Transformations, transmutations, and kernel functions [Volume 2] 9780582091092

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Transformations, transmutations, and kernel Junctions Volume 2

Main Editors H. Brezis, Universite de Paris R.G. Douglas, State University of New York at Stony Brook A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of Cambridge A. Bensoussan, INRIA, France S. Bloch, University of Chicago B. Bollobds, University of Cambridge S. Donaldson, University of Oxford J. Douglas Jr, Purdue University R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, Universite de Paris K.P. Hadeler, Universitat Tubingen D. Jerison, Massachusetts Institute of Technology K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York at Stony Brook W.F. Lucas, Claremont Graduate School S. Mori, Kyoto University L.E. Payne, Cornell University G.F. Roach, University of Strathclyde B. Simon, California Institute of Technology S.J. Taylor, University of Virginia

Pitman Monographs and Surveys in Pure and Applied Mathematics 59

Transformations, transmutations, and kernel functions Volume 2 Heinrich Begehr Freie Universitttt Berlin and Robert P Gilbert University of Delaware

@

CRC Press Taylor & Francis Grou p Boca Raton Londo n Ne w Yor k

CRC Pres s is an imprint of th e Taylor & Franci s Group, an informa busines s A CHAPMA N & HAL L BOO K

AMS Subject Classification: (Main)35-02, 35J25, 35J40 (Subsidiary)30G30, 31B30, 33A45, 35A20, 35J55, 35K22, 35M05, 45E05 First published 1993 by Longman Scientific & Technical Published 2021 Chapman & Hall/CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1993 by Taylor & Francis Group, LLC Chapman & Hall/CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN 13: 978-0-582-09109-2 (hbk) ISSN 0269-3666 This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http ://www.taylorandfrancis.com and the CRC Press Web site at http ://www.crcpress.com British Library Cataloguing in Publication Data A catalogue record for this hook is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this hook is available

Dedicated to our children

Birgit, Astrid, Fabian and

Jennifer

Preface Volume one of this book gives some applications of complex analytic methods in plane elasticity and for boundary value problems of elliptic equations, presents some recent results on Hele Shaw flows and the transmutation theory and finally reports on the Bergman-Vekua treatments of elliptic equations with analytic coefficients. In this second volume the last four chapters are included. Chapter VI continues the development of the theory of the kernel function begun in Chapter II, considering in the first section kernel functions for higher-order systems. In section 2 the theory of bianalytic functions due to Hua Loo Keng and his students Lin Wei and Wu Ci-Quian is capsuled. The theory of first-order systems of composite type, founded by A. Dzhuraev, and apparently independently in a thesis by a Ph. D. student [Vidi69] of W. Haack, is an important subject in its own right. This material provides an interesting application of the Vekua theory of generalized analytic functions. Section 4 reports on some of the work by Kiihnau and his students on generalized Bel tram i equations, including applications to quasi conformal mappings. Section 5 includes further results on the theory of kernel functions and recent results due to Gilbert a.ncl Lin Wei concerning first-order systems with analytic coefficients. As one of the main tools of complex analytic methods in partial differential equations in the plane is the method of singular integral equations, it is natural that a section is devoted to this material. The numerical treatment of the integral equations with Cauchy kernels has become an active field due to the efforts of Profidorf, Wendland, their students, and others. In the final section of this chapter the classification problem of first-order elliptic systems in the sense of B. Bojarski [Boja66] are presented. Chapter V I I contains a detailed discussion of the envelope method for the locating of singularities of elliptic equations plus some applications of the method to physical mathematics. The work is then divided into two subareas; one is devoted to a study of the singularities of certain special classes of differential equations, and the other to the singularities of infinite series of special functions. Many of the results in one of these categories may be transformed into results in the other category, and vice versa. The Nehari theorem concerning the singularities of Legendre series and many of its generalizations by Gilbert, Howard, McCoy, Walter, and Zayed are given. Indeed, a table indicating the type of series, the generalizations with respect to Nehari's result and one of the authors may be found in this chapter. In a sense the material of Chapter VIII should be quite different from the preceding material; however, it is surprisingly similar. Again our approach is function theoretic, thai is analytic. It is shown how many representations of solutions for the meta- and pseudoparabolic equations may be obtained using analytic techniques that were primarily elliptic in nature.

The last chapter reviews of some new results in Clifford analysis, a topic of importance in mathematical physics [HesoS4], [Chco86] and quickly developing in recent years. For a basic book see [Brds82]. As for volume one the chapters for this volume had to be completely retyped because the wordprocessors in Newark and Berlin turned out to use not compatible languages. This was done by Annette Elch Miihlenfeld, nee Link, who also retyped volume one. She, together with Ute Fuchs, prepared the figures on computers and made the index. We would like to thank both of them for their excellent work as well as the staff at Longman Group UK Ltd. for their comprehension and patience. Berlin and Newark, Delaware May, 1993

Heinrich Begehr Freie Universitat Berlin

Robert P. Gilbert University of Delaware

Table of Contents VI Systems of Elliptic Equations 1. Kernel functions for higher-order systems

1 1

2. Bianalytic functions

11

3. Systems of first-order equations of composite type

28

4. Kernel functions for a complex first-order equation

56

5. Systems of first-order equations with analytic coefficients

71

6. Numerical treatment of singular integral equations

81

7. Remarks and further references

92

VII Singularities of Solutions to Elliptic Equations 1. Introduction 2. The envelope and pinching methods

95 95 104

3. The Bergman-Whittaker operator: singularities of harmonic functions 107 4. Singularities of elliptic equations in the plane

116

5. Singular partial differential equations

128

6. Solutions having distributinal boundary values

138

7. Remarks and further references

142

VIII Evolutionary Equations

IX

151

1. One space dimension

152

2. Two space dimensions

158

3. Systems

173

4. Boundary value problems for pseudoparabolic systems

184

5. More than three space variables

192

6. A hyperbolic differential equation

206

7. Remarks and further references

209

Clifford Analysis

215

1. A concise introduction to Clifford Analysis

215

2. Remarks and further references

237

Heinrich Begehr and Robert Gilbert •

References and Further Reading

241

•

Index of Names

259

•

Index of Subjects

263

VI 1.

Systems of Elliptic Equations Kernel functions for higher-order systems

In Chapter II we investigated various properties of the kernel function for second-order equations of the form &u-F(x)u = Q,

xeDcIRn

(1.1)

and actually considered, for x £ JR 2 , the case of systems where F was an (n x n) matrix and u a vector, u := (1/1, • • • , w n ) T - In this section, which is based on the Technical Report by [Cogi79], we consider L to be a linear matrix differential operator of the form L[U] = Apu + (-l)"Qu, x£DclRn, p > l , (1.2) and L* to be the formal adjoint, i.e. L*[v] = A p v + (-l) p Q*v

(1.2*)

where Q* is the conjugate transpose of Q, i.e. Q* = QT. For n = 2, the fundamental singularity for (1.2) or (1.2*) is of the form S = ap2r2p-2 log -I + s, r 2p where a£2 = -1 TFTy matrix and s is a r rrnri I is the identit J v C (D] matrix -- I)!]2 ' except at r = 0, where it is C2p~l(D).

For n > 2, the fundamental singularity is of two different types depending on whether n is even or odd, and whether n > 2p :

{ (ii)

n — odd. n = even, and n > 2p,

S = a£r 2 < p -*)logi/ + a , f o r n = 2fc,

fc 0 is scalar, the inner products (1.3) are positive definite, scalar valued, and one has a Schwarz inequality, namely {w,v}2 < {u,u}{v,v}.

(1.4)

The matrix analogue of the Gutzmer formula ([Nico36], p. 26) is , / (u*Apw - [Apu*] v) dx = JD

dD

•A'w da, (1.5)

AV

which holds for p even or odd. However, the even and odd dimensional cases are more effectively treated separately. For p = 2k, one has I u*tfkvdx JD

=

/ A*u* JD k-i

(1.6) da.

(In the above two equations da is the induced surface measure.) From (1.6) one obtains (p = 2k) {u,v} =

=

f (A VAS; + u'Qv) dx JD r T[u })*vdx n. j // (//(L* JD

(1.7)

2i-ll i

"

• u'&'v \ da.

'

Putting in a fundamental singularity of L*, 5»(P, Q) into (1.5) for u, and performing the usual residue computation, one obtains Boggio-type representations ([Nico36], p. 27) for v namely

v(Q) = dD

(1.8) •S:(P,Q}tiv(P}\

dap,

3

Systems of elliptic equations

(L8)

where p = 2k or p = 2k + 1. Doing this same computation for (1.7) one obtains for p = 2k v(Q) = {S.(P,Q),v(P)}

(1.9) In what follows let use assume Q > 0, i.e. we exclude the case of (vector) polyharmonic functions, which will be treated later. For Q > 0 (strictly positive definite) Conlan and Gilbert [Cogi79] define the Green and Neumann matrices G (P, Q), N (P, Q) to be those fundamental singularities of L [u] = 0 which satisfy respectively the boundary conditions:

G(P,Q) = (L10)

and

= ^—A 2A: ~ 1 A r (P,Q) = 0,

(PedD).

In an analogous manner we may define G f *(P,Q) and N*(P,Q). This definition of the Green matrix is similar to the Green functions; see, for example, Vekua [Veku67], p. 183 and [Agmo65], p. 90. Let ft be the class of C2p(D(JdD), and ft0 C ft the subspace of functions satisfying on aD Finally, let E C ft be the class of strong solutions of L [u] = 0 in D. Putting S* = N* into the representation (1.9) yields v (Q) = {N+, v} for all v G ft ,

(1.13)

and replacing 5* by G* yields v (Q) = {G,, v} for all v G 0° .

(1.14)

4

Heinrich Begehr and Robert Gilbert

On the other hand, (1.9) leads to EE.

(1.15)

The Boggio representation yields the following two surface integral representations for solutions to the Neumann and Green problems respectively: **•-*•

f

= V / A'-'-'«;(p,

72(0 7 C-2 C ',

if

0 < "2 ,

if n2 < 0.

(2.42) From Re {z-n2(fn2(9) (z)} = 0 on dJD , = Re 2 > cikZ

Re

if 0 < n 2 , on dJD ,

if n 2 < 0,

A:=0

where

it follows that Re {z~

- Re

Re {z~

= Re

, if 0