Regularity Problem for Quasilinear Elliptic and Parabolic Systems
The smoothness of solutions for quasilinear systems is one of the most important problems in modern mathematical physics. This book deals with regular or strong solutions for general quasilinear second-order elliptic and parabolic systems. Applications in
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Alexander Koshelev
Regularity Problem for Quasilinear Elliptic and Parabolic Systems
Springer
Author Alexander Koshelev Faculty of Mathematics and Mechanics St. Petersburg University Bibliotechnaja 2 198904 St. Petersburg, Russia
Cataloging-in-Publication Data.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Koselev, Aleksandr I.: Regularity problem for quasilinear elliptic and parabolic systems / Alexander Koshelev. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Tokyo: Springer, 1995 (Lecture notes in mathematics; 1614) ISBN 3-540-60251-8 NE:GT
Mathematics Subject Classification (1991): 35DlO, 35K55
ISBN 3-540-60251-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479536 46/3142-543210 - Printed on acid-free paper
Contents Preface
vii
Introduction
viii
List of Notation
xix
Chapter 1 Weak solutions and the universal iterative process 1 1.1 Quasilinear elliptic and parabolic systems and systems with bounded nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1.2 Universal iterative process and weak solutions; non degenerate systems with bounded nonlinearities 5 1.3 Degenerating elliptic systems with bounded nonlinearities . 12 16 1.4 Regularization of the universal iterative process . . . . . Chapter 2 Regularity of solutions for non degenerated quasilinear second order elliptic systems of the divergent form with bounded nonlinearities 2.1 Some functional spaces and preliminary results. . . . . . . . 2.2 Estimates for ordinary differential operators . . . . . . . . . 2.3 Holder continuity for weak solutions of the nondegenerated elliptic second order systems with bounded nonlinearities in interior domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Holder continuity in the entire domain . . . . . . . . . . . . . . . 2.5 The sharpness of the regularity conditions for solutions of second order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3 Some properties and applications of regular solutions for quasilinear ellliptic systems 3.1 The Liouville theorem 3.2 The Korn inequality in weighted spaces . . . . . . . . . . 3.3 Holder continuity of displacements for elasto-plastic media wi
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