use the following commutative rule for possible indefinite expressions: 0 . (.tO3) = 0, -co + 00 = -0. 2. Main Results

We prove the following results.

Theorem 2.1. Let a E C be a fixed point; G c C \{a} be a finely open set with non-polar fine boundary af G ; p E tlJz; h : G t be a finely meromorphic function for which hG,f (2)

< p(1z

-

a

1)

vz E afG\{a>.

Denote z1 := a , z2 := 03 and suppose that for each s = 1 , 2 (independently from each other) one of the conditions (A,zs) or (B,z,) is satisfied. Let ? !3 be the set of all poles of h in G. Then

Theorem 2.2. Let a E C be a fixed point; G c C \{a} be a finely open set with nonpolar fine boundary df G; Q be a set contained in FG and containing the points a and 03; p E 9X; h : G -+ C be a finely meromorphic function. Denote z1 := a , z2 := 00 and suppose that for each s = 1 , 2 (independently from each other) one of the conditions (A,z,) or (B,z,) is satisfied. Let (p be the set of all poles of h in G. Denote A

X(x) := logp(x)

vx

> 0,

120

:= 1%

Ih(c)I

c

gG(P,

-

c) . k ( h , P ) c E G .

PEP

Suppose t h a t f o r every finely connected c o m p o n e n t T of G t h e following condit i o n s are satisfied: Q i s nearly negligible relative t o T and iiT,f(Z)

< 03

GT,~(z5 ) X(lz

vz E t+T\{a};

-

al)

v.z E afT\Q.

T h e n (3) holds. Notice that in formulations of Theorems 2, 8.1, 8.2 of' the following obvious correction should be done: Q must be assumed t o be contained in FG instead of

8fG. Theorem 2.3. L e t G c C be a finely open set; p E !.TI; h : G n C f u n c t i o n finely m e r o m o r p h i c in G and satisfying t h e condition

I h ( z ) - h ( 0 I<

< I)

c E afG, c #

--t

be a

(4) L e t y be t h e s e t of all poles of h in G n C. L e t o n e of t h e conditions (A,m) or (Bo,m) be satisfied f o r t h e restriction of h o n t o G (instead of h). Suppose also t h a t for every finely connected c o m p o n e n t T of G t h e restriction of h o n t o (T n C )\?j3 i s finely continuous. U n d e r these a s s u m p t i o n s w e have P(I

1

-

vz,

121

Theorem 2.6. L e t G , p, h satisfy all a s s u m p t i o n s of T h e o r e m 3. If 00 E G , additionally suppose t h a t h c a n be extended t o 00 in s u c h a w a y t h a t t h e extended f u n c t i o n h satisfies t h e following hypothesis: t h e restriction h l p of h t o T i s finely continuous (and f i n i t e ) a t 03 for every finely connected c o m p o n e n t T of G w i t h 00 E T . U n d e r these a s s u m p t i o n s t h e inequality (13) i s t r u e .

3. Proof of Theorems

We need to recall a number of notations from' . Let L be the class of all functions X : (0, +m) + [-03, +m) for each of which the set I , := {x : X(x) > -co} is connected and the restriction of X to Ix is concave with respect to log 2. Let L* be the class of all X E L for which Ix is non-empty. For X E L* let us denote by xi and z,: respectively, the left co. When and the right ends of the interval I x . Obviously, 0 5 xi 5 x: A(.) runs through the classes L or L*, the function expX(.) runs through the classes !7X or !7X*, respectively. When xX < x:, the mentioned concavity condition is equivalent to the combination of the following conditions: the function X(z) is concave with respect t o log 2 (and therefore continuous) in the interval (xi,2:) and lower semicontinuous on I x . For X E L the limits

<+

XO

:= lim z+o

:= lim

~

X(x)

~

logz'

z++cc

log 2

exist, and we have X0

2 Am,

X0

> - 00, Am < + co.

For every fixed a E R, 6' E R the function X(x) := a l o g x and for it we have Xo = ,A = a . Let a E CG be a fixed point.

+ 6' belongs to L * ,

122

c and a

Let G be a finely open set in u : G + [-m, +co] let us denote:

E

CG be a fixed point. For any function

4C) fine lim sUPC-ta, CEG lloglC-tll when

0

when

a E afG,

a $ dfG.

Let X E L. Consider the following conditions: (A‘, co) 00 E b(CG) and for every finely connected component T of G with 00 E b(T) there holds uFf < 00; 03 6b(CG) and there exist a constant t E Iw and a fine neighbour(B’, co) hood V of co for which

+

u(C) 5 X(lC

-

+

a [ ) t VC E G n V ;

a E b(CG) and for every finely connected component T of G with (A‘, a) a E b(T) there holds u ? , ~< co; a 6b(CG) and there exist a constant t E R and a fine neighbourhood (B’, a) V of a for which

+

u(C) 5 X(lC

-

aI)

+t

V< E G n V.

The validity of Theorem 1 follows from Theorem 2. In [I,pp. 355-3571 the following statement was actually proved, although not formulated. Lemma 3.1. Let a E @. be a fixed point; G\{a} be a finely open set; p E i7X; h : G + be a finely meromorphic function. Denote z1 := a , z2 := 00 and suppose that for G , h, p and some s = 1 , 2 one of the conditios ( A ,z,) or ( B ,z,) is satisfied. Then for G , the functions log lh(C)l, X(x) := logp(x) and the same s the corresponding of the conditions ( A ’ , z , ) or (B‘,z,) is fulfilled, and z, is finely distincted from the set of all poles of h in G and h is finely holomorphically extendable into z,.

Notice that the presenting in’ fine holomorphicity of h in the context of the given statement was used only in a fine neighbourhoods of the points a and 00,not in the whole G.

Proof. [Proof of Theorem 21 Applying Lemma 7 t o the set G and the function h we obtain, that for G , the functions u(<):= loglh([)l, X(x) := logp(x), and the same s the corresponding of the conditions (A’, z,) or (B’,z,) is fulfilled.

123

We omit from consideration those connected components of the set G at each point of which the series C gG(p,<) . k(h,p)diverges. Therefore, without loss PmnG(6)

of generality, we shal assume that it converges on the set G\F. Assume now that Z is an arbitrary finite set of zeroes of the function h in G. We consider in G the function

5 E G.

gG(W,< ) k ( h , w ) ,

vZ(<):= u(<)$zEZ

The set QZ := Q U ( F G ) , is nearly negligible relative t o every finely connected component T of G. The function V Z ( ( ) is finely hypoharmonic in G and i i ~ , f ( Z )=

(Cz)~,f(z)

VZ

E

(dfG)\Qz.

So, the function VZ(<) fulfils all assumptations of Theorem 8.2 from’

, and we

obtain

and finally

r

1

1

0. k ( h , w )

SG(W,

V< E G\

F.

Since 2 is an arbitrary finite subset of the set of zeroes of h, from here we obtain

(8). Proof. [Proof of Theorem 31 Fix an arbitrary point w E (d fG)T.Let us check that we may apply Theorem 2 with w as a , and the function

4“)

:= h(C)- h(w)

(C E G)

instead of h(C). Indeed from (4) we get (3) with w as a , and 4 as h.Since for every finely connected component T of G the function h l ( ~ ” rP~ is, finely continuous and w E ( d f G ) , c b(CG), therefore h satisfies the condition (A,w), and the same is true for 4 . One of the conditions ( A ,m) or (Bo,00) is assumed t o hold for h,and it implies the same condition for 4 . Hence, Theorem 2 is applicable in the situation mentioned above. Since w E ( d f G ) , , we have G # C \ {w}and the exceptional case of Theorem 1 is impossible in the situation under consideration. Thus we get (3) for q5 instead of h, and the estimate of ( 5 ) with w as z .

124

Because of the choice of w, i t gives us (5). Theorem 3 is proved. T h e proofs of Theorems 4-6 use our Theorem 3 a n d a r e similar to t h e proofs of Theorems 4-6 in' .

References 1. P.M. Tamrazov, Finely holomorphic and finely subharmonic functions in contour-solid problems. Annales Academia Scientiarum Fennica. Mathematica, Vol. 26, 2001, 325360. 2. P.M. Tamrazov and T. Aliyev, A contour-solid problem for meromorphic functions taking into account zeroes and nonunivalence. Dokl. Akad. Nauk SSSR 288 (1986), No. 2, 304-308 (Russian); English translations in Soviet Math. Dokl. Vol. 33(1986), No. 3, 670-674. 3. T.G. Aliyev and P.M. Tamrazov, A contour-solid problem f o r meromorphic functions, taking into account nonunivalence, Ukrainian Math. Zh. 39 (1987), 683-690 (Russian). 4. T . Aliyev and P. Tamrazov, Contour-solid theorems for meromorphic functions taking multivalence into account. In: Progress in Analysis. Proceedings of 3-d International ISAAC Congress, Vol. I, 469-476. World Scientific, Singapore, 2003. 5. T.G. Aliyev, Irregular boundary zeroes of analytic functions in contour-solid theorems, Bull. SOC.Sci. Lettres Lodz 49 Ser. Rech. Deform. 28(1999), 45-53. 6. T.G. Aliyev, Contour-solid theorems, Preprint 85.84, Math. Inst. Acad. Sci. Ukrainian SSR, Kiev, 1985 (Russian). 7. M. Brelot, On Topologies and Boundaries in Potential Theory. -Lecture Notes in Math. 175, Springer-verlag, 1971. 8. B.Fuglede, Finely harmonic Functions.-Lecture Notes in Math. 175, Springerverlag, 1971. 9. B.Fuglede, Finely holomorphic Functi0ns.A survey.- Rev. Roumaine Math. Pures Appl. 33, 1988, 283-295.

125

A GENERALIZED SCHWARTZ LEMMA AT THE BOUNDARY T. ALIYEV AZEROGLU University Department, University N a m e , Gebze Institute of Technology,Department of Mathematics, Faculty o f Sciences Gebze, Kocaeli, T U R K E Y 141-41400 aliyevQgyte. edu. tr

BULENT N.ORNEK University Department, University N a m e , Gebze Institute of Technology,Department of Mathematics, Faculty of Sciences Gebze, Kocaeli, T U R K E Y 141-41400 nornekogyte. edu.tr

In the recent decade in publications of Burns and Krantz [1,pp. 662-6641, Dov Chelst’ and a few other authors papers there was studied the uniqueness portion of the Schwarz Lemma namely: Let the function q5 : D + D be a holornorphic in the unit disk D. From property q5(%) =

+ o(lz

-

114),

+

1,

E

D,

(1)

at the point 1 follows that 4 ( z ) = z , z t D . In the present work this circle of problems are generalized in the following aspects: a) more general majaront are taken instead of the habitual power majorant in (1); b) in (1) condition z + 1 was usually treated as tending from inside of the unit disk; we take the behavior of 4 at the point 1 along the boundary instead; c) instead of f ( z ) = z in the right-hand side of ( I ) , the finite Blaschke product is taken(see2). Let M be the class of functions p : (0,+m) + (0, +oo) for each of which logp(z) is concave with respect to logz (see3). For each function p E M the limit

exists, and

-a2

< po

< +m

Let T be the class of sets with zero capacities.

126

Let d ( z , E ) and U ( z ,r ) be the distance of the point z from the set E and open disk with centre z and radius r respectively. We prove the following results. Theorem 1. Let p E ~, po > 3; 4 : D t D be a holomorphic function that is continuous on D n U(1,SO) for some SO > 0 and it satisfies condition:

4(0 = c + 0 (P(lC

-

c

Then 4(c) = on D . Theorem 2. Let Q E W,p E m; po > 3; which satisfies the following condition l i m s u ~14(C) - CI 6 C+z,

,412 -

c E d D , c + 1.

11)

4: D + D

11)

(2)

be a holomorphic function,

Vz E (dD\Q) n U(1,So);

CEO

c

for some 60 > 0.Then 4 (() = on D . Theorem 3. Let f : D + D be a finite Blaschke product which equals r E d D on a finite set A f c d D and 4 : D t D be a holomorphic function that is continuous on on { z : d ( z , A f )< S O } for some So > 0; p l , p z E !JX, p; > 3 , p i > 2 . Suppose the following conditions are satisfied (i) for a given yo E A f 4(z) = f(z) + W ( 1 z

(ii) for all V y E A f

4(z)

+70,

-

"/OD),

z E dD, z

-

TI)),

z E d D , z+y.

- (70)

= f (z)

+ 0(P2(12

f(c)

Then #(<) = on D . Theorem 4. Let f : D t D be a finite Blaschke product which equals r E d D on a finite set A, c d D and 4 : D -+ D be a holomorphic function; Q E T i , p l , p2 E 9X, pk > 3 , pi > 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(<)= f ( < ) on D . For proof of our theorems we use the following propositions: Lemma 1. (Hopf's lemma on the disc) [4, p.341 Let u be a nonconstant real-valued harmonic function in D. Let y E d D be such that: (i) u is continuous at y; (ii) u(y) 2 u ( z ) for all Z E d D .

127

of u at y,if it exists, satisfies the strict

Then the outer normal derivative inequality

Remark 1.2 Let f : D t D be analytic function and have a continous limit at some y E d D , and let f(y) = 1. Then f is not o(z - y). Proof. of Theorem 1.Let the assumptions of Theorem 1 is satisfied. By (2) there exists a number C1 > 0 such that

I4(<)-51

KEaDnU(L60).


Let us denote k and C2 as follows; k :=

SUP

I< - 11 = 60

I4(<) -

<EG

It can be easily seen that for all boundary points of the set D inequality

n U(l,60) the

I4(0 -
Applying Theorem 3 of (see also5 function $(<) - one receives

I4(0 - 51 6 C2P (I<

,6

- 11)

and7) to the set D

n U(1,do) and the

v< E D n U ( l >60).

From pa > 3 follows that there exist some positive constants min(60,l) such that

and logp(x) 6 (3

+

€)

logx, vx E (0,g).

In other words p(x) 6 x3+€, vx E (0,g). Now from (5) and (6) we obtain

(5) E

and

g

<

128

We introduce the harmonic function

The function -maps the disk D to the right half plane and hence the first term of w is nonnegative, the second term is zero on aD\{l}. Consequently, lim inf

c 4 z,c E D w(C) 2 0

VJZ E aD\{l}.

By means of elementa,ry transform; one can receive that

From Remark follows that, the deminator of the last fraction can decrease no more quickly than O(lc - 112)at the point 1. According (7), the numerator approaches 0 as 0 - 113+'). Thus, w ( C ) is 0 (I( - ll"') in some neighborhood of 1. We obtain from the maximum principle (see, for example [8, p.481) either

(

4c) > 0

ye

E

D,

<

or w 3 0. If w is not a constant, it takes minimum at the point = 1, and it is 0 - 11"') there, as well. This contradicts Hopf's lemma statement. Consequently w = 0. This implies 4(c) =. <. Theorem 2 is generalization of Theorem 1, and its proof is similar t o the proof of Theorem 1 given above. Theorem 4 is generalization of Theorem 3 .

(Ic

Proof. of Theorem 4. Due t o (3), there exist numbers C3

Let us denote k and

C4

> 0, 60 E (0,1) such that

as follows;

C,

:= max

{ -, k

~

3

}

P1 ( 6 0 )

It can be easily seen that for all boundary points of the set D n U(y0,So) the inequality limsup C-2,

is satisfied. Applying Theorem 3 of yields

CED

Id0 - f(0l 6 c4d (12 - 701)

to the set D n U ( r o ,SO)and to the function $(<) - f ( < )

14(0 - f(0l 6 C4P1 (15 - ,701)

ye

€

D n U(Y0,bo).

(8)

129

From po > 3 follows that there are some positive constants E and (r < min(S0,l) such that inequality (6) is satisfied . Combining (8) and (6), we obtain

I+(<) - ~

(IC

K ) GI c4

-

7013+e).

w E D n u ( y 0 ,c).

(9)

Analogously, for any point y E Af\{yo}, from conditions pg > 2 and (4) we have

I$(<)

- f(0l 6 c 5 (IC - TI2+') .

K E D n V(T,05)

(10)

with some constants C5 and us. We introduce the harmonic function

Since a finite Blaschke product f is analytic function on 1 on d D , we have that the second term of v is zero on d D

-

D

and that

If1

=

A f . The first term

of v is nonnegative. Consequently, when taking limits to any boundary point in (aD\Q) - A f , one always obtains a nonnegative(possib1e infinite) value. Now, let's examine behavior of the function w at points of set A f . Let us represent v(c) in the form

Let's take any point y E Af\yo. Due to (lo), the numerator of the last fraction

( more quickly than O(l< approaches zero as 0

-

11"').

From Remark the dominator can decrease no

11'). Thus, v must have a liminf at y.As in the proof of

Theorem 1, from (9) we obtain v(c) is 0 (I< - y0l1+') in some neighborhood of yo. Thus, from the Phragmen-Lindelof principle (see, for example [8, p.2321) we obtain, either v(C) > 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(<) Tn(l
U(t) =

Lo

Sn(t - s)?(s) da,

V t € cllIO,

and of (2) in the form

S,(t

- S)T(S)

da,

+

J

tani

S,(t

- S)T(S)

da,

V t E clA(c),

where 7,r are unknown densities (or moments). Since t is a parameter in our problem, we find convenient to get rid of its presence in the domain of integration of the simple layer. It turns out that a good way to do so is to introduce the function p E Co>"(dIo)and the pair of functions (11,~)E Co@(dIli)x Co>"(dJio)by setting

Then we have

ii(t)=

Lo

S,(t

- s ) ~ ( sda, )

V t E clI",

133

and

+ i v ( t )+ dU

-(t)

1

= --p(t)

2

8UO

+

/

an i

ui(t). DtS,(t

-

b't E

s ) q ( s )do,

dIi,

.h.

u"(t) . D&(t - s ) p ( s ) dos

where Dt denotes the gradient with respect to the t-variable. By plugging (4)-(6) into the boundary conditions of (2), one obtains a system of two integral equations in the unknowns 7, p, which depends on the parameter E €10, EO[. We rewrite such a system into the abstract form

Mk,17, PI = 0 ,

(7)

where M is a suitable nonlinear map of 10, ~O[xC~,"(diTz) x Co>"(dlI")to Co@(dlIi) x C0>"(aII0). In many instances M may be seen t o be the restriction of some regular map (still denoted M ) of ] - E O , E O [ X C ~ , "x (Co@(dIo) ~IZ) to Co>"(dIi) x Co>"(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.

139

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.

140

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

141

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.

142

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.

143

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(<) 5 t } for any t E (0,1]. Let rt(z)= {< E C : w(<) = t } denote the boundary of !&(z). The function v is called the generating function for a given normal system {Gt(z)} (see, e.g., [Ill). Denote

These values r ( z , t ) and R ( z , t ) are equal, respectively, to the minimal and the maximal radii of the neighborhood G t ( z ) . The limit

is called the regularity parameter of the family {Gt(z),O < t 5 l}. Any such system {Gt(z)}is called the regular normal system, provided A(z) < 00. Let now f : G + G* be a homeomorphism of two bounded domains in C, and let { G t ( z ) }be a normal system of neighborhoods of z E G. One can introduce similarly the minimal and the maximal radii for the image of Gt(z) by

and

We now present an essential strengthening of the above theorems. Let p be a fixed real number such that 1 5 p < 00. Denote by I p ( z , t )and O,(z, t ) the following quantities

and put

where B ( z , h ) is the disc {C E C : dimensional measure of the set A.

I<

-

zI < h } , and m A is the Lebesgue two-

151

Theorem 1. Let a function f ( z ) be continuous and locally univalent in a domain G, and let f o r almost every point z E G there exists a normal regular system of neighborhoods { & ( z ) } c G such that either the inequality 2-P

lini sup I p ( z ,t ) 5 [ O ( z ) ]2 ,

(4)

t-0

holds for 1 5 p

52

o r the inequality lim sup ~ , ( z t, ) 5 [ ~ ( z q )],

(5)

t+O

-

holds for 2 5 p < 00. T h e n either f ( z ) or the conjugate function f ( z ) is analytic in G. We shall establish the assertion of the theorem for the function f (2) itself, assuming that f is orientation preserving. The proof for f ( z ) is accomplished in a similar way. Proof of Theorem 1. The proof is based on the following lemmas given in [3]. Lemma 1. Under the assumptions of Theorem 1, the function f (2) is differentiable

almost everywhere i n the domain G , and f o r a n y Bore1 set E

J’ I f ’ ( z ) I 2 d z d y <

c G, we have

03.

E

Lemma 2. I f f is orientation preserving and satisfies the condition of Theorem 1, then f o r almost all points z E G , we have the equality lfzl

+ lfzl

lfzl -

=1

lfzl

Here fi and f i denote the formal derivatives of f at z . Lemma 3. T h e function f is ACL, i.e., absolutely continuous o n lines. Lemma 4. For any rectangle lJ @ G with the sides parallel t o the coordinate axes, we have the equality

f

f ( z ) dz = 0

an Now the proof of Theorem 1 is completed in the following way. Lemma 1 and Lemma 2 yield together that the function f ( z ) is C-differentiable in the domain G and has there the derivative f ’ ( z ) ; moreover, this derivative is square integrable over G. In addition, f satisfies by Lemma 3 the N-property on almost all lines parallel to the coordinate axes. By Lemma 4 we have the equality (6) for arbitrary rectangle II. It follows now from the classical Morera theorem that f ( z ) is complex analytic on the domain G.

3. Generalizations of the Bohr and Menshoff theorems for continuous functions. The assumption of univalence of continuous functions was essential in

152

the investigations by Menshoff and by other mathematicians. The further investigation of analyticity of continuous functions has been made by Trokhimchuk. He used the notion of the interior mapping and the fundamental theorem of interior mappings obtained by Stoilow. Another ingredient of the method of Trokhimchuk is the theory of the sets of monogeniety. Following [13], we call a continuous mapping w = f ( z ) of the domain G interior if it satisfies the conditions: (i) no continuum of the domain is mapped onto a point, (ii) the image of an open set of the z-plane is open in the w-plane. We now consider the mappings preserving orientation for the case of continuous functions. Let zo be a point of a domain G, where the function w = f ( z ) is continuous, and let 200 = f ( 2 0 ) . The point zo will be called a U-point of the mapping w = f ( z ) if there exist two sequences { z i } , { z i ’ } (i = I,2 , . . .) of points converging t o zo, such that the semitangents t’, t” at zo lie on the different straight lines and all the points wi = f ( z i ) , w” = f ( z i ’ ) are different from wo = f (zo). We say that the mapping w is orientation preserving at the U-point zo if the sequences { w ; } , {w;} have semitangents T’, T” at wo with the following property: if 0 < {t‘;‘t”} < T then 0 5 (T’;T’’} < T . Yu. Yu. Trokhimchuk [14] gave the generalizations of the above Bohr and Menshoff theorem for arbitrary continuous functions. Theorem E. Let w = f ( z ) be a continuous mapping of the domain G with constant

stretching at almost every point z in G , and let this mapping be orientation preserving at almost every U-point. Then the function f ( z ) is analytic in G. In addition, if there are no U-points o f f in G , then f ( z ) G const. In this section we give an extended version of Theorem 1 for arbitrary continuous functions w = f ( z ) . We use the local geometric characteristics introduced in (3). Since in the case of the continuous functions the characteristic neighborhood Gt(z) can be shrunk to a point or t o an arc, the quantities (3) are not determined in the general case unlike the case of univalent functions. But for analytic functions the values I p ( z ,t ) and O,(z, t ) are finite a1 least for sufficiently small t . Theorem 2. Let a function f ( z ) be continuous in a domain G . Assume that for almost every point z E G there exists a normal regular system of neighborhoods {Gt(z)} C G such that the quantities I,(z,t) and O,(z, t ) are finite for suficiently small t > 0 , and let f be orientation preserving at every U-point. If either the inequality (4) holds for 1 5 p 5 2 or the inequality (5) holds for 2 5 p < 00. Then the function f (2) is analytic in the domain G and this function is not constant. The proof of Theorem 2 is based on certain estimates of ( 3 ) obtained in Lemma 3 (see, [3]).We also apply Theorem 9 of Trokhimchuk [14, p. 601 for interior mappings. 4. Generalized quasiconformal mappings in IW”. The above results can be extended t o quasiconformal mappings (even t o the mappings quasiconformal in the

153

integral sense) of multiply-dimensional domains. Let A : Rn -+ Rn be a linear bijection. Consider the quantities (see, e.g., [4],

[a1

Here 1(d4)= min IAhI, L ( A ) = max ( A h (and detA is the determinant of A. Ihl=l

JhJ=l

For a = n, the values of HI,,(A) and H O , ~ ( Acoincide ) with the inner and outer dilatations of A, respectlively (see, e.g., [15]). Let G and G* be two bounded domains in Rn,n 2 2 , and let a mapping f : G + G" be differentiable at a point z E G. This means there exists a linear mapping f'(s): R" i Rn, called the (strong) derivative of the mapping f at 5 , such that, f ( z + h ) = f ( z ) + f ' ( z ) h + w ( z , h)lh(,where ~ ( zh), + 0 as h -+ 0. Now we consider the homeomorphisms f differentiable almost everywhere in G and fix the real numbers n,B satisfying 1 5 Q < /3 < 00. Define

where H I , ~ ( . Zf ,) = H ~ , ~ ( f ' ( z Ho,o(z, )), f ) = N o , p ( f ' ( z ) ) We . call these values the inner and the outer mean dilatations of a mapping f of a given domain G. Define for the fixed real numbers a , P , y , G such that 1 _< a < ,6 < 00, 1 5 y < 6 < 00, the class B ( G ,G*,a , p, y,6) of such homeomorphisms f : G i G* which satisfy: (iii) f and f - l are ACL-homeomorphisms, (iv) f and f-' are differentiable with the Jacohians J ( z , f ) # 0 and J ( y , f - ' ) # 0 a.e. in G and G*, respectively, (v) the inner and the outer mean dilatations H l o , p ( f ) and HO,,b(f) are finite. In other words B(G,G*,a,P,y,d) consists of mappings with finite mean dilatations. The case /3 = S = n was explicitly studied in [6]. It follows that for any f E B ( G ,G*,a, p, y,a), we have the equalities

Yu. G. Reshetnyak [ll]has investigated quasiconformal mappings of the space domains using the radii of the normal regular system of neighborhoods. He called a mapping f quasiconformal at a point 3: E G if there exists a normal regular system {'&(.)} of neighborhoods of z such that A(z:>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)
-

a'(%)= lim inf @(Q) h+Od(Q)
-

where Q ranges over all open cubes and open balls such that z E Q c G and d(Q) = Q. Here mA denotes the n-dimensional Lebesgue measure of the set A. Due to [lo], these derivatives have the following properties: (vi) and g ( x ) are Borel's functions; (vii) @(x) = g ( x ) < ca a.e. in G; (viii) for each open set V c G, s F ( z ) dx 5 @ ( V ) .

v(z)

V

Using these set functions, we define for the fixed real numbers a , P , y , S such that 1 5 ct < /3 < ca and 1 5 y < 6 < 00, the class Z ( G , G * , c t , P , y , b )of homeomorphisms f : G + G* which satisfy: (ix) there exist @, @ E F in G, (x) for any point x E G there exists { G t ( x ) }c G, (xi) the inequalities

hold for all points x E G at which the derivatives W ( z ) and W(z) exist; B ( x ,h ) = {y E IW" : Iy - 21 < h}. 5. Equivalence of analytic and geometric descriptions. We now are enable to establish that the classes B(G,G*,a , P , y, 6) and Z ( G ,G*,a , p, y, 6) are coincide. Lemma 5. Let f E 'H(G,G*,a,P,y,6), then f is ACL-mapping, which i s dzflerentiable a.e. in G. Sketch of the proof. First, we show that f is ACL-mapping in G. Fix for each x E G a normal regular system {Gt(x)} of neighborhoods such that Gt(z) C G for any t E (0,1]. Consider an arbitrary point a = ( a l , . . . , a,) E G and h > 0 so that the cube Q = &(a, h) belongs to G, where & ( a ,h ) = {z : Ixi-aiI < h,i = 1 , .. . , n } . Denote by Ck, k = 1 , .. . , n, the intersection of Q with the plane P k ( a ) = {z E Iw" : xk = a k } , and let p ( x ) be the segment Izk - a k l 5 h/2 of the line passing through z (x E Ck) parallel to the kth coordinate axis xk. Similarly, let p ( A ) denote the union of all segments p ( x ) , when A c Ck and z E A. We show that for almost all x E Ck (with respect to (n-1)-dimensional Lebesgue measure) the restriction of f to p ( z ) admits the Lusin N-property. Further, for x,Z E G,Z # x, we define

155

Then k ( z ) < 00 for almost all z E G, and for any open set E C G k ( z ) is integrable with the powers P / ( P - n + 1) and y. This conclusion allows us to the classical Stepanov theorem [12] on differentiability and we obtain that f is differentiable almost everywhere in G. Lemma 6. Let f E X ( G , G * , a , p , y , 6 ) ,then H I a , p ( f ) and HOr,b(f) are finite. Lemma 7. I f f E B (G ,G*,a , P , y,6), t h e n f E Z(G,G * ,a , P , y,6 ) . The proofs of Lemmas 6 and 7 are similar to the corresponding lemmas from

PI. Now we can formulate the following theorem: Theorem 3. T h e classes B ( G , G * , a , Py,S) , a n d Z ( G ,G * , a , P , y , G) coincide. Thus we can characterize geometrically in terms of radii of normal neighborhood systems many well-known classes of mappings (quasiconformal, quasiconformal in the mean, enc). For the planar mappings which are quasiconformal in the mean, we refer to [7]. References 1. H. Bohr, Ueber streckentreue und konforme Abbildung, Math. Ztschr. 1 (1918), 3-19. 2. A. Golberg, Geometric characteristics of mappings with first generalized derivatives, Revue Roumaine de Math. Pure et Appl. 47 (2002), 671-680. 3. A. Golberg, O n generalization of Menshoff’s theorem, Israel J. Math. 156 (ZOOS), 243-254. 4. V. Gol’dshteyn, L. Gurov, and A. Romanov, Homeomorphisms that induce monomorphisms of Sobolev spaces, Israel J . Math. 91 (1995), no. 1-3, 31-60. 5. E. Goursat, Sur la dkfinition ge‘ne‘rale des fonctions analytiques d’apre‘s Cauchy, Trans. Amer. Math. Soc., 1 (1900), 14-16. 6. V. I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. 130 (1986), no. 2, 185-206. 7. V. S. Kud’yavin, Local structure of plane mappings that are quasiconformal i n the mean (Russian), Dokl. Akad. Nauk Ukrain. SSR 1991, no. 3, 10-12, 164. 8. D. Menshoff, Sur la representation conforme des domaines plans, Math. Ann. 95 (1926), 640-670. 9. D. Menshoff, Sur une generalisation d’un theoreme de M . H . Bohr, Mat. sb. 2 (1937), 339-356. 10. T. Rado and P. Reichelderfer, Continuous Transformations in Analysis, SpringerVerlag, 1955. 11. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Amer. Math. Soc., Providence, RI, 1989. 12. V. V. Stepanov, Sur les conditions de l’existence de la differentielle totale, Mat. sb. 30 (1924), 487-489. 13. S. Stoilow, Lecons sur les princzpes topologiques de la the‘orie des fonctions analytiques, Paris, 1938. 14. Yu. Yu. Trokhimchuk, Continuous mappings and conditions of monogeneity, Translated from Russian, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., New York, 1964. 15. J. Vaisala, Lectures o n n-dimensional Quasiconformal Mappings, Springer-Verlag, 1971.

156

SEPARATELY QUASI-NEARLY SUBHARMONIC FUNCTIONS JUHANI RIIHENTAUS

Department of Mathematics, University of Joensuu P.O. Box 111, FI-80101 Joensuu, Finland [email protected]. org After recalling some properties of separately subharmonic functions, we present the corresponding counterparts for separately quasi-nearly subharmonic functions. In addition, considering the subharmonicity of functions subharmonic in the first variable and harmonic in the second, we generalize some results of Cegrell and Sadullaev and Kolodziej and Thornbiornson.

Keywords: Separately subharmonic, harmonic, quasi-nearly subharmonic, Harnack, integrability condition.

1. Introduction 1.1. Separately subharmonic functions. Solving a long standing problem, Wiegerinck [Wi88], see also [WZ91, Theorem 1, p. 2461, showed that a separately subharmonic function need not be subharmonic. On the other hand, Armitage and Gardiner [AG93, Theorem 1, p. 2561 showed that a separately subharmonic function u on a domain R of Rm+n, m 2 n 2 2, is subharmonic provided $(log+u+) is locally integrable, where $ : [0, fca) -+ [0, +GO) is an increasing function such that

Ts

(n- 1 / ( m - 1

( $ ( s ) ) - ' / ( ~ - ' ) ds

< +a.

(1)

1

For related previous results of Lelong, Avanissian, Arsove and Riihentaus, see e.g. [Le45], [Le61], [Le69], [Av61], [Ar66], [He71], [Ri89] and the references therein. One of these previous results was ours: Theorem A. ([Ri89, Theorem 1, p. 691) Let R be a domain in Let u : R -+ [-m, +co) be such that (a) for each y E Rn the function R(Y) 3 x

is subharmonic, (b) for each x E Rm the function R(x) 3 Y is subharmonic,

t+

-

U(Z,Y)

E

1-03>

+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.

<

2.3. A function $ : [0,+m) + [0, +m) is permissible, if there exists an increasing (strictly or not), convex function $1 : [0, +m) -+ [0, +m) and a strictly increasing surjection $2 : [0, +m) -+ [0, +a) such that $ = $20$1 and such that the following conditions are satisfied: (a) $1 satisfies the A2-condition. (b) $T1 satisfies the &-condition. (c) The function t + is quasi-decreasing,i.e. there is a constant C = C($2) > 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”(<) # 0 at least in one point := zel ye2 ze3 E Qc, then in this case the basic elements e l , e2, e3 satisfy the condition

+

<

+

+

+

ef+ei+ez=~.

(2)

168

We say that an algebra A is harmonic if in A there exists a triad of linearly independent vectors { e l , e2, e3) satisfying the equality (2) provided that e: # 0 for j = 1 , 2 , 3 . We say also that such a triad {el, e2, e 3 ) is harmonic. In the paper [3] it is proved that there does not exist the three-dimensional harmonic algebra over the field of real numbers.

3. Monogenic functions in an infinite-dimensional harmonic algebra Consider as harmonic algebra A an infinite-dimensional commutative associative Banach algebra

00

over the field

IR with

the norm llgllp :=

and the basis { e k } e l , where the

lckl k=l

multiplication table for elements of basis is of the following form:

enel = en,

1

e2n+1e2n = 2 e4n ' i n 2 1 ,

-

It is evident that here e l , e2, e3 form a harmonic triad of vectors. : QE

Theorem 3.1. In order that a function

IF

of

the form

00

+ ye2 + ze3) = C u ~ (Y,z ).,

ek,

(3)

k=l

where Uk : Q + IR, be monogenic in the domain QE c E3, it is necessary and suficient that the functions U k , k = 1 , 2 , , . . , be dzfjerentiable in the domain Q , and that the conditions dUl(X, Y, ). dY

-

1 dU2(X, Y , ). 2

dX

,

169

auk(X>

dz

Y,')

- -

2

auk-2(X, dX

Y>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 .

+

-

+

<

+

+

-

Theorem 4.1. [6]. If Dc is a proper domain in p , then every monogenic function Q, : DC IH[c i s expressed in the f o r m

7

where y is a n arbitrary closed Jordan rectifiable curve in D , that embraces the segment connecting the points z = fi,(<) and Z, and the function @O i s u monogenic function taking values in the ideal To.

172

Note that under the conditions of Theorem 4.1 the integral in the equality (10) is the principal extension of analytic function F = A@ into the domain Dg. Thus, the algebra of monogenic functions in Dc can be decomposed into the direct sum of the algebra of principal extensions of complex-valued analytic functions a.nd the algebra of moriogenic functions taking values in the ideal lo. In the paper [6], for every function F : D , + C analytic in a proper domain D , we constructed explicitly the expansion in the basis { e 2 k - 1 } r ? l of the principal extension of F into the domain Dc:

~

1 2ni

J’k Y

where

C = zel +re3 and z = z f i r for (z, r ) E D , and the curve y has the same properties as in Theorem 4.1, and J ( t - z ) ( t - 2 ) is a continuous branch of this function analytic with respect to t outside of the segment mentioned in Theorem 4.1. Note that for every z E D, with I m z = 0, we define J ( t - z ) ( t - 2 ) := t z . ~

Theorem 4.2. [6]. If D , is a proper domain in C and F is an analytic function in D,, then the first and the second components of principal extension (11) of function F into the domain Dc generate the solutions y and $ of system (9) in D by the formulas Y(2,T) =

ul(x,r), $ ( x , r ) = T U z ( x , r ) .

(13)

From the relations (11) - (13) it follows that the functions

F(t) dt, ( t - z ) ( t - z)

Y ( x , Y )= 2na Y

are solutions of system (9) in the domain D.

5. Integral expressions for axial-symmetric potential and the the Stokes flow function We generalized integral expressions (14) and (15) for the axial-symmetric potential and the Stockes flow function, respectively, to the case of arbitrary simply connected domain symmetric with respect to the axis Ox. In what follows, D is a bounded simply connected domain symmetric with respect to the axis Ox of the plane xOr. For every z E D , with I m z # 0, we fix an in D , which connects the points z and 2 . In arbitrary Jordan rectifiable curve

rzr

173

d(t

this case, let - z ) ( t - 2 ) be a continuous branch of this function analytic with respect to t outside of the cut along TZz.

Theorem 5.1. [8].Suppose that the axial-symmetric potential p(x, r ) is even ,with respect to the variable r in the domain D . T h e n there exists the unique function F analytic in the domain D , and satisfying the condition

-

F(2)= F ( z )

V Z E D,

(16)

such that the equality (14) is fulfilled for all (x,r ) E D . Theorem 5.2. [8].Suppose that the Stokes flow function $(x,r ) is even with respect t o the variable r in the domain D and satisfies the additional assumption

$(x,0) = 0 ‘d (x,0) E D .

(17)

T h e n there exists a function Fo analytic in the domain D , such that the equality (15) is fulfilled with F = Fo for all ( x , r ) E D . Moreover, any analytic function F which satisfies the condition (16) and the equality (15) for all (x,r ) E D is expressed in the f o r m F ( z ) = Fo(z) + C , where C is a real con,stan,t. Note that the requirement (17) is natural. For example, for the model of steady flow of an ideal incompressible fluid without sources and vortexes it means that the axis Ox is a line of flow. Using integral expressions (14) and (15), we developed a method for effective solving boundary problems for axial-symmetric potential fields (see [S]).

References 1. M. A. Lavrentyev and B. V. Shabat, Problems of hydrodynamics and theirs mathematical models, Moskow: Nauka, 1977, 408 p. [in Russian] 2. F. Klein, Vorlesungen uber die entwicklung der mathematik i m 19 jahrhundert, T . l , Berlin: Verlag von Julius Springer, 1926. 3. I. P. Mel’nichenko, On expression of harmonic mappings by monogenic functions, Ukr. Math. J., 27 (1975), no. 5 , 606-613. 4. I. P. Mel’nichenko, Algebras of functionally-invariant solutions of the three-dimensional Laplace equation, Ukr. Math. J., 55 (2003), no. 9, 1284-1290. 5. I. P. Mel’nichenko, On a method of description of potential fields with axial symmetry, in: Coritemporwi-y Questions of Real and Complex Analysis, Kiev: Institute of Mathematics of Ukrainian Academy of Sciences, 1984, 98-102. [in Russian] 6. I. P. Mel’nichenko and S. A. Plaksa, Potential fields with axial symmetry and algebras of monogenic functions of vector variable, 111, Ukr. Math. J . , 49 (1997), no. 2, 253-268. 7. S. Plaksa, Algebras of hypercomplex monogenic functions and axial-symmetrical potential fields, in: Proceedings of the Second ISAAC Congress, Fulcuoka, August 16-21, 1999, Netherlands-USA: Kluwer Academic Publishers, 1 (2000), 613-622. 8. S. Plaksa, Singular and Fredholm integral equations for birichlet boundary problems for axial-symmetric potential fields, in: Factotization, Singular Operators and Related Problems: Proceedings of the Conference in Honour of Professor Georgii Litvinchuk, Funchal, January 28-February 1 , 2002, Netherlands-USA: Kluwer Academic Publishers, 2003, 219-235.

174

THE PARAMETER SPACE OF ERROR FUNCTIONS OF THE e?"'dw FORM a

:s

SHUNSUKE MOROSAWA Department of Mathematics and Information Science, Faculty of Science, Kochi University 780-8520, Japan [email protected]. ac.jp In this note, we consider the parameter space of error functions of the form a J( e-w2 dw Some properties of hyperbolic components of the parameter space are shown.

Keywords: Complex dynamics; transcendental entire function; error function; hyperbolic component.

1. Introduction Let f be a transcendental entire function. The maximal open set in C where the family { f n } is normal is called the Fatou set of f and its complement in is called the Julia set of f . When we consider complex dynamics of transcendental entire functions, the simplest one is an exponential map Ex(.) = Xe" (A # 0). It has only one asymptotic value 0 as a singular value. Behavior of its orbit under iterate decides the dynamics. The parameter space for Ex was first studied by Baker and Rippon' and Devaney2 independently. In this note, we consider error functions, which are defined in Section 2. The error function has two asymptotic values. We investigate a parameter space of a certain subclass of error functions. Elementary facts on dynamics of transcendental entire functions we use in this note can be found in Ref. 4. 2. Results An error function is a transcendental entire function given by the form

+

aEr(z) b with a E @ \ (0) and b E @, where

+

b and has no other singular value. A certain It has two asymptotic values h & / 2 property of Julia sets of error functions given by the form

+

a E r ( z ) id5

175

with a E Iw \ (0) and B E R are shown by Morosawa3 . In this note, we investigate a subclass of error functions given by the form fa(z) =a W z )

with a E C \ (0). We note that fa is a one complex parameter family. We say that fa is hyperbolic if the orbit of each asymptotic value accumulates to attracting cyclic points. A connected component of the set of parameters for which fa is hyperbolic is called a hyperbolic component. It is known that hyperbolic components are open. We define subsets in the parameter space of fa as follows:

A

= {a I

fa

has a completely invariant component.},

B, = { a I f a has only one attracting cycle with the period an.}, D, = {a I for n E

fa

has two attracting cycles with the period n.},

N. We have the following theorems.

Theorem 2.1. E v e r y hyperbolic c o m p o n e n t i s contained in o n e of A, B, a n d D,. Furthermore, A i s also described by { a I la1 < l}. E a c h of B1 and D1 consists of only o n e component. Theorem 2.2. E v e r y hyperbolic c o m p o n e n t except A i s simply-connected and u n bounded. Theorem 2.3. E a c h of B, and D , contains a c o m p o n e n t which i s t a n g e n t t o A

3. Proofs In this section, we give proofs for the theorems given in Section 2 .

Proof. [Proof of Theorem 2.11 First, we note that fa(-)

Hence, if there exists a cycle Furthermore, we have

{21,z2,.

= -fa(z).

. . z n } , then {-%I,

-z2,.

. . - z,} is also a cycle.

(f,")' (4= (f,")'(-4. Thus, if it is attracting, repelling or indifferent, then so is the corresponding one, respectively. Therefore, if fa has only one attracting cycle with the period n 2 2, then n is even. If fa has completely invariant component, then it has only one attracting fixed point or only one parabolic fixed point. In either case, the fixed point is the origin. If fa has only one parabolic fixed point, then a = 1. Since we have f1(%) = E T ( 2 )

=z

23 - -

3

+'.' ,

176

Fig. 1. The parameter space of f a ( z ) . The range shown is 1 Real 5 2 , I Imal 5 2. The disk in the center is A. Hyperbolic components of B, are colored white and those of D , are colored black.

f a has two petals at the origin. Because Fatou sets of transcendental entire functions have at most one completely invariant, component, fa has no completely invariant component. If fa has only one attracting fixed point at the origin, then the immediate basin of the attracting fixed point contains at least, one asymptotic values. From the symmetry, both a.symptotic values are contained in it. It follows that fa has a completely invariant component. Hence fa has a completely invariant component if and only if fa has an attracting fixed point at the origin. It is clear that fa has an attracting fixed point at the origin if and only if a satisfies la1 < 1. If f a has a fixed point # 0, then a is given by ( ‘ / E T ( ~Furthermore, ). if 5 is at’tracting, then it satisfies I<ed’/Er(c)I < 1. Hence, for fa with a sufficiently small multiplier of C, is in the sectors

<

<

and the same holds for a. It follows that D1 consists of only one component. Similarly, we see that p-31 consists of only one component.

177

Proof. [Proof of Theorem 2.21 The arguments here have already been given in Refs. 1,2. It is clear that gk(a) = f,"(aJ;;/2) is a n entire function of a.Let D be a component of D,. We see that { g n k ( u ) } converges uniformly on compact sets in D to one of the attracting cyclic point. Hence D is simply-connected. Next, we consider

It satisfies that which converges t o the multiplier X(a) of the cyclic point in 0. IA(a)\ < I in D and IX(a)l = 1 on d D . Since X(a) # 0 in D , we see that D is unbounded. Similarly, we obtain the claim for B,. Proof. [Proof of Theorem 2.31 From Theorem 2.1, A is described by {a 1 (a1 < I}. For a = ,ae (0 E R) on dA, the origin is indifferent fixed point. As we saw, f l has two petals a t the origin. The hyperbolic component D1 is tangent to A at 1. For 8 = 27rk/n with k , n E N,k < n and ( k , n ) = I, we have f a b )

=z

- pa2

n+l +

for pa # 0, if n is even and f a ( z ) = z - paz2,+l

+ .. .

for pa # 0, if n is odd. Hence, if n is even, then f a has n petals at the origin and has only one cycle with the period n. If n is odd, then f a has 2n petals at the origin and has two cycles with the period n. By the argument similar to those in the proof of Theorem 3 in Ref. 1, (1

+ e)ei2+h-(z)

has an attracting cycle for sufficiently small

t

> 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(<,Y

)WO

Hence we have, for any measurable and positive function f on D ,

N n f ( z )=

s,

4n(z,Y)f(Y)dA(Y).

Let GN = 00 N n be the potential kernel associated to N . We have the following classical results: N.GN = G N . N , N.GN I = G N . Note that if f : D + R be a measurable function on D , we put N f := N f + - N f - if { N f + = +03} n { N f - = +00} is empty.

+

Lemma 4.1 (2). L e t h be a T-invariant f u n c t i o n o n X . T h e n , f o r j , k { 1,2}, j # k , t h e f u n c t i o n f j

=

{

~ lj:

E

- GNRjh o i k , if h o i j < +00 , if h 0 i j = +00

is positive a n d locally A- integrable on { h o i j

< 00}

Proposition 4.1. If h i s a h a r m o n i c f u n c t i o n o n X s u c h t h a t h 0 i j ; j = 1 , 2 are locally A-integrable, t h e n f o r a n y U E U a n d f o r a n y z E X s u c h t h a t B ( n ( z ) , r ( n ( z )c) n ( U ) , w e h a v e T h ( z )= h ( z ) ,f o r a n y z E X Proof. Let j , k E {l,2}, j # k and U E U . We have h o i j = H,(U)hoij+Kfl(,)ho i k . Hence h o i j - K” h o i k is harmonic on n ( U ) . So, from [7, lemma 4.121, .(U)

N ( h 0 ij - K,”tu)h0 ik)(z)= h 0 i

j ( ~)

K t ( u ) h0 i k ( z ) ,

for any z E D such that B ( z , r ( z ) c ) .(U). Hence

hoi

j ( ~= )

Nh 0i

j ( ~)

NKfl(,)h

0

ik(z)

+ K:(,)h

0

ik(z),

for any z E D such that B ( z , r ( z ) c ) .(U). It remains to prove that

’J

Rjh 0 i k ( z ) = K?,(,) h 0 ik(z)- N(K:iU,h 0 ik)(z)

185

for any z E D such that B ( z , r ( z ) )c .(U).

From the definition of R j , we have

Rjh o i k ( z ) = K g h 0 i k ( X ) - N ( K g ( h0 i k ) ( z ) ) . Or

+

K g h o ik (x) K,"Iu)h 0 ik (x) H,(u) ( K z h 0 i k ) (X). Hence

(1) Rj h 0 i k (x)= K;iu) h 0 ik (X) -N(H,(U)K: Since H , ( u ) ( K z h k ) is harmonic on

+ H,(u) (K:

h0

k)

(x)- N (K;iq h 0 i /c ) (x)

h 0 ik)(z).

~ ( uthen ) , from [7,lemma 4.121,

( 2 ) N ( H , ( U ) K z h 0 ik)(X)= H,(,)(Kgh

0

ik)(z).

So, from ( 2 ) and (1) we have the result.

< 00;j #

Theorem 4.2. Any T-invariant function h on X such that K E h 0 ik j , k E {1,2}, i s harmonic with respect to (S)uEu.

Proof. On the one hand, G N ( K( h ~o i k ) ) then,

G N N ( K( ~ h0 i k ) )

+ ( K z( h

0

< +oo. Indeed, since G N N + I

k;

=GN,

i k ) ) = G N ( K( ~ h 0 i k ) ) . (*)

from the remark 4.1.3, and [7, lemma 1.11, we have, N ( K g ( h o i k ) ) is locally bounded. Thus, from the [7, lemma 1.51, G"(K;(h 0 i k ) ) is locally bounded. Hence, from (*) and the hypothesis, we have G N ( K( h~ o ik)) < +co. On the other hand, since K E ( h o ik) < +co, it results from the remark 4.1.3 that R j h o i k =K:hoik

-N(Kc(hoik));

and

GNRjh 0 ik

+ G"(K2

h 0 i k ) = GN(K: ( h 0 i k ) ) .

Therefore,

G N R j h o i k + G N N ( K g h o i k ) +K z h o i k = G ~ ( K E ( h o i k ) ) + K : h o Z k . Using again the relation GN = G N N

GNRjh o ik

+ I , we have,

+ G N ( Kh~o ik) = G N ( K( ~h

0

ik))

+K 2h

0

ik.

Hence

GNRjhoik = K z h o i k Since N ( h o i j - G N R3. h o i k ) = N ( lim N n h o i j ) = lim N"+lhoij, then h o i j n+oo

n+oo

GNRjh o ik is N-invariant. Hence, it results from [7, theorem 4.141 that, H , ( q ( h o i j - G ~ R j h o i k )= h o i j - G ~ R j h o i k

186

Hence,

Therefore, h 0 ij = H,(u)h

oij

+K,

' j

(h 0 i k ) - H , ( u ) K z ( h o i k )

so, h 0 i j = H,(qh

0

ij

+ K,i"iU)h i k 0

U E U.

Thus h is harmonic with respect to ( S U ) U ~rnU . Corollary 4.1. Let D be a Green domain in Rd, d >_ 2, and r : D + R; be a measurable function such that there exists positive real number E and a lipshitizian function p on D , with Lipschitz constant 1, dominated b y d i s t ( . , d D ) and E P 5 r 5 (1 - E ) P . Then, for any measurable and positive couple of functions ( h l ,h2) o n D satisfying the restricted m e a n value property; the function h defined o n X b y h o i j = hj ;j = 1 , 2 i s harmonic with respect to ( S U ) U ~ U . References 1. A. Benyaiche, Mesures de repre'sentation sur les espaces biharmoniques, in Potential Theory. I C P T 91, E d : E m i l e M.J. B e r t i n , p : 171.178, (Kluwer Academic Publishers, Netherlands, 1994 ) 2. A. Benyaiche and S. Ghiate , Proprie'te' de m o y e n n e restriente associe'e d un systbme d%.D.P., Rendiconti Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. 121, Vol. XXVII, fasc. 1, p: 125-143 (2003). 3. A. Benyaiche and S. Ghiate , M a r t i n boundary associated w i t h a s y s t e m of P D E . , Comment. Math. Univ. Carolin. 47, 3, p: 399-425 (2006). 4. J . Bliedtner and W. Hansen , Potential T h e o r y - A n analytic and Probabilistic Approach t o balayage ( Universitext, Springer, 1986). 5. A. Boukricha , Espaces biharmoniques, in The'orie d u potentiel, Proceedings, O r s a y 1983, E d s : G. Mokobodzki and D . P i n c h o n , p : 116-149, Lectures N o t e s in Mathematics. 1096 ( Springer- Verlag, 1984). 6. M. Brelot , Ele'ments de la the'orie classique d u potentiel, 4th edn (Centre de Documentation Universitaire, Paris, 1969). 7. A. Cornea and V. Vesely , M a r t i n compactification for discrete potential theory and the m e a n walue property, Potential Analysis 4, p: 1-21 (1995). 8. W . Hansen , Restricted m e a n value property a n d h a r m o n i c f u n c t i o n s , Potential TheoryI C P T 94, Eds: J. K r d , J. Lukei, I. Netuka and J. Vesel9, (Walter de Gruyter and Co., Berlin. New York 1996). 9. W. Hansen, Modification of balayage spaces by transitions with application t o coupling of P D E ' s , Nagoya Math. J.169, p: 77-118 (2003). 10. R. M. Her& and M. HervC , L e s f o n c t i o n s surharmoniques associe'es ri un ope'rateur elliptique d u second ordre d coeficients discontinus, Ann. Inst. Fourier, 19, 1,p: 305-359 (1968). 11. R. R. Phelps, Lectures o n Choquet's T h e o r e m ( Van Nostrand, 1966). 12. E.P. Smyrnelis , A x i o m a t i q u e des f o n c t i o n s biharmoniques, premibre section, Ann. Inst. Fourier, tome 26, 1, p: 35-98 (1976).

187

AN IMPLICIT FUNCTION THEOREM FOR SOBOLEV MAPPINGS I. V. ZHURAVLEV Mathematical Department, Volgograd State University, Volgograd, Russia [email protected] A nonsmooth variant of the implicit function theorem is proved. Keywords: implicit function; normalized Jacobi matrix; Clarke derivative.

F. H. Clarke proved the inverse function theorem for Lipschitz mappings more then forty years ago [ 1, 21 and as a consequence, the implicit function theorem for the class of Lipschitz mappings was obtained [ 31. Several years ago M. Cristea [ 41 proved the inverse function theorem for more wide class of mappings. We present an implicit function theorem for mappings with Sobolev's derivatives. Our method uses normalized Jacobi matrix. Let m, n 2 1be integers. Denote by Mm," the linear space of the (m,n)-matrices with real elements. For an arbitrary matrix A E Mm," we put I( A I/ = (tr(A.AT))'/2 (T is the transposition of matrices ), I A I = sup (Axl. Let D be a domain in

Rn-tmand F : D

+ Rm be

(x(=l

a continuous function of variables x E R", y E Rm. Assume that the function F ( z , y ) belongs t o the Sobolev's class Wt,l,,c(D)[5, 61. For the function F ( z , y ) the formal Jacobi matrix F ' ( x , y ) is defined on some set D * ,D* c D , almost everywhere in D . Everywhere below we suppose IF'(x,y)l > 0 a. e. in the set D . For a point a E D we denote by N ( F , a ) the convex hull in Mm,n+m of a set {B E Mm,n+m: there exist a sequence z p = (xp,yp)E D* ( p E N ) such that z p + a and + B } . The set N ( F ,a) is an analogue of the Clarke derivative [ 1, 21 and an analogue of the normalized Jacobi matrix [ 71. Let N,(F, a) is a projection of the set N ( F , u ) c Mm,"fm onto Mm,m (last m columns of a matrix B E N ( F , a ) are considered as a square m-matrix ). Let B " ( z , r ) be the ball in R" with center at the point, IC and radius T > 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


d2

8x2,

.

200

Defination 2. s ( R ~ )= {'p E c - ( R ~ ) I VQ, VP E Z$ , SUP,EWd Ixa8S'p(z)I < m}. Defination 3. S'(Rd) denotes the dual space of S(Rd). Proposition 1. Let u be a linear form from S to C. Then the following statements (i) and (ii) are equivalent. (i) u E S ' ( @ ) , (ii) 3c > 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
g(weit)dt,

WET.

The function OP+ ( 9 , .) : [0,ca) + [0, co) defined by l
OP,W( 9 , 6 , := suP{II

- gh I I L P ( T , w ) , h

The improvement obtained in7 can be formulated in the following way: If the boundary L of the domain G is a smooth Jordan curve, then

for n 2 2, where 02,1+;l(.,

i) is the integral modulus of continuity in space L2(T,

I $A I)

1

for cp’ [$o ( w ) ] . ($6 ( w ) ) ~From . this result in particular it follows that if G is a finite domain with a smooth Jordan boundary, then

which improves the estimation (1). Later in8 , under some restrictions of the boundary L , the improvement (2) and hence (3) has been extended t o the finite Radon domains. In this work, developing the idea used in7-’ we shall obtain an estimation for the approximation of c p p , 1 < p < 00, by means of the generalized Bieberbach ~ . estimation in case of p = 2 appears simpler than (2). polynomials T ~ ,This We begin with the following definition.

let

Definition 1 Let G be a domain with a smooth boundary L , 1 < p , r aP:= (cpb 0 $0) o n T.The function

4fP, 6) := SUP II @’p(weih)

-

@ p ( w ) IILP(T)=:

%(@p,

< co and

61,

Ihl3

is called the generalized integral modulus of continuity f o r cpb. Our main result is the following.

Theorem 1 Let G be a finite domain with a smooth boundary L and let p Then

II c p p - T n , p

llGi

cl&w;+E($2,

$1,

P = 2;

czn-l’pw;+,($P,

k),

p

c3n1/P-1+&

*

WP+&((PP,;),

> 2;

1< P

< 2,

> 1.

217

for every

E

> 0 and

with the constants ci = ci

(E)

> 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)<

c

WT+E(@P,

lln)

> 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
,

(ILp(G)

> 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,<

II

c 7 ~ n Pn IIL,(G)

and

11 pk - n i , p IILP(G)< C8P:,,

220

with s o m e p o s i t i v e c o n s t a n t s c7 and cs, w h e r e {an} 7, {Pn} L

and (7, := a n . P n } \ .

If in a d d i t i o n , t h e r e e x i s t s a sequence of i n d e x e s Ynk+l

for s o m e

E

E

5 EYnk

an,+,

{nk}

I csank,

s u c h that

k = 1 , 2 , ...

( 0 , l ) a n d a c o n s t a n t c 2 1, t h e n

II '

~p nn,p

l l ~ CYn. I

3. Proof of Main Result Proof of Theorem 1 . We apply the traditional method (see, for example' in case of p = a), its modification in case of 1 < p < 00 given in6 , and also the idea developed in the papers7-' . The proof goes following to the procedure drawing by Lemma 3 with an suitable choice of a, from Lemma 2, of Pn from Lemma 1, and defining n k := 2k in Lemma 3. References 1. F. G. Abdullayev, : Uniform convergence of the generalized Bieberbach polynomials in domains of complex plane, In: Approximation Theory and its Applications, Pr. Inst. Mat. Nats. Akad. Nauk Ukr., Kiev (1999), 5-18. 2. V. V. Andrievskii, : Uniform convergence of Bieberbach polynomials in domains with piecewise-quasiconformal boundary (Russian), In: Theory of Mappings and Approximation of Functions, 3-18. Kiev: Naukova Dumka (1983). 3. E. M. Dyn'kin, : T h e rate of polynomial approximation in the complex domain. In Complex analysis and spectral theory (Leningrad, 1979/1980), pp. 90-142. Berlin: Springer 1981. 4. D. Gaier, : O n the convergence of the Bieberbach polynomials in regions with corners. Constr. Approx. 4 (1988), 289-305. 5. D. M. Israfilov, : O n the approximation properties of the extremal polynomials, Dep. VINITI, No:5461 (1981) (in Russian). 6. D. M. Israfilov,: Uniform convergence of some extremal polynomials in domains in dom a i n s with quasi conformal boundary. East Journal on Approx., 4 (1998), 527-539. 7. D. M. Israfilov,:Approximationby p - Faber - L a w e n t rational functions in the weighted Lebesgue space L p ( L , w ) and the Bieberbach polynomials, Constr. Approx. 17 (2001), 335-351. 8. D. M. Israfilov,:Bergman type kernel function and approximation properties of some extremal polynomials o n quasidisks, East Journal on Approx. vol 9, 2 (2003), 157-173. 9. D. M. Israfilov,: Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation, Journal of Approximation Theory, 125 (2003), 116-130. 10. M. V. Keldych,: Sur l'approximation e n m o y e n n e quadratique des fonctions analytiques, Math. Sb., 5 (47)(1939), 391-401. 11. S. N. Mergelyan,: Certain questions of the constructive theory of functions (Russian), Trudy Math. Inst. Steklov, 37 (1951), 1-91.

221

12. I. I. Privalov,: Introduction to the Theory of Functions of a Complex Variable, Nauka, Moscow, (1984). 13. P. K. Suetin,: Polynomials ortogonal over a region and Bieberbach polynomials, Proc. Steklov Inst. Math., Providence, RI: American Mathematical Society, 100 (1974). 14. J. L. Walsh,: Interpolation and Approximation by Rational Functions in the Complex Domain, Third edition, Rhode island, (1960).

222

HARMONIC TRANSFINITE DIAMETER AND CHEBYSHEV CONSTANTS N. SKIBA Rostov State University of Transportation, Rostov-na-Donu, Russia skibanimail. 1 7 ~ VYACHESLAV ZAKHARYUTA Sabanci University, Orhanli, 34956 Tuzla/Istanbul, Turkey tahasabanciuniu. edu For a given compact set K in R p f 2 the harmonic transfinite diameter d h ( K ) ,directional harmonic Chebyshev constants T ~ ( K 0) , and the main harmonic constant T ~ ( Kare ) introduced by analogy with the corresponding characteristics for sets in C n (Leja, Zakharyuta). It is proved that always d h ( K ) 5 T ~ ( K Unlike ). the cme of seneral complex variables (considered in"), the equality d h ( K ) = T ~ ( Kis)proved under some restrictions on K . For the spheroid Q r , obtained by rotating of the ellipse with the half-axes sinh u and cosh 0 around the main axis, the following formula is obtained: d h ( K ) = T h ( K )=

sinh u (coth ~

);

fi

1. Introduction In" the second author introduced the directional Chebyshev constants r ( K ,0) of a compact set K c Cn and proved that the main Chebyshev constant r ( K ) ,defined as the integral geometric mean of the directional constants, coincides with the Fekete-Leja transfinite diameter d ( K ) introduced in;6 this fact may be considered as a natural generalization of the classical result of Fekete-Szego (see, e.g.,4). The existence of the usual (instead of upper) limit in the definitions of the directional Chebyshev constants and of the transfinite diameter in multidimensional case is based on some arithmetic multiplicative properties of the monomial basis, ordered in a natural way. This was the reason for the successful applications of ideas from" in studying of capacities in arithmetic geometry (see, e.g.,7>s) Here, by analogy with the classical approach in we define the harmonic transfinite diameter d h ( K ) of a compact set K in RP+2, p 3 0. Following," we introduce a natural notion of the m a i n harmonic Chebyshev constant r h ( K )defined as the integral geometric mean of directional Chebyshev constants T~ ( K ,0) of K . We discuss some interconnections among these characteristics in general case and

223 cosh

show, in particular, that d h ( K ) =

7 h(

K)=

(coth

0

5 )2for

the prolate

tl

, 0 < n < 00. The last cosh2a sinh 0 computation is based on the asymptotics of associated Legendre functions suggested in13 . Since the natural basis of harmonic polynomials (see below) has, contrary to the monomial basis in C",no explicit arithmetic properties the results for harmonic characteristics are considerably weaker than those from" . Nevertheless the equality d h ( K )= 7 h( K ) for the spheroidal case encourages our hopes that there might be some hidden recurrent "arithmetic" properties of the basis (3) which may confirm the following (tl,t2,t3)E

R3 :

~

Conjecture 1.1. T h e equality d h ( K ) =

7 h(

K ) holds for each compact s e t K in

RP+2.

2. Preliminaries 2.1. Notation

+

B y RP+2 we denote the ( p 2)-dimensional Euclidean space, p 2 1; B,(u) stands for the open ball in RP+2 centered at a with the radius T ; B, := BT(0);S = SP+l is the unit sphere in RP+2. For every set F and each function x : F + C we denote 1x1~:= sup{Ix(t)I : t E F}.The sign ~f means a continuous linear embedding. The open p-dimensional simplex

<

is denoted by C. Notice that, in particular, C := (01 : -1 < 01 1} for p = 1. If K is a compact set in RP+2, then h ( K ) stands for the space of all harmonic germs on K endowed with the standard inductive limit topology (for more details about spaces of harmonic functions see, e.g.,12) . Denote by h2 (B) the harmonic &-Hardy space in the unit ball B,i.e. the Hilbert space obtained as a closure of the set of all harmonic polynomials in the space L2 (S, n ) with the Lebesgue measure n on the sphere S.

2.2. Harmonic polynomials For each n Z+ consider the standard system of homogeneous harmonic polynomi) RP+2(see, e.g.,3 , 11.2). als of degree n of the variables t = ( t l , .. . , t p , t p + l , t p + 2E Namely, let M , be the set of all vectors m = ( m l ,. . . , mP)with integer coefficients such that

224

and r k defined by the formula 0 , 1 , . . . , p . Then the functions

Tk

=

&+,+ t,,, 2

+ . . . + t:+, ,

with m E M n , form the complete system of s ( n ) = s ( n , p ) = (an

where k =

+ p ) ( n +pp! n- ! l)!

linearly independent harmonic polynomials of degree n;here C,”(lc) is the Gegenbauer (ultraspheric) polynomial of degree n and order v ( ~ e e , ~ ) .

Definition 2.1. The standard (complete orthonormal in the space h2 (B))system of harmonic polynomials ei = e i ( t ) , i E N, is obtained from the system (3) by normalizing and enumerating in the strict lexicographic order with respect to the indices mo, m l , . . . , m p - l ,mp (therewith, the last index mp is ordered as usual: 0, -1,1,. . . , - m p , m p ) :

+

where (mo (i) , m (i)) is the enumeration, determined by the order malizing numbers N (mo,m ) are defined in3 , 11.3(5)).

< and the nor-

The chosen order provides, in particular, that the degree n (i) = mo ( i ) of the the set of all polynomial ei is a non-decreasing function of i. We denote by i

harmonic polynomials represented in the form: p ( t )=

C cj

ej

(t).

j=1

For the case p = 1, in the spherical coordinates ( r ,$, cp), we have

where P,” are the associated Legendre functions.

3. Harmonic characteristics of cornpacta 3.1. Harmonic transfinite diameter Let K be a compact set in RP+2.Given & E K , j = 1 , 2 , . . . , i, we define the Vandermondian: ( [ 1 , [ 2 , . . . ,ti)= det (ew(,$,))L,u=l,where {ei}iEN is the standard complete orthonormal system of harmonic polynomials (4). Introduce the numbers

yh = v,” ( K ):= SUP { IVh(&,&,. . . ,C)l : [ j

E

K , j = 1 , 2 , .. . ,i} , i E N.

(6)

[ii),

One can easily prove that there exist extremal points E F ’ , . . . , <ji) (in general, not uniquely defined) for which the supremum in (6) is attained.

Definition 3.1. The harmonic transfinite diameter of K is determined by the formula & ( K ) = limsup[Vh n t m

T(n)

]&,

n

where 1 ( n )=

n

C v . s ( v )and r ( n )= C s(v) is w=l

the number of all harmonic polynomials (3) of degree

< n.

v=O

225

Followingg we say that a compact set K is unisolvent if there is a sequence {ti);" c K such that V h([I, ( 2 , . . . ,En) # 0 for each n E N. Notice that if K is not unisolvent then there is io such that yh ( K )= 0 for all i 2 io.

3.2. Harmonic Chebyshev constants Given a compact set K c E%P+2, consider for each i E N the minimal deviation of harmonic polynomial p E K,,with the leading coefficient equal 1, from the origin in C ( K ) : 2-1

M,

=

M , ( K )=: inf{lplK : p ( t ) = e,(t)

+CcJeJ(t)).

(7)

J=1

It is a common place (see, e.g., [l])that there exists a Chebyshev harmonic polynomial 2-1

T, ( t )= e,(t)

+ C cl"ee,(t))

(8)

e,=1

(maybe not unique) such that M 2 = ITzIK.The numbers (6) can be estimated through the numbers (7) in the following way.

Lemma 3.1. Let K C Rp+2 be a compact set such that Then

I n particular, if K is unisolvent, then (9) holds for all i

(y), p ,. . . , I'-'(

providing V h

V,!,

=

Yh, ( K ) #

0.

= 2 , 3 , . . ..

((i"'), [t-l),. . . , EtP1

(2-1))

=

I&,

belongs to

3-1, and has the leading coefficient equal 1. Hence M , 5 lPzlK.On the other hand, since V h (($'-'),[:-'),. . . ,[z-l( 2 - 1 ) , t ) 5 v,", t E K , we have lPzlK 5 Thus

&.

the left hand side of (9) is proved. Let us prove the right hand side. Taking the extremal points (:I, (:I, . . . , [!'), it is easily seen that the right hand side of the equality Y h = det[ep((p))]L,u=lis not changed if we substitute the last row of

([p)))'

, where T, is

det ( r , ( ( ~ ' ) ) ' by the row (T2 u=l

a Chebyshev polynomial

V=l

(8). Therefore, expanding the resulting determinant along the last row, we obtain the desired estimate:

226

Definition 3.2. Denote

(w)

P k=l

E C with

m k

72

= r," (K) =: ( M i ) l / n ( i )and O ( i ) = (& ( i ) ) =:

( i )and n(i)from Definition 2.1. The number

T~(K,Q = limsupTi, ) I9 = (&) E C,

(10)

e(i)+e

is the harmonic Chebyshev constant of the compactum K for the direction (3 . Lemma 3.2. The function r h ( K ,19) i s upper semicontinuous on C .

Proof. Given 6"') C, choose a sequence O ( j ) E C, j E W, so that d ( j ) @'), I9(j) # and limsupTh(K,0) = lim T ~ ( K , O ( ~ In) )accordance . with Definition --f

e-o(o)

3.1, for each j E

3-00

N we find i ( j ) such that T~(K @)) ,

-

l/j 5

~ q j ) 1 , 0 (i

( j ) )- O(j)I

< l/j.

Then, because of 6 (i (j))4 @'), we obtain the required estimate: lim sup P ( K ,e ) = Jim rh( K ,

) 5 lim sup T ~ (5~ ) P (K,o(') ).

3-00

e-e(O)

j-03

Since the function In T ~ ( K 0) ,is bounded from above and, by Lemma 3.2, measurable with respect to the induced Lebesgue measure X on C, we are ready to give the following Definition 3.3. The (main) harmonic Chebyshev constant of a compactum K c RP+' is defined as the geometric mean of the directional Chebyshev constants ( lo), namely,

it is assumed here that T ~ ( K=) 0 if the integral equals -a. It is easy to see that

dh (a,)= T h ( B T )= Th(B,,O)= T , I9 E Lemma 3.3. Let K be a compact set in Rip+',

Then lim sup

5 T~ (K).

v-03

Proof. 1nt.roduce the rectangles

c.

7~= 7," (K)

and

227

and denote by S, the union of all Ai with define the function on the set S , 2 C: T,

( K , 0 )= {

if

~i

Q

E

T(V

- 1) < i

5

T

(v). For each n E W

Ai, ~ ( -v 1) < i 5 ~ ( v ) .

It is clear that (14) and limsupTu(K,O) 5 T " ( K , ~if) 0 E C. Therefore, taking into account that U-03

ln7, (K,0) dX(0) 5 0 and using the Fatou Theorem, we obtain

limsup u-00

S"\C

3.3. Relation between the characteristics Theorem 3.1. The estimate d h ( K ) 5 T ~ ( Kholds ) for any compact set K

c RP+2.

This theorem follows from Lemma 3.3 combined with the following

Lemma 3.4. For any unisolvent compact set K lim inf n-00

c RP+2 the following

is true:

5 dh ( K ) 5 lim sup F ~ . n-00

Proof. First we derive from the estimate (9) that n

n

r(n)

11(7u)us(u) 5 u=l

qn) < (n)!n

5

( T p )< T

- T

i= 1

n n

r(n)

( T i p

(n)!

i=l

(7,)us(u) ,

u=l

where 7, is defined in (13). Then consider the sequence cj =:

In70 = 0 1nFu

if if

j =1 r(v-I)<jLr(v)

lnr(n)! and the upper limit by Cesaro does not Taking into account that limw exceed the usual upper limit, we obtain, by Lemma 3.3, the estimates:

The left part of (15) can be obtained in an analogous way.

228

Unlike the case of CN-characteristics,we can prove the coincidence of the haronly ) under certain quite restrictive assumpmonic characteristics dh ( K )and T ~ ( K tions about K . Fortunately, these conditions can be checked in some important concrete cases (see, section 4).

Theorem 3.2. Suppose that in (10) the usual limit exists f o r each Q E C and liminf ' ~ >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,

<

<

with 0 rj < co,0 6 6 7r, -7r < p 6 n-. Let G, be the Hilbert space defined as the completion of the set of all complex valued harmonic polynomials with respect to the norm IldIG, =

(1 [ dp

,.(.I

6,P)i2sinddd

-7r

)

, 1/2

where .(q, t9,'p) = z ( t ) .The system of harmonic polynomials

with n E Z+, m E

Z,-n < m < n, forms an orthogonal basis in each Hilbert

space G,, 0 < o < 00. The norms of polynomials (18) are expressed by the following formula (,5 item 247):

I I u ~ , ~ ~ ~ G =, PimI(coshg),

0 < c < 03.

(19)

Lemma 4.1. The system (18), after the same enumeration of indices ( n ( i ) ,m ( i ) ) as in Definition 2.1, can be represented in the form:

229

with

(an(i))! c,z =

2 4 z ) n ( i ) ! ( (i) n - m (i))! .

Proof. We apply the result (proved in2 and mentioned implicitly long ago, e.g., in,5 chapter X) about the representation P,"(coshv) P,"(cosd) = x k n , m , l rlPlm(cos$). lsn

Then the system f , is represented in the form (20) with c,, = kn(z),m(z),n(z). So, applying the expression for kn,m,n from,2 we obtain the desired representation.

Lemma 4.2. Let I( = aU,o > 0. Then f o r any I9 E C = (-1, 1 ) there exzsts a usual lzmat zn (10) and the equalzty

holds, where

F ( o , y )=

(sinho)Y(cosho

(ycosho

+ Jy2 + (sinho)2), 0 < y < l .

+ d y z + (sinho)2)Y

(22)

Proof. For any harmonic polynomial p we have the estimates

Therefore, using the extremal property of partial sums of expansions by an orthogonal system, we obtain the estimates

It was proved in13 that

with F ( o , y ) defined above. From here, taking into account Lemma 4.1, we obtain that

-

(1+ y ) Y F ( o , y ) e Y ( l- Y)l-Y - ( 1 + Y)? F(o, y), eY(1 - y ) l - ~. 2 2

Therefore, by the estimate (23) and continuity of the function F ( o , y ) , we come to the equality

T ( ~ ) ( @ 0)~=,

w F ( r , 101) for any I9 E C = (-1,l).

230

Theorem 4.1. For any spheroid (17) we have

dh (@,)

= Th

sinh 0 (coth

(a,) = -

4)

fi

Proof. By Lemma 4.2 the conditions of the Theorem 3.2 are met, hence d h ( K )= T~ ( K ) .Putting (21) into (lo), we obtain

The direct calculation of the integral gives the formula (24). 5. Final remarks The following questions, which have been solved positively for the corresponding characteristics in Cn (") , are open for a n arbitrary compact set K in Rp+2.

(1) Does there exist a usual limit in (lo)? ( 2 ) Does the equality dh ( K )= T~ ( K ) holds?

( K ,0) continuous on C? T~ ( K )invariant under translations? In particular, would the equality (12) hold if B, be changed to I5, (a)?

(3) Is the function

T~

(4) Are the characteristics d h ( K )and

References 1. N. I. Akhiezer, Lectures on the theory of approximation, 2nd revised ed., Nauka, Moscow, 1965 (English transl. of the 1st ed., New York. 1956). 2. H. A. Buchdahl, N. P. Buchdahl and P. J. Stiles, J.Phys. Gen. 10, 1833 (1977). 3. A. Erdelyi, Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York, 1953. 4. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, 26,1969. 5. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmon.ics, Cambridge Univ. Press., Cambridge, 1931. 6. F. Leja, Colloq. Math. 7, 153 (1959). 533 (1994). 7. R. Rumely and Chi Fong Lau, Math. Z., 8. R. Rumely, Chi Fong Lau and R. Varley, Existence of the sectional capacity, Memoirs of AMS 145 (690), 2000. 9. J. Siciak, Trans. Amer. Math. Soc. 105, 322 (1962). 10. N. Skiba and V. Zahariuta, Israel Mathematical Conferences Proceedings 15, 357

(2001). 11. V. P.Zahariuta, Math USSR Sbornik 25, 350 (1975). 12. V. Zahariuta, Spaces of harmonic functions, in Lect. Notes i n Pure and Appl. Math., Dekker, NY 150 (1993),pp. 497-522. 13. V. Zahariuta, Indiana University Math. J., 50 (ZOOl),1047.

231

ON PROPERTIES OF MODULI OF SMOOTHNESS OF CONFORMAL MAPPINGS OLENA W. KARUPU D e p a r t m e n t of Higher and Numerical M a t h e m a t i c s , National A v i a t i o n University, K y i v , 1 K o m a r o v ave, Ukraine [email protected] e t www.n a u . edu. u a We consider some new estimates for the uniform, local and integral moduli of smoothness for the function realizing conformal mapping of the unit disk onto the simply connected domain.

K e y w o r d s : conformal mapping; modulus of smoothness; finite difference smoothnesses.

1. Introduction

Suppose G is a simply connected domain in the complex plane bounded by a smooth Jordan curve r, T = T ( S ) is the angle between the tangent to r and the positive real axis, s = s(w) is the arc length on I?. Suppose UJ = cp(z) is a homeomorphism of the closed unit disk D = { z : IzI 5 1) onto the closure of the domain G, conformal in the open unit disk D. Kellog in 1912 [l]proved the theorem in which it had been established that if 7 = ~ ( ssatisfies ) Holder condition with index a , 0 < a < 1, then the derivative p’(eie) satisfies Holder condition with the same index a. Afterwards this result was generalized in works by several authors. S.E. Warshawski received generalization of Kellog’s theorem for higher order derivatives of the function w = p(z) [a] and for domains with nonsmooth boundary [3]. J.L. Geronimus [4] and S. J. Alper [5] received results for general moduli of continuity satisfying some integral conditions. P. M. Tamrazov [6] obtained solid reinforcement of these results. Some close problems were investigated by E.P. Dolzenko [7] and V.A. Danilov [8]. Generalization of Kellog’s theorem for the moduli of smoothness of the second order satisfying Holder condition was received by R.N. Kovalchuk [9]. L. I. Kolesnik obtained inversion of this result [lo]. Results by S.E. Warshawski, R. N. Kovalchuk, L I. Kolesnik and author [ll] for moduli of smoothness of the 3-d order were proved by means of method due to S.E. Warshawski based on the introduction of additional point. But the men-

232

tioned method contains a roughering step in the replacement of finite differences (and moduli of smoothness) of order k ( k 2 2) by finite differences (and moduli of smoothness) of the 2-nd order. And as a result the less sharp inequalities obtained by means of this method do not posess any property important for applications and have essentially a restricted range of applications. P. M. Tamrazov [la] solved the problem of estimating for finite difference smoothnesses of composite function. These results gave posibility to receive generalizations and inversations of Kellog's type theorems for general moduli of smoothness of arbitrary order.

2. Estimates for uniform moduli of smoothness of arbitrary order The localization Z = Z(k, E , z , 6) is called the rule which to every ordered collection ( k ,E , z , 6), that consists of nonnegative integer k , set E c C, point z E C and positive number 6, corresponds the unique set Z(k, E , z , 6) c Ek+'. Let the finite function w = f ( z ) be defined on a curve y C C . Let (20, ...,Z k ) be the collection of the points on the curve y and [ZO,..., z k ; f , zo] be the finite difference of order k for the function w = f ( z ) . P. M. Tamrazov [6] defined on rectifiable curves the uniform curvilinear moduli of smoothness of order k for the function is the set of collections (20, ..., z k ) such that curvilinear (with respect to the curve y ) < distances between points zo, ...,z k E y satisfy the condition p (zir zi+l)/p ( Zj,Zj+l) N , ( N E [l,+a)) and , p ( z i ,w)5 6 ( i , j = 1,..., k )

Theorem 2.1 ( O.W. Karupu, [13] ). Let modulus of smoothness w k ( r ( s ) , 6 ) of order k ( k E N ) f o r the function r ( s ) satisfy the condition W k ( ' r ( S ) , d ) = 1

0 [w (S)] (S t 0) where w ( 6 ) is normal majorant satisfying the condition

q d t<

0

foo.

T h e n the modulus of smoothness wk(cpr(eiR), 6 ) of the same order k for the derivative p'(eie) of the function cp ( z ) on d D satisfies conditions wrc(argp'(eiR),5) = 0 ( p (6)) (6 + 0) , Wk(logcp'(eie),S) = 0 (v (6)) (6 t O ) ,

where

233

In partial case when modulus of smoothness wk ( ~ ( s )6), of order k for the function ~ ( ssatisfies ) Holder condition Wk(T(S),

S) = 0 (6") (6 4 0) , o < a < k ,

the modulus of smoothness wk,d(cp'(eis),6) of the same order k for the derivative cp'(eie) of the function cp(z) on d D satisfies the condition wk(cp'(eie),6) =

o (6") (S + 0)

with the same index a. 3. Estimates for local moduli of smoothness of arbitrary order

Let the finite function w = f ( z ) be defined on the set E c C. Let l ( k , E , z , 6) be the maximal Euclidian localization described by nonnegative integer number k , positive number 6 and point z E C . Then local modulus of smoothness of order k for the function w = f ( z ) is determined by formula W 6 , E ( f ,z

SUP

,4 =

l [ ~ O , . . . , z k ; f , ~ O l >I

(20,,..,tk)El(k,E,Z,6)

where [zo, ...,z k ; f , 201 is the finite difference of order k for the function w = f ( z ) . Let consider the noncentralized local arithmetic modulus of smoothness w k , + ( f ( z ) , 6 )of order k ( k E N ) for the function w = f ( z ) on a curve y, that is Wk,+(f(Z),4=

SUP

I[z0, "'1 z k ;

(zO>...?k)ETw,&(N)

f , zO]l >

234

where yur,8( N ) is the set of collections (20, ..., z k ) such that curvilinear (with respect to the curve y ) distances between points 20, ...,z k E y satisfy the condition P (zi,z i + ~ )(/ ~ < N ( N E [l,+oo)), and p ( z i ,w) 5 6 ( i ,j = 1,...,k ) . z j , %+l)

-

Theorem 3.1 (O.W. Karupu, [14] ). If the local modulus of smoothness wk(r(s),S) of order k f o r the function r ( s ) at the point wo = w ( S O ) o n the curve r satisfies the condition 0 ( d o ) (6 + 0)7

wk,so ( 7 ( 5 ) 76 )

then the local modulus of smoothness wk,o(cp’(ei*),6) of the same order k f o r the derivative cp’(eie) of the function p(z) at the point zo = eiso o n d D satisfies the condition wk,oo((p’(eis),6)= o ( h a ) (6 + 0). Theorem 3.2 (O.W. Karupu, [14] ). If the local modulus of smoothness wk,~(p’(eie), 6 ) of order k f o r the derivative cp’(eio) of the function p(z) at the point zo = eioo o n d D satisfies the condition Wk,O0

(cp’(ei6),6)= 0 ( 6 7 (6 + 0)7

then the local modulus of smoothness of the same order k f o r the function r ( s ) at the point wo = w (so) o n the curve I? satisfies the condition

(’ +

=

wk,S0(7(s)76)

Theorem 3.3 (O.W. Karupu, [14] ). Let the local modulus of smoothness of order k f o r the derivative cp’(eis) of the function p(z) at the point zo = eioo o n d D satisfy the condition Wk,Qo (cp’(eio)76 )

=

3

o

6 log - (6 + 0).

(1)

T h e n the local modulus of smoothness wk(r(s),S) of the same order k f o r the function r ( s ) at the point wo = w (so) o n the curve r satisfies the condition

Proof. We will consider at first the case k = 1. After the wellknown Lindelof’ theorem on d D the formula argcp’(eio) = r

7r

o

s (0) - B - 2

takes place. Consequently w1,00(7 0

s (0),6) = qoo(argcp’(eio),6 ) + ~ 1 6

(3)

235

where constants C1, C, and Cs do not depend on 6. so,

wl,Q,(Tos(0),6)=0

(5)

As

wl,so(T

where constants

Cd

( s )1 6 ) 5

C4wl,6'0(7

s (0) b 6 ) , 1

and b do not depend on 6, then W,s,(.(S),6)

=0

(61% 3 (6 -

and theorem is proved for k = 1. Let consider now the case k 2 2. Let condition of theorem be satisfied for k (3), that for all m 2 1 the identity

+

0)

1

(6)

2 2. Then it follows from equality

w m , e o ( ~ o s ( e i e ) l= ~ )wm,eo(argcp'(eie),6)

(7)

is true. It follows from the supposition of the theorem, that Wk,e"(argcp'(eie), 6) 5

1

bidk 1% S l

where constant b l does not depend on 6. So, 1 ( Slog ~ -))(6 + 0). (8) 6 Applying inequalities from [la],we receive the estimates of the local modulus of smoothness W ~ , ~ , ( T ( S )6) , via the local moduli of smoothness w ~ , Q , ( T o s ( e i e ) ,6) and wk,e,(s(eie)l6). We have W~,O,(T

w k , s o ( 7 ( s )6) ,

5 Wk,Q,(T

o s ( e i s ) , 6 )= O

0

s(eie)lS)

+ b26-k(k+1)'2

k-1 ~ W j , S 0 ( 0T s(eiO), b6) j=1

k-1 X T 1 >... >

k(k-1)/2

c

(9) q=l

Tk(k-l)/zLl

r l + ...+r k ( k ~ l ) / z = k ( k + 1 ) / 2 - j

where constant

where constants

b2

does not depend on 6. We will also use inequalities:

B ( m ) ,( m = 1,...l k

-

l ) ,do not depend on 6.

236

As for the local moduli of smoothness of the function ~ ( s the ) estimates, that are the analogues of Marchaud inequalities, take place, then it follows from the inequalities (9) and (lo), that for all j = 1,..., k - 1 the estimates

where constants b i j 1 ,

( j = I, 2, ...,k - 1) do not depend on 6, are true. So,

where constant C, does not depend on 6. Theorem is proved. 4. Estimates for integral moduli of smoothness of arbitrary order

Let ~ k ,f (~ z )( ,6) be a noncentralized local arithmetic modulus of smoothness of order k ( k E N ) of the function w = f ( z ) at a point z on the curve y. Then the integral modulus of smoothness of order k for the function w = f ( z ) on the curve y is introduced by the formula

where X = X(z) is the linear Lebesgue’s measure on the curve. These integral moduli is the special case of integral moduli of smoothness introduced by P. M. Tamrazov in 1977. He defined integral moduli of smoothness as averaging on arbitrary measure on the curve of the respective local moduli of smoothness. Difference between these moduli and traditional integral moduli of smoothness, introduced as the least upper bound of averaging absolute values of finite differences, is that the operators of averaging and taking of least upper bound are applied in reverse order. Theorem 4.1 (O.W. Karupu, [16] ). Let r ( s ) E L,k [O,Z], 1 5 p < +co, k E N . Let integral modulus w k ( r ( s ) ,6 ) p k of smoothness of order k for the function r = r ( s ) satisfy condition

where w ( 6 ) is normal majorant satisfying the condition

237

T h e n integral modulus of smoothness of order k of the derivative p'(eie) f o r the function p ( 2 ) o n d D satisfies condition: Wk((p'(eio), 6 ) p

= 0 [v (S)]

(6 + 0) >

where

Theorem 4.2. If integral modulus of smoothness of order k of the m - t h derivative of the function T = T ( S ) satisfies Holder condition with index a , where m E N ( m < k ) and

O
then integral modulus of smoothness of order k of the ( m + 1)-th derivative of the function 'p(eis) satisfies Holder condition with the same index a: wl,('p(m+l)(eio),

6)pk

= 0 (do)

(6 + 0) ,

If

then wk((p("+')(eis), 6 ) p k = 0

Proof of this theorem is similar to the proof of theorem 4.1. References 1. 0. D. Kellog, Harmonic functions and Green's integral, Truns Amer Math. Soc., 13 (1912), 109 - 132. 2. S.E. Warshawski, Uber einen Satz von 0. D. Kellog, Nachr. Ges. Wzss., Gottingen, (1932), No 1, 73 - 86.

238

3. S.E. Warshawski, On differentiability at the boundary in conformal mapping, Proc. A m e r Math. SOC.,14 (1961), No 4, 614 - 620. 4. J.L. Geronimus, On some properties of functions continuous in closed disk, Soviet mathematics Dokl., 98 (1954), No 6, 889 - 891 [In Russian]. 5. S. J . Alper, On uniform approximation of functions of complex variable in closed domains, Soviet mathematics Izv., Ser. Math., 19 (1955), No 6, 423 - 444 [In Russian]. 6. P. M. Tamrazov, Smoothnesses and polynomial appoximations, Kiev: Naukova dumka (1975), 270 p. [In Russian]. 7. E.P. Dolzenko, Smoothness of harmonic and analytic functions at boundary points of domain, Soviet mathematics Izv., Ser. Math., 29 (1965), No 5 , 1069 - 1084 [In Russian]. 8. V.A. Danilov, On construction of the smooth quasiconformal automorphism of the plane transforming the given smooth curve into the circle, Siberian. Math. J., 14 (1973, No 6, 1341 - 1345 [In Rassian]. 9. R. N. Kovalchuk, On some generalization of Kellog’s theorem, Ukr. Math. J., 17 (1965), No 4, 104 - 108 [In Russian]. 10. L I. Kolesnik, Inversation of Kellog’s type theorem, Ukr. Math. J., 21 (1969), No 1, 104 - 108 [In Russian]. 11. 0. W. Karupu, O n finite difference smoothnesses of conformal mappings, Kiev: Institute of mathematics of Ukrainian Academy of sciences, (1977), 1 2 p. [In Russian]. 12. P. M. Tamrazov, Finite difference identities and estimates for moduli of smoothness of composite functions, Kiev: Institute of mathematics of Ukrainian Academy of sciences, (1977), 24 p[In Russian]). 13. 0. W. Karupu, On moduli of smoothness of conformal ma,ppings, Ukr. Math. J., 30 (1978), No 4, 540 - 545 [In Russian]. 14. 0. W. Karupu, , Local moduli of smoothness of conformal homeomorphisms, in: Metrical questions of functions theory, Kiev:Naukova dumka, (1980), 49 - 53. [In Russian]. 15. 0. W. Karupu, On some properties of moduli of smoothness of conformal mappings, Acad. Nauk Ukrain. Works Inst. Mat., 31, Kiev, 2000, 237-243 [In Ukrainian]. 16. 0. W. Karupu, Integral moduli of smoothness of conformal mappings, Kiev: Kiev Air Force Institute Publishers (2000), 12 p. [In Ukrainian]. 17. 0. W. Karupu, On some chracteristics of conformal mapping, in: Proceedings of the second world congress, Ukraine, Kiev, 19 - 21 September, 2005, 5 , (2005), NAU Publishers. 586 - 591.

239

STRICT STABILITY CRITERIA OF PERTURBED SYSTEMS WITH RESPECT TO UNPERTURBED SYSTEMS IN TERMS OF INITIAL TIME DIFFERENCE COSKUN YAKAR University Department, University Name, Gebze Institute of Technology,Department of Mathematscs, Faculty of Sciences Gebze, Kocaeli, T U R K E Y 141-41400 cyakarpenta.gyte. edu. t r

In this paper we investigate the strict stability criteria between an unperturbed differential system and a perturbed differential system that have different initial positions and an initial time difference. Keywords: strict stability, initial time difference, perturbed differential systems, Lyapunov’s stability.

1. Introduction The method of Lyapunov’s stability has been very useful and fruitful techniques in the qualitative theory of differential equations since it is a practical tool in the investigation of the behavior of solutions of differential equations. It has been applied to investigate the relationship of unpertubed and perturbed systems in nonlinear differential systems [a, 4 , 5 , 6 , 7 , 8 ]the boundedness, stability properties of perturbed the study of initial value problems with an inisystems. Recently in [l,5,6,7,8], tial time difference has been initiated and the corresponding theory of differential inequalities has been investigated. As in all measurements, errors occur in measurements of starting time and the solutions of an unperturbed differential system may start at an initial time that is different than the starting time of the perturbed differential system. In real situations it may be impossible to have only a change in the space variable and not also in the initial time. We investigate strict stability criteria between unperturbed systems with different initial conditions and perturbed systems with different initial conditions. We not only consider the change in initial position, but also an initial time difference. Previously, the investigation of initial value problems in differential equations has been restricted to a perturbation in the space variable only with the initial time unchanged.

240

2. Definition and Notation

Consider the differential systems

-‘ z = f ( t- 7 , z), ~ ( 7 0 = ) zo for t 2 TO,TO E R+ N

N

and the perturbed differential system of (2.1)

where f , F E C [R+x Rn, Rn] and 7 = TO - t o . A special case of (2.3) is where F ( t , y ) = f ( t , y ) R ( t , y ) and R ( t , y ) is the perturbation term. Assume that the existence and uniqueness of the solution z ( t ) = z ( t ,to, 20)of (2.1) for t 2 t o and y ( t )= y ( t ,70,yo) of (2.3) for t 2 t o . Let us give the definition of the strict stability criteria with initial time difference.

+

Definition 2.1: The solution y ( t ,7 0 ,yo) of the system (2.3) through (TO, yo) is said to be initial time difference strict stable with respect to the solution z (t - 7 , t o , z o ) , where z ( t ,t o , 20)is any solution of the system (2.1) for t 2 70 2 0,to E R+ and 7 = TO - t o . If given any €1 > 0 and TO E R+ there exist 61 = 61 ( E ~ , T O ) > 0 and 62 = 62 ( E ~ , T O ) such that

whenever Ilyo - zo11 < 61 and € 2 < min {ST, 6;) such that

170 -

to1 < 62 and, for 6; < 61and 6; < Sz there exist

Definition 2.2: If 61,62and E Z in Definition 2.1 are independent of TO, then the solution y ( t ,TO,yo) of the system (2.3) is initial time difference uniformly strict stable with respect to the solution z (t - 7 ,t o , 2 0 ) for t 2 TO. Definition 2.3: The solution y ( t ,70,yo) of the system (2.3) through (TO,yo) is said to be initial time difference strictly attractive with respect to the solution z ( t - 7 ,t o , z ~ ) where , z ( t ,t o , z o ) is any solution of the system (2.1) for t 2 TO 2 0,to E R+ and 7 = TO - t o . If given any a1 > 0 , y l > O , E ~> 0 and TO E R+, for every a2 < a1 and yz < 71, there exist €2 < t l , T ~= TI ( E ~ , T O ) and TZ= TZ( € 1 , ~ such that

)

241

whenever (Iyo - z01I > a2 and 170- t o \ > 7 2 . If Tland T2 in Definition 2.3 are independent of T O ,then the solution y(t, TO,yo) of the system (2.3) is initial time difference uniformly strictly attractive stable with respect to the solution z(t - 77, t o ,50)for t 2 70. Definition 2.4: The solution y ( t , ~ o , y O )of the system (2.3) through ( ~ 0 , y o ) is said to be initial time difference strictly assimptotically stable with respect to the solution z (t - 7 ,t o , 2 0 ) if Definition 2.3 satisfies and the solution y ( t ,T O ,yo) of the system (2.3) through ( T O , yo) is initial time difference stable with respect to the solution z ( t - 7, t o , I C O ) . If Tland T2 in Definition 2.3 are independent of T O ,then the solution y ( t , TO,yo) of the system (2.3) is initial time difference uniformly strictly assimptotically stable with respect to the solution z(t - 77, t o , 2 0 ) for t 2 TO. Definition 2.5: For any function V, E C[R+ x Rn,lW+],we define the Dini derivative of the function with respect to the solution of the system (2.1)

D + V ( t ,z)

=

1 lim sup - [V ( t h-O+ h

+ h, 5 + h f ( t ,z)) -

and 1

D - V ( t , z) = lim inf - [V ( t + h, z h i O h for ( t , s )E R+ x

+ h f ( t ,z)) - V ( t ,z)]

R".

-

Definition 2.6: K is said to be the class K set of functions such that K := [ a l , b l , c l : a l , b l , c l E C [ O , p ] R+, increasing and al(0) = O,b1(0) = O,c1(0) = 01 and S, = { z E R".: llzll 5 p } . 3. Main Results

In this section we obtain the strict stability concepts with initial time difference parallel to the Lyapunov's results. Theorem 3.1: Assume that (Al) for each 77, 0 < 7 < p, V, E C[R+ x S,, R+] and V, is locally Lipschitzian in z and for ( t , z ) 6 R+ x S, and llzll 2 7,

242

v@

(Az) for each 8,0 < 8 < p, E c[R+ x in z and for ( t , z ) E R+ x S, and Ilzll 5 0,

s,, R+]and v@is locally Lipschitzian

where z ( t ) = y ( t , ~ o , y o )- x ( t - 77, t0,zo) for t L T O , V ( ~T,O , Y O ) of the system (2.3) through ( T O , yo) and z ( t - q , t o , zo), where z ( t ,t o , zo) is any solution of the system (2.1) for t 2 70 2 0,to E R+ and 7 = TO - t o . Then the solution y ( t , ~ o , y o )of the system (2.3) is the initial time difference uniformly strictly stable with respect to the system (2.1) for t 2 TO 2 0, t o E R+ and 77 = TO - t o .

Proof of Theorem 3.1: Let 0 <

€1

<

p and

TO

E R+. Let us choose

61

=

61(€1,T O ) > 0 such that

Then we claim that

whenever llyo - zgII < 61 and /TO- to1 < 6 2 . If (2.7) is true, then there exist t1 > t2 > 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

<

+ TI) 2 T , it follows that

This completes the proof. Then the solution y(t,To,yo) of the system (2.3) through ( T O , yo) is initial time difference uniformly strictly asymptotically stable with respect to the solution z(t - 7 ,t o , zo), where z ( t ,t o , 5 0 ) is any solution of the system (2.1) for t 2 TO 2 0,to E R+ and 7 = TO - to. Before we prove the general result in terms of the comparison principle. Let us consider the uncoupled comparison differential systems,

The comparison system (3.11) is said to be strictly stable: €1 > 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

<

c lz

-u j ~ ~ p ( ~ j ) - ~ j - ~ ,

(9)

j=1

where the constant c does not depend on z . Also the following estimation is fulfilled

254

pj := zj - A;(uj) - E and the constant c does not depend on z . Taking into account estimations (9), ( l o ) , and also inequalities ~j

+ Ap(

~ j)

A; ( a j ) + u - 1 - 2~ > -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] -

256

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<- ir Cjzl Rk-jflpj, k = 1,.' .,r,

<

-

i

C Bk-TRT-jfl j=1

k

C

-i

B k - j C p j r ,k = r

PjT

+ 1,.. ., N

j=r+l

is the solution of the r-modified Crank-Nicholson difference schemes for the approximate solutions of Cauchy problem iuk-uk-1

2-

uk-uk-1

A

+ H(Uk

f Uk-l)

= p k , 'Pk = f ( t k

+Auk = p k , p k = f ( t k -

$),tk

-

$),tk

= kr,1

= k7,

T

+ 15 k 5 N ,

5 k 5 r,ug = <.

Here

For

uO,

the three cases should be examined separately: ( i ) rr 2 A, ( i i ) rT < X and TT 2 A. Then, using formula (2.1) and the

3 E Z+,(iii) TT < A and 3 $ Z+.Let

258

we get 1

E

= Q(I

+ i i o ~ ) ~ 1 ira(I 6 + i i o ~C) ~ l - j + l p+~p - iiovz. -

j=1

Since the operator I

-

a ( I + iZ0A)R' has an inverse 1

-ira(I

+ ii,,~C ) ~ l - j + l p+~p - ii0pl j=1

where

+

T, = ( I - a ( ~~ z ~ A ) R ~ ) - ~ .

5

Let T r < X and E Z+.Then, using formula (2.1) and the uo = aul + p condition, the following formula is obtained:

Bl-'RTuo - i

2B

1 2-T

RT - j + 1

j=1

Since the operator I

uo = T,

{

-

aB1-'RT has an inverse ( I - C Y B ' - ~ R ' ) it - ~follows , that T

RT-j+'pj - iar

iarB'-'

5

Bl-T-jC

pT+j+p

j=1

j=1

where

1

>

(2.3)

T, = ( I - aB1-TR )-'. Let T r < X and condition

@ Z+.In this case if uo = a ( I

T

<

1 + illA)-(ul + 2

the following formula is obtained

[5] , then using formula (2.1) and the + p - ill^

UI+~)

259

Since the operator ( I - a ( I + i l l A ) i ( I+ B)B1-'RT)has an inverse it follows that

(I+B)

T

C B1-TRz-i+lpj j=1

+

1

C

B1-'Cpj +Cpi+i

j=T+l

where

I - a ( I + iZlA)-(I 1 2

If r =

+ B)BlPTRT

)-l.

[a], then in a similar manner one establishes

So, for the solution of problem (1.2)we have the following formulas

260

where

1

( I - a ( i + ~ z ~ A ) Rrr~ 2) -A,~

TT =

( I - aB1-rRr)-17rr < X,

p E Z+,

(2.5)

( I - a ( I + i Z l A ) ~ ( I + B ) B " - ' . R P ) - l , r T< A , ? @ Z+.

Theorem 3.1.The solution of the difference schemes (1.2) obeys the stability inequality

Proof. Using the estimates IIRIlH+H

F

1 7 IIBIIH+H 5 17 IlCIIH+H < - 1

(2.7)

and formula (2.4), we can obtain

Using the spectral representation of the self-adjoint positive defined operators one can establishes l/TTl/H+H

5 c(cy)'

The three cases should be examined separately: (i) rr 2 A, (ii) r r (iii) r r < X and @ Z+. Let rr 2 X.Then

3

Let r r < X and

(2.9)

< X and $ E Z+,

3 E Z+.Then -1

Let rr

< X and

9 @ Z+.Then

261

Now for estimate lluOllHthe three cases should be also examined separately: ( i ) TT 2 X , ( i i ) TT < X and E Z+, (iii) TT < X and f Z+.Let TT 2 A. Then by using formula (2.4), the triangle inequality and estimates (2.7), (2.9) the following estimate is obtained:

+

3

The proof of estimate (2.6) for difference scheme (1.2) is based on the last estimate and estimate (2.8). Let TT < X and E Z+,then by using formula (2.4), the triangle inequality and estimates (2.7), (2.9) the following estimate is obtained:

3

262

3

[$I,

Let TT < X and 4 Z+. If T = then by using formula (2.4), the triangle inequality and estimates (2.7), (2.9) the following estimate is obtained:

If T < [$] , then by using formula (2.4), the triangle inequality and estimates (2.7), (2.9) the following estimate is obtained:

Note that if the stability estimate of Theorem 2.1 and the passing to the limit for T + 0 is considered, one can recover Theorem of papers [ 2 ] , [3] on the stability of the nonlocal boundary value problem (1.1) . In the next section abstract Theorem 2.1 is applied in the investigations of two nonlocal boundary value problems.

263

3. Applications

First, the nonlocal boundary value problem for one dimensional Schrodinger equation

i ~ -t (u(z)u,),

+ 6~ = f ( t , ~ ) ,

u ( 0 , z )= au(X,z)

+ p(z),

05

2

0 < t < T , 0 < z < 1,

5 1, 0 < x 5 T ,

(3.1)

u(t,O) = ~ ( lt ),, uz(t,O)= u,(t, l ) , 0 5 t 5 T is considered. Problem (3.1) has a unique smooth solution u ( t , z ) for the smooth a(.) 2 a > 0 (z E (0, I ) ) , p ( z ) ( z E [0,1])and f ( t ,z) (t E [O, TI, z E ( 0 , l ) ) functions and 6 > 0 constant.This allows us to reduce the mixed problem (3.1) t o the nonlocal boundary problems (1.1)in a Hilbert space H = &[0, 11 with a self-adjoint positive definite operator A defined by (3.1). The discretization of problem (3.1) was carried out in two steps. In the first step the grid set

is defined. To the differential operator A generated by the problem (3.1) we assign the difference operator A: by the formula

acting in the space of grid functions uh(z)= { u m } f ,satisfying the conditions - U N - ~ .With the help of A: we arrive at the nonlocal boundary-value problem uo = U M , U ~- uo = U N

[i v+ uh(t,a:

A;uh(t,z)= fh(t,z),O< t < T , z

E [0,l]h,

(3.3)

for an infinite system of ordinary differential equations.

264

In t

\

To formulate the result, one needs to introduce the space Lah = La([O,l ] h ) of the all grid functions @(z) defined on z E [ O , l ] h , equipped with the norm

Theorem 3.1. Let T and Ihl be a sufficiently small numbers.Then the solutions of difference scheme (3.4) satisfy the following stability estimate:

(3.5) Here C ( a ) does not depend on 7, h, p h ( z ) and cp,h(z), 1 5 k 5 N . The proof of Theorem 3.1 is based on the abstract Theorem 2.1 as well as the symmetry properties of the difference operator A; defined by the formula (3.2) in L2h.

Second, let R be the unit cube in the n-dimensional Euclidean space R" (0 < 1, 1 5 k 5 n ) with boundary S and 6 = R U S. In [O,T]x 6 the nonlocal boundary-value problem for multidimensional Schrodinger equation

xk

<

2%

-

c;==,

(a&)uzJz,

u ( 0 , z )= Q U ( A , X )

=f(t,X),

+ p ( x ) , X E 6,

0


E 0,

(3.6)

u ( t , ~=) 0, x E S,0 5 t 5 T is considered. The problem (3.6) has a unique smooth solution u ( t ,z) for the smooth

a,(z) 2 a > 0 (z E R), p ( z ) ( z E 6) and f ( t , z )(t E [O,T],z E 0) functions. This allows us to reduced mixed problem (3.6) to the nonlocal boundary value problem

265

(1.1) in a Hilbert space H = L2(R)of the all integrable functions defined on R , equipped with the norm

with a self-adjoint positive definite operator A defined by (3.6). The discretization of problem (3.6) was carried out in two steps. In the first step the grid sets

0

5 m, 5 N,,h,N,

-

= L,r = l , . . . , n } ,

-

R h = R h n R , Sh = Rh n S are defined. To the differential operator A generated by the problem (3.6) we assign the difference operator A; by the formula

acting in the space of grid functions u h ( z ) ,satisfying the conditions u h ( z ) = 0 for all z E Sh.With the help of A;I we arrive at the nonlocal boundary-value problem

266

To formulate the result, one needs to introduce the space L2h = L2(flh) of the all grid functions cph(x) = (cp(hlrn1,.. ., h,rn,)} defined on flh, equipped with the norm /

Theorem 3.2.Let 7 and 1/ = Jh: + . . . + h; be a sufficiently small numbers.Then the solutions of difference scheme (3.4) satisfy the following stability estimate:

Here C ( a ) does not depend on 7, h,p h ( x ) and cpk(x),1 5 k 5 N . The proof of Theorem 3.2 is based on the abstract Theorem 2.1 as well as the symmetry properties of the difference operator A: defined by the formula (3.7) in

4. Numerical Analysis

In this section, the numerical solutions of the nonlocal-boundary value problem

1

. au(t %)

+ u(t,x ) = exp(itr2)sin r z , O < t ,x < 1, u ( 0 , x ) = $ u ( l , x ) + cp, cp = (1 2 jexp(ir2))s i n m , o 5 x 5 1, at

-

Pu(t,z)

-

(4.1)

u(t,0) = u ( t ,1) = 0, 0 5 t 5 1

for Schrodinger equation by using r-modified Crank-Nicholson difference schemes (1.2) are investigated. The exact solution of this problem is

u ( t ,x) = exp(itr2)sin rx. For the approximate solution of problem (4.1), the set [0, 1ITx grid points depending on the small parameters T and h

[O,l]h

of a family of

is defined. Applying (1.2) for the approximate solutions of the problem (4.1), we

267

- i uk-uk--l T

-

-

+ 15k 5 N

. u k -uk--l { -2

-

7

0 - 1 N U, - ZU,

un+l -2u;+u;-1

-

1,15n

u;;; -2u;-1+u;::

5M

+

u:+1-2u;+u;-l h2

-

Uk

+ ip(~,), 15 n 5 M

1,

=f(tk -

1; U ; =

,;

z,),

15 k

5 T , 1 5 n 5 M - 1,

UZ= 0,0 5 k 5 N ,

f ( t , ~=)exp(itr2)sinm,ip(z) = (1 - + e x p ( i r 2 ) ) s i n m . (44 So, we have (N + 1) x (M + 1) system of linear equations. This system one can write in the following equivalent form

where

a=---

1 b= 2h2 ' d=--

(- - + - + - i) ,

i

l h2

c=-+-+T

l 2

1 e = - - i f = - i+ - + +2 . h2 ' r' r h2

Then (4.3) can be written as the matrix form

{

cp: =

(1 -

$ exp(in2)sinm,,

f(tk -

;,z,),

k = 0,

15 k 5 N

l 2'

268

A=

0 d 0 O...O 0 O...O 0 0 0 0 d O...O 0 O...O 0 0 ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 O...d 0 O...O 0 0 0 0 0 O...a a O...O 0 0 0 0 0 O . . . O a a ... 0 0 0 0 0 0 O . . . O 0 a...O 0 0 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

0 0 0 O . . . O 0 O...a a 0 0 0 0 0 ... 0 0 0 ... O a a 0 0 0 o...o 0 o...o 0 0

B=

e f 0 O...O 0 O...O 0 0 e f O...O 0 O...O 0 ... ... ... ... ... ... ... ... ... ... ... 0 0 0 o... f 0 o . . . o 0 0 0 0 0 ... b c O ... 0 0 0 0 0 O . . . O b c...O 0 0 0 0 O . . . O 0 b...O 0 ... ... ... ... ... ... ... ... ... ... ...

0 0

... 0 0 0 0 ...

... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 O . . . O 0 O... b c 0 0 0 0 O...O 0 O...O b c

o...o

1 0 0

0

o...o

0

and

C = A,

-

-

1 0 o...o 0 0 1 o...o 0 0 0 1...0 0 D= ... ... ... ... ... ... 0 0 o... 1 0 - 0 0 o . . . o 1-

>

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
1 < n < M - 1.

270

From that it follows

Using formulas (4.7) and we can compute a, and 1 5 n M - 1, starting from

P,. After that we obtain U,,

and using recursive equation

U, = a,+lU,+1

+ Pn+1,

n=M

-

1,...,2 , l .

Now, we will give the results of the numerical analysis. In order t o get the solution of (4.2) we use MATLAB programs. The numerical solutions are recorded for different values of N = M and ti: represents the numerical solutions of these difference schemes at ( t k , x,). For their comparison, the errors computed by

E = m a x 1 5 k 5 N1 5 n 5 M I ~ ( t k , x , )-u,1. k Tables 1 give the error analysis between the exact solution and the solutions derived by difference schemes. Table 1 is constructed for N = M = 60,80, and100 respectively.

Table I N=M=60

N=M=80

N=M=100

M.C.N. r = l

0.0310

0.0175

0.0112

M.C.N. r=2

0.0413

0.0234

0.0150

M.C.N. r=3

0.0522

0.0297

0.0191

lStorder accuracy

0.4704

0.3866

0.3267

Method

Note that the second order of accuracy r- modified Crank-Nicholson difference schemes are more accurate comparing with the first order of accuracy difference scheme.

271

References 1. M.E. Mayfield, Non-reflective boundary conditions for Schroedinger’s equation, PhD

Thesis,University of Rhode Island (1989) 2. D.G. Gordeziani and G.A. Avalishvili, T i m e - nonlocal problems f o r Schrodinger type equations: I. Problems in abstract spaces, Differential and Equations 41(2005), no.5, 703-711 3. D.G. Gordeziani and G.A. Avalishvili, Tame- nonlocal problems f o r Schrodinger type equations: II. Results for specific problems, Differential and Equations 41(2005) no.6, 852-859 4. H. Han, J. Jin, X. Wu, A finite difference method for the one-dimensional Schrodinger equation o n unbounded domain, Computers and Mathematics with Applications 50 (2005), 1345-1362 5. X. Antoine, C. Besse and V. Mouysset, Numerical schemes f o r the simulation of the two-dimensional Schrodinger equation using non-reflecting boundary conditions, Mathematics of Computation 73 (2004), no.248, 1779-1799 6. A. Ashyralyev, An estimation of the convergence f o r the solution of the modified CrankNicholson difference schemes f o r parabolic equations with nonsmooth initial data, Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk (1989), no.1, 3-8. (Russian). 7. A. Ashyralyev, S. Piskarev and S. Wei, O n well-posedness of the difference schemes for abstract parabolic equations in Lp([O,11,E ) spaces, Numerical Functional Analysis and Optimization 23(2002), no.7-8, 669-693 8. R. Rannacher, Discretization of the heat equation with singular initial data, Z. Angew. Math. Mech. 62(1982), no.5, 346-348.

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PART B

OPEN PROBLEMS

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275

SOME OLD (UNSOLVED) AND NEW PROBLEMS AND CONJECTURES ON FUNCTIONAL EQUATIONS OF ENTIRE AND MEROMORPHIC FUNCTIONS CHUNG-CHUN YANG The Hong Kong University of ScienceTechnology, Hong Kong mayang @ w t .hk

Introduction The purpose for posing these old and new problems and conjectures arised in studying the Nevanlinna’s value distribution theory and its applications is t o stimulate peers’ interest to study further of the theory and properties of entire and meromorphic functions of one complex variable. Also be reminded that one can carry the studies t o similar problems and conjectures entire and meromorphic functions of several complex variables as well as that of p-adic complex variables. For simplicity , here we shall address the problems and conjectures for (non-constant) entire or meromorphic functions of one complex variable only, and most of them were proposed by the present author and his co-workers earlier. We refer the reader t o [ 3 ] for the Nevanlinna’s theory and its associated notations. Problems and conjectures will be stated in three different topics : 1.Factorization and fixed points of entire functions, 2. Dynamics of two permutable entire functions, and 3. Functional equations of Diophantine type over functions field. Sources, definitions, and results relevant t o these problems and conjectures will be stated or cited, with references. 1. Factorizations and fixed points of entire functions Definition. Let F, f and g be non-constant entire functions. The expression or composition: F = f ( g ) (or fog) is called a factorization of F , with f and g the left and right factor, respectively. F is called a prime (pseudo-prime ) function iff whenever F has a factorizaion : F = f ( g ) implies that either f or g is a linear function (polynomial ), see, e.g.,[l]. There have been many sufficient conditions or criteria t o judge whether or not a certain function is prime or pseudo-prime, but no necessary conidtion or criterion for such a verification has been found yet. Question 1.1. What is a necessary condition for an entire function t o be prime or pseudo-prime ?

276

Conjecture 1.2 ( [l],p.181). Let F be a pseudo-prime entire function and P a non-constant polynomial. Then F ( P ) remains t o be pseudo-prime. Note that one can exhibit some examples to show that, in general, P ( F ) may not be psudo-prime. Definition Let f , F, g , G , h, H , k and K denote entire functions. h is called a (right)factor of H if and only if H = k ( h ) and will be noted as hlH. And K is called a greatest common (right factor) of F and G iff K ( F , KIG and if h is any other common factor of F and G, then KI h. K is called a least common multiplier of F and G iff F l K and G I K , and if H is any other function satisfying HIF and HIG, then K I H . It has been shown [2] that any collection of entire functions E = { F a } , under a slightly general definition of a (right) factor, there always exists a common factor for E , which may be a linear function. However, by example of two functions: exp(z) and exp(iz) in [9], it follows that a common multiplier of two functions may not exist. Question 1.3. What is a necessary and condition for two entire functions F and G to be relatively prime ? That is the only common factor of F and G is a linear function. It has been known that F ( z ) = z + p ( z ) expa(z) is a prime function, where p is a polynomial ($ 0) and a a non-constant entire function. Which immediately shows that if neither f nor g is linear, then a transcendental function of the form f ( g ) must have infinitely many fixed points. In fact, it had been a conjecture for long while that such a function f(g) must have infinitely many fixed points, see, e.g., [8]. As a further study of the quantitative estimate of the numer of fixed points of composite functions, the following conjecture was raised: Conjecture 1.4 ( [ 7 g a nonlinear function

N

((Tl

1,

[ 8 ] ). Let f be a transcendental meromorphic function,

, u ( z ) a small function of f . Then l l ( f ( 9 ) - a ) ) # 4 1 ) T(T1f ( 9 ) ) as

T

+ 0.

2. Dynamics of two permutable entire functions It has been a well known result that if two polynomials p , q are permutable, then p , q have the same Julia and Fatou sets. Thus far,the same conclusion holds for two permutable transcendental entire functions f,g that are of bounded type, that is the sets of finite singularities (critical values and asymptotic values ) of both f and g are bounded , see, e.g.,( [ 6 ] ). Question 2.1 . What are the relations between two permutable transcendental

277

+

entire functions f and g other than the following two situations : 1. f = a g b, with a , b constants and 2. f = h, and g = h, for some transcendental entire function h, where h k denotes kth-iteration of h. Remark. One may find the example in [6] which shows f and g are permutable but may satisfy neither situations 1 and 2 above. 3 . Functional equations of Diophantine type Conjecture 3.1 ( [ 10 ] ). If a Diophantine equation : F(z,y) = 0, where F is an irreducible polynomial of degree higher than 3 , with rational numbers as the coefficients, has none or finitely many rational solutions, then the correspondening equation F ( f ,g ) = 0 has none or finitely many non-constant transcendental meromorphic solutions f and 9. Here the two pairs of solutions ( f , g ) and ( f ( h ) , g ( h ) ) for any non-constant entire h are said to be equivalent. Conjecture 3.2. Let P denote a non-constant polynomial. The only transcendental entire solutions for the equation of the form : f 2 ( z ) + p ( z ) g 2 ( z ) = 1 are pairs :(*f, h g ) , with f = cos and g = sin

m

m/m.

Definition. Let P ( z ,f , g ) be a polynomial in f and g, with entire or meromorphic functions of z as the coefficients. A pair of meromorphic solutions ( f ,g ) of the equation P ( z ,f ,g ) = 0 is called admissible if all the coefficients of the equation are small functions of f and g. See [4] for some results on the existence of admissible solutions of functional equations of the form :

f"

+ a i f n - , + bl

= c(gn

+ a2gn-, + b 2 ) ,

where a i ( i = 1,2), bi(i = 1,2) and c are meromorphic functions and none of them is identically zero. Note, in the above result, n is assumed to be greater than 8, with m 2 2 and n > 2m+3. It seems to be quite difficult to deal with the above functional equations when the degree of f or g is less than 8. Conjecture 3.3( [4] ). Let p , q be two polynomials both having at least three distinct zeros, and a(.) be any non-constant meromorphic function. Then the functional equation: p ( f ) = a ( z ) q ( g ) has no admissible solutions f and g. Conjecture 3.4 ([ 5 ] ). The functional equation: f 2 = g3 + b ( z ) g + c, where b is a non-constant meromorphic function and c a constant, has no pair of admissible solutions.

278

References 1. Chi-Tai Chuang and Chung-Chun Yang, Fix-points and factorization of meromorphic functions, World Scientific , 1990 . 2. A. Eremenko and L.A. Rubel, The arithmetic of entire functions under composition, Advances in Mathematics, 124 (1996) ,334-354 . 3. W.K. Hayman, Meromorphic Functions, Oxford Press, 1964 . 4. Ping Li and Chung-Chun Yang, Admissible solutions of functional equations of Diophantine type, Preprint. 5. Ping Li and Chung-Chun Yang, Decompositions of meromorphic functions over small functions fields , This Proceedings . 6. Liang-Wen Liao and Chung-Chun Yang, On the Julia sets of two permutable entire functions, Rocky Mountain J. of Math. , vol. 35 no. 5 2006, p.1659-1674 . 7 . Chung-Chun Yang, On the zeros of f ( g ( z ) )- a(.), Proc. Nankai Conference, International Press, 1994, 261-269 . 8. Chung-Chun Yang and Jian-Hua Zheng, On the fix-points of composite meromorphic functions, J. D’Analyse Mathematique, vol. 68 (1998), 55-93 . 9. Chung-ChunYang, On common right factor and common right multiplier of two entire functions, Chinese J. of Contemporary Math., vol. 18, no. 3, 1997, 347-375 . 10. Chung-Chun Yang and Ping Li, Some further results on the functional equation P ( F ) = Q ( G ) , Value distribution theory and related topics, ACAA series, Kluwer 2004, p. 219-231 .

279

AN OPEN PROBLEM ON THE BOHR RADIUS LEV AIZENBERG Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel aizenbrg @math.biu. ac.il

Keywords: Hausdorff operator, Fourier transform, real Hardy space

Let G c C be any domain. A point p E dC: is called a point of convexity if p E aG. Here G is the convex hull of G. A point of convexity p is called regular if there exists a disk U c G so that p E aU. In' the following result is proved.

Theorem 1. If the function

k=O

is such that f (Ul) inequality

c G I with G #

CC and U1 the unit disk, then for IzI

< 1/3 the

03

C lckzkl < dist(c0,dG) k=O

is valid. The constant 1/3 cannot be improved if dG contains at least one regular point of convexity. Open problem: if G # C and no regular point of convexity exists in a G , figure out whether it is possible or not to improve the constant 1/3 in the above theorem. A similar problem concerns multidimensional analogs of that theorem (see').

References 1. L. Aizenberg, 'Generalization of results about the Bohr radius for power series', submitted t o Studia Mathem.

280

OPEN PROBLEMS ON HAUSDORFF OPERATORS ELIJAH LIFLYAND

Department of Mathematics, Bar-Ilan University, Ramat- Gan, 52900, Israel E-mail: [email protected] After 2000 an interest in Hausdorff operators grew. Here we give a list of open problems in the subject along with history and prerequisites.

Keywords: Rower series, Hausdorff operator, Fourier transform, Hardy space.

The aim of these notes is to present open problems concerning Hausdorff operators. The idea appeared during the International Conference on Complex Analysis and Potential Theory in Istanbul, Turkey, September 8-14, 2006. The author’s talk on recent results in the subject and further discussions with participants of the conference confirmed that there is an interest in these problems. We will give minimal information on the topic, enough to formulate the problems and make the paper more or less self-contained. Further details can be found in the relevant literature from the list of references. Hausdorff means, the Ceshro means among them, are known long ago in connection with summability of number series. We are not going to touch this subject, though we do not exclude that there are interesting open problems there. 1. Power series. Ceshro means for power series from the Hardy space H1 in the unit disk were considered by A. Siskakis. The idea is to substitute the coefficients f E H I , for their CesAro means

co

/

k-1

\

k=O

\

P=o

1

2;

(1)

for an elegant proof of the boundedness of the corresponding operator in H1, see Ref.14 Later, already after appearance of the paper Ref.g in 2000, general Hausdorff matrices were considered in Ref.5 (we are also aware of continuation of the latter work by P. Galanopoulos and M. Papadimitrakis) as follows. Let A be the forward difference operator defined on scalar sequences a =

281

by A a , = a,

-

an+l and set

H ( a ) is defined by ( H ( a ) ) i j=

Cn,k =

0,

( ; ) A n p ku ,k i < j

cij, i

k 5 n. The Hausdorff matrix

. It induces two operators on spaces of

2j

power series which are formally given by

and

Necessary and sufficient conditions on a = a ( p ) are found in order that H ( a ) and A(a) induce bounded linear operators on the Hardy spaces HP, 1 5 p < M. Problem 1. a) Find multivariate versions of all aforementioned results. b) Study partial sums of (1) as well as those of (2) and (3).

2. Ceshro means for Fourier transforms. The Fourier transform setting for the Cesaro means was considered in Ref.;7 this is a partial case of general Hausdorff operators (4) below by letting p(z) to be the indicator of [ O , l ] . . For a special interesting case p 5 1 a complete solution for the C e s k o means was given in Ref.12 Problem 2. Study the boundedness of the Hausdorff operators

(4) i n

HP for

P51.

3. One-dimensional Hausdorff operators. In fact, general Hausdorff means of a Fourier-Stieltjes transform were introduced in Ref.,' but only in L1. In the real Hardy space on R,for the Hausdorff operator defined, by means of 'p E L1(R), as

the boundedness of this operator taking H1(R) into H1(R) was proved in Ref.g and 13 similar problems are considered for Observe that, as in Ref.,' in the Hausdorff operators defined by suitable measures, while (4) is the partial case of absolutely continuous measures. Here we restrict ourselves to the latter case for the sake of brevity and convenience; we are aiming at a different generality, more transparent in a simpler setting. Extensions to arbitrary measures go through as in the cited papers. Similarly to the case of power series, the point is as follows. Since, generally speaking, the inverse formula

282

does not take place for f E L1(R) as well as for f E H1(R), where transform o f f ; expected is that

f^ is the Fourier

behaves better and characterizes f properly, in a sense. Though these operators were considered in various spaces, the most important is the case of Hardy spaces; see, for example, Refsg or 12. Some open problems that will be given below in the multivariate setting are also open for dimension one. Here we present a very general problem which first of all deserves investigation on the real axis.

Problem 3. Prove or disprove that the real Hardy space H1(R) can be characterized via Hausdorff operators in the sense that if (4) f o r an integrable function f is in H1(R) then f is necessarily in H1(R). I f not (this is very probable), a counterexample should be given. 4. Multidimensional Hausdorff operators. In the multidimensional case the situation is, as usual, more complicated. The Ceshro means in Ref.8 and special Hausdorff means in Ref.1° were considered in dimension 2 only for the so-called x R). In the recent paper Ref.16 a slight exproduct (mixed) Hardy space tension was made in the same direction. In Ref.l the boundedness in H1(Rn) of the Hausdorff type operator

where x E R", was proved. But defining the operator by one-dimensional averaging does not seem t o be natural for the multivariate case. Similar to Refs.3 and 13, we generalize (4) by defining a Hausdorff type operator as

where A = A(u) = ( c ~ i j ) : ~ ==~ ( c ~ ~ j ( u )is)the ~ ~n=x ~n matrix with the entries aij ( u )being measurable functions of u. This matrix may be degenerate at most on a set of measure zero; zA(u) is the row n-vector obtained by multiplying the row n-vector z by the matrix A. By this, each Hausdorff operator is defined by a couple (a,A) satisfying at least the condition

283

II@IIL~

=

J'W" I@(u)II detA-'(u)l

du

< m,

which provides the boundedness of the operator in L1(Rn). In Ref.13 the boundedness of such operators in H1(Rn) is proved for a very special case of diagonal matrices A with all entries on the diagonal equal t o one another. But, defining Q, E LL by

with IlAll = IIA(u)lI = maxj(Jalj(u)I+ ... + lanj(u)I),we proved in Ref.ll that the boundedness on H1(IWn) for general matrices A is provided by @ E Llp1.

(6)

Since the dual approach is used in Ref.,ll the key ingredient in the proof is a lemma on the behavior in u of the BMO-norm of f ( z A ( u ) )This . also allows us to get conditions for the boundedness of the operator in B M O ( R " ) . Moreover, similar results are obtained for the adjoint operator as well. These lead to Problem 4. a) Prove (or disprove) the sharpness of the condition (6) for the boundedness of the Hausdorff operator in H 1 ( R n ) . b) Find a different proof of the boundedness in H1(Rn) of the Hausdorff operator satisfying (6). 5. Mixed Hardy spaces. Besides "usual" H' space of importance are the

so-called product (mixed) Hardy spaces considered in detail by R. Fefferman (see, e.g., Ref.*). Results on Hausdorff operators in very special mixed Hardy spaces are mentioned above. Problem 5. Find connections between (a,A ) Hausdorff operators and mixed Hardy spaces an terms of belonging of @ to a space defined b y A. 6. Partial integrals. It is known (see Ref.3) that

where (A-')T is the transpose of A-l. Taking

284

using (7), and applying Fubini's theorem we arrive at

'tr,vf(z) =

ln@(.)I

det A-ll d u

e--i(z-uA-l

("),Y))dy.

The last integral is well-known (see, e.g., a formula from the proof of Theorem 3.3 from Chapter IV in Ref.15), and we get (27rN)-"/2'tr,vf(z) = JR2,, @(u)(detA-l(f(w)(z - W A - ~ ( U ) ( - " / ~ J , / ~ ( NwA-l(u)()dudw, (~

where Jn12 is the Bessel function of order n / 2 , with f either from L1 or from H1. In dimension one it is of the following form: with 4 E L1

Problem 6 . S t u d y

'trN f

o r maybe 'trfllfvf= SUPN I%,vfl.

7.More problems. Certain problems worth being studied become apparent in discussions during the conference, first of all with 0. Martio. One of such problems is to consider the spaces defined by general singular operators rather than those defined by the Hilbert or Riesz transforms. We mention also the question the author was asked after his talk about compactness of Hausdorff operators. References 1. K.F. Andersen, Boundedness of Hausdorff operators o n Lp(Rn),H1(Rn), and B M O ( R n ) , Acta Sci. Math. (Szeged) 69 (2003), 409-418. 2. G. Brown and F. Mbricz, T h e Hausdorff and the quasi Hausdorff operators o n the spaces L p , 1 5 p < 03, Math. Inequal. Appl. 3 (2000), 105-115. 3. G. Brown and F. Mbricz, Multivariate Hausdorff operators o n the spaces Lp('R"),J. Math. Anal. Appl. 271 (2002), 443-454. 4. R. Fefferman, S o m e recent developments in Fourier analysis and H P theory and product domains. 11, Function spaces and applications, Proc. US-Swed. Semin., Lund/Swed., Lect. Notes Math. 1302 (1988), 44-51. 5. P. Galanopoulos and A.G. Siskakis, Hausdorff matrices and composition operators, Ill. J. Math. 45 (2001), 757-773. 6. C. Georgakis, T h e Hausdorff m e a n of a Fourier-Stieltjes transform, Proc. Am. Math. SOC.116 (1992), 465-471.

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7 . D.V. Giang and F. Mbricz, T h e Cesdro operator is bounded o n the Hardy space H 1 , Acta Sci. Math. 61 (1995), 535-544. 8. D.V. Giang and F. Mbricz, T h e two dimensional Ceshro operator is bounded o n the multi-parameter Hardy space N1(Rx R),Acta Sci. Math. 63 (1997), 279-288. 9. E. Liflyand and F. Mbricz, T h e Hausdorff operator is bounded o n the real Hardy space H1(W),Proc. Am. Math. SOC.128 (2000), 1391-1396. 10. E. Liflyand and F. Mbricz, T h e multi-parameter Hausdorff operator is bounded o n the product Hardy space Hl1(R x R),Analysis 21 (2001), 107-118. 11. A. Lerner and E. Liflyand, Multidimensional Hausdorff operator o n the real Hardy space, t o appear in J. Austr. Math. SOC. 12. A. Miyachi, Boundedness of the Cesdro Operator in Hardy spaces, J. Fourier Anal. Appl. 10 (2004), 83-92. 13. F. Mbricz, Multivariate Hausdorff operators o n the spaces H1(R") and B M O ( R n ) , Analysis Math. 31 (2005), 31-41. 14. A.G. Siskakis, T h e Cesdro operator is bounded o n H1, Proc. Am. Math. SOC. 110 (1990), 461-462. 15. E. M. Stein and G. Weiss, Introduction to Fourier Analysis o n Euclidean Spaces, (Princeton Univ. Press, Princeton, N.J. 1971). 16. F. Weisz, T h e boundedness of the Hausdorff operator o n multi-dimensional Hardy spaces, Analysis 24 (2004), 183-195.

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AUTHOR INDEX

Aizenberg, L. 279 Aliyev Azeroglu, T. 116, 125 Ashyralyev, A. 256 Begehr, H. 84 Benyaiche, A. 178 Bojarski, B. 66 Dittmar, B. 54 Fern6ndez Arias, A. 140 Golberg, A. 148 Israfilov, D. M. 214 Karupu, 0. W. 231 Krushkal, S. 3 Kuhnau, R. 46 Lanza de Cristoforis, M. 131 Li, P. 17 Liflyand, E. 280 Malinnikova, E. 207

Miklyukov, V. M. 33 Morosawa, S. 174 Oka, Y. 198 Ornek, B. N. 125 PBrez Alvarez, 3. 140 Plaksa, S. A. 166 Riihentaus, J. 156 Sirma, A. 256 Skiba, N. 222 Tamrazov, P. M. 116 Vasil’eva, Yu. V. 249 Yakar, C. 239 Yang, C. C. 17, 275 Yoshino, K. 191, 198 Zakharyuta, V. 222 Zelinskii, Yu. B. 145 Zhuravlev, I. V. 187