ARIATIONAL PRINCIPLES AND FREE-BOUNDARY PROB LEMS
AVNER FRIEDMAN
AVNER FRIEDMAN
VARIATIONAL PRINCIPLES AND FREE-BOUNDARY PROBLEMS
AVNER FRiEDMAN
4 Nørthwrium Uk*y
A PUBLICATION
JOHN WILEY & SONS
Mew York Cbichester
AVNER FRiEDMAN
4 Nørthwrium Uk*y
A PUBLICATION
JOHN WILEY & SONS
Mew York Cbichester
Copyright © 1982 by Jobn Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of part of this work beyond that permitted by Section 107 or M of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley &. Sons, inc.
01 Congress Cataloging in Pubilcatine Data
Friedman, Avner. Variational principles and free-boundary problems.
(Pure and applied mathematics. ISSN 0079-8185) “A Wiley-Interscience publication.” Includes bibliographical references and index. I. Roundaiy-value problems. 2. Variational
1. Title. II. Pure and applied mathematics (John Wiley & Sons)
QA379.F74 1982 51 5.3’5 82-8654 ISBN G.471-86849-3 AACR2
Printed in the United States of America JO 9 8 7 6 5 4 3 2 1
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of part of this work beyond that permitted by Section 107 or M of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley &. Sons, inc.
01 Congress Cataloging in Pubilcatine Data
Friedman, Avner. Variational principles and free-boundary problems.
(Pure and applied mathematics. ISSN 0079-8185) “A Wiley-Interscience publication.” Includes bibliographical references and index. I. Roundaiy-value problems. 2. Variational
1. Title. II. Pure and applied mathematics (John Wiley & Sons)
QA379.F74 1982 51 5.3’5 82-8654 ISBN G.471-86849-3 AACR2
Printed in the United States of America JO 9 8 7 6 5 4 3 2 1
PREFACE
Important developments in the study of free-boundary problems have been achieved in recent years by introducing variatiorial approach wherever possi- ble. This enables one to conclude without great efforts that a solution to the free-boundary problem exists in some “weak” sense. One can proceed to establish the regularity of the solution and then, hopefully, study the smooth- ness of the free boundary itself. In fact, within the last five years significant new methods have been developed for analyzIng the free boundary and the theory has now reached a certain stage of maturity; its future looL even more exciting. An increasing number of physical and engineering problems are becoming accessible to this growing body of methods.
It is therefore an appropriate time to discuss the main developments in the field in a systematic and self-contained manner. Since some of the main developments were motivated by physical models, we have kept close contact between the general theory and applications to physical examples.
In order to make the book more readable by graduate students afld nonspecialists, we have included in the text the detailed slatanents of the standard theory of elliptic and parabolic operators that is being used. This is done more systematically in the first two chapters.
We have problems at the end of each section and remarks at the of each chapter.
I would like to thank Luis A. Caffarelli, Hans Wilhelm Alt, and Joel Spruck for several useful conversations, and Ms. Leslie Hubbell for an excellent job of typing the manuscript.
Ecøjtston. Illinois June 1W
Important developments in the study of free-boundary problems have been achieved in recent years by introducing variatiorial approach wherever possi- ble. This enables one to conclude without great efforts that a solution to the free-boundary problem exists in some “weak” sense. One can proceed to establish the regularity of the solution and then, hopefully, study the smooth- ness of the free boundary itself. In fact, within the last five years significant new methods have been developed for analyzIng the free boundary and the theory has now reached a certain stage of maturity; its future looL even more exciting. An increasing number of physical and engineering problems are becoming accessible to this growing body of methods.
It is therefore an appropriate time to discuss the main developments in the field in a systematic and self-contained manner. Since some of the main developments were motivated by physical models, we have kept close contact between the general theory and applications to physical examples.
In order to make the book more readable by graduate students afld nonspecialists, we have included in the text the detailed slatanents of the standard theory of elliptic and parabolic operators that is being used. This is done more systematically in the first two chapters.
We have problems at the end of each section and remarks at the of each chapter.
I would like to thank Luis A. Caffarelli, Hans Wilhelm Alt, and Joel Spruck for several useful conversations, and Ms. Leslie Hubbell for an excellent job of typing the manuscript.
Ecøjtston. Illinois June 1W
Variational principles and free-boundary problems.