Generalized Cauchy-Riemann Systems with a Singular Point PDF
Preview Generalized Cauchy-Riemann Systems with a Singular Point
Pitman Monographs and Surveys in Pure and Applied Mathematics 85 t Generalized Cauchy-Riemann systems with a singular point Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board H. Amann, University of Zurich R. Aris, University of Minnesota G.I. Barenblatt, University of Cambridge H. Begehr, Freie Universitat Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta RP. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York at Stony Brook B. Moodie, University of Alberta S. Mori, Kyoto University L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University Pitman Monographs and Surveys in Pure and Applied Mathematics 85 Generalized Cauchy-Riemann systems with a singular point ZDUsmanov Institute ofM athematics, Tajik Academy ofS ciences Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK First published 1997 by Addison Wesley Longman Limited Published 2019 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1997 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN 13: 978-0-582-29280-2 (hbk) 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. 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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 The right ofZ afar D Usmanov to be identified as author oft his work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. AMS Subject Classifica1ions: (Main) 35J, 35C, 45P (Subsidiary) 308, 45E, 53C ISSN 0269-3666 British Library Cataloguin1 in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Usmanov, Zafar Dzuraevich. Generalized Cauchy-Riemann systems with a singular point/ Zafar Dzuracvich Usmanov. p. cm. - (Pitman monographs and surveys in pure and applied mathematics, ISSN 0269-3666 ; ???) ISBN 0-582-29280-8 (alk. paper) 1. CR submanifolds. 2. Singularities (Mathematics) I. Title. II. Series. QA649.U86 1997 516.3'62-DC20 96-31177 CIP Contents Introduction 1 Chapter 1 Interrelation between sets of general and model equation solutions 6 1 The method of constructing a general integral operator 6 2 Properties of the functions fii and ÎÎ2 9 3 Properties of a general operator 14 4 The general integral equation 19 5 Additions to Chapter 1 21 6 Unsolved problems 21 Chapter 2 The model equation 23 1 Basic kernels and elementary solutions of the conjugate equation 23 2 Cauchy generalized formula 25 3 Sequences of continuous solutions for the model equation 27 4 ${z) representation by some series. Analogy of the Liouville theorem. Uniqueness theorem 27 5 Regularity of solutions at a singular point 31 6 Analogy of Laurent series 34 7 Generalized integral of Cauchy type 35 8 Cases of a one-to-one correspondence between the sets {*(*)} and MC)} 37 9 Riemann-Hilbert problem for solutions of the model equation 39 10 Conjugation problem for solutions of the model equation 53 11 Additions to Chapter 2 57 Chapter 3 The general equation 60 1 Behaviour of solutions at a singular point 60 2 Solutions bounded on the plane 65 3 The canonical form of the Riemann-Hilbert problem 70 4 The Riemann-Hilbert problem with a zero index 72 5 The Riemann-Hilbert problem with a positive index 78 CONTENTS 6 The Riemann-Hilbert problem with a negative index 80 7 The conjugation problem 84 8 The method of constructing additional operators 87 9 Unsolved problems 90 Chapter 4 Modified generalized Cauchy—Rie man n systems with a singular point 91 1 The general integral equation 91 2 Elements of the theory of the model equation 95 3 Boundary value problems for solutions of the model equation 100 4 The general equation 103 Chapter 5 Generalized Cauchy-Riemann system with the order of the singularity at a point strictly greater than 1 104 1 Constructing general solutions of an inhomogeneous model equation 104 2 Properties of the functions fii and i^2 109 3 Properties of an operator 118 4 Relation between solutions of general and model equations 121 5 Investigation of the model equation 123 6 Boundary value problems for solutions of the model equation 134 7 Properties of the solutions of the general equation at a singular point 144 8 The Riemann-Hilbert problem for solutions of the general equation 145 9 The conjugation problem 155 10 Unsolved problems 159 Chapter 6 Infinitesimal bendings of surfaces of positive curvature with a flat point 161 1 Equations of infinitesimal bendings of a surface with positive curvature 162 2 Conjugate isometric system of coordinates on the model surface 163 3 Conjugate isometric system of coordinates on the general surface 166 4 Equations of infinitesimal bendings of the general surface 171 5 Geometric significance of boundary conditions 175 6 A general expression of infinitesimal bendings for the model surface 176 7 Surfaces of types So and S\ 178 8 Properties of infinitesimal bendings of class Cp for surfaces of type S\ 180 9 Properties of smallness in deformations of class Cp for the general surface 181 10 Representation for 6L,8M,8N variations through component ((x,y) of a bending field 183 11 Determination of deformation classes for the general surface in conjugate isometric coordinates 185 CONTENTS 12 General results for the model surface 187 13 Study of an analytic problem for the general surface 190 14 The influence of a flat point on infinitesimal bendings of the surface with boundary conditions 196 15 Surfaces with a more complicated structure in a neighbourhood of a flat point 198 16 Additions to Chapter 6 204 17 Unsolved problems 206 Supplement Generalized Cauchy-Riemann systems with a singular line 208 References 214 Preface In this monograph a theory of generalized Cauchy-Riemann systems with polar singularities of order not less than 1 is presented and its application to the study of infinitesimal bendings of positive curvature surfaces with an isolated flat point is given here. It contains results of investigations obtained recently by the author and his collaborators. In this monograph special attention is paid to the description of formal methods of constructing general integral operators which are a natural extension of the classical apparatus of generalized analytic function theory. The monograph is written not only for specialists in complex analysis, geometry and mechanics but also for student-mathematicians who could use it as a manual. Professor Zafar D. Usmanov Institute of Mathematics of the Tajik Academy of Sciences ul. Aini 299, Academgorodok Dushanbe 734 063 TAJIKISTAN
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