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Lectures in APPLIED MATHEMATICS Volume 3A Partial Differential Equations American Mathematical Society LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957 Volume I Probability and Related Topics in Physical Sciences BY MARK KAC with G. E. UHLENBF.CK, A. R. HIBBS, and BALTH. VAN DER POL Volume II Fluid Mechanics By SYDNEY GOT DSTE.TN with J. M. BURGERS Volume III Partial Differential Equations By LIPMAN BERs, FRITL JOHN, and MARTIN SCHECHTER with LARS GARDING and A. N. MILGRAM Volume IV Solid Mechanics By R. S. RIVI.IN with W. PRAGER LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957 AMERICAN MATHEMATICAL SOCIETY EDITORIAL COMMITTEE ALSTON S. HOUSEHOLDER, Chairman Oak Ridge National Laboratory MARK KAC Rockefeller University H. J. GREENBERG International Business Machines Corporation VOLUME III PARTIAL DIFFERENTIAL EQUATIONS By LIPMAN BERS FRITZ JOHN MARTIN SCHECHTER Courant Institute of Mathematical Sciences, New York University WITH SPECIAL LECTURES BY Lars Girding University of Lund, Lund, Sweden and the late A. N. Milgram University of Minnesota. AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 Copyright Q 1964 by John Wiley & Sons, Inc. All Rights Reserved This book or any part thereof may not be reproduced in any form without the written permission of the publisher. SECOND PRINTING, OCTOBER, 1966 BY JOHN WILEY & SONS, INC. THIRD PRINTING, 1971 BY THE AMERICAN MATHEMATICAL SOCIETY International Standard Book Number 0-8218-0047-3 Library of Congress Catalog Card Number 63-19664 Printed in the United States of America ARTHUR N ORTON MILGRAM June 3, 1912-January 30, 1961 Arthur Norton Milgram died suddenly on January 30, 1961. He was a penetrating mathematician and an inspiring teacher. The surviving authors, who were his friends, dedicate this volume to his memory. Foreword This is the third volume of the Proceedings of the Summer Seminar on Applied Mathematics, sponsored by the American Mathematical Society and held at the University of Colorado over the four weeks beginning June 23, 1957 The purpose of the Seminar was tutorial, to present to mature mathematicians the current status of the theory in several fields covered, and to pose for them some of the more pressing and interesting mathematical problems lying open in these fields. It could be described as an endeavor to promote cooperation of mathematicians with theoretical physicists. The publication of these volumes is intended to extend the same information to a much wider public than was privi- leged to actually attend the Seminar itself, while at the same time serving as a permanent reference for those who did attend. The program of the Seminar was organized by a committee of the American Mathematical Society with the following membership: P. R. Garabedian A. S. Householder Mark Kac R. E. Langer C. C. Lin Wm. Prager J. J. Stoker M. H. Martin, Chairman Local arrangements, including the social and recreational program, were organized by a committee of the Department of Applied Mathematics, University of Colorado, as follows: J. R. Britton R. Ben Kriegh L. W. Rutland L. C. Snively K. H. Stahl C. A. Hutchinson, Chairman The indefatigable energy and enthusiasm of the chairmen, and the cooperation of other members of the university staff, contributed viii FOREWORD immeasurably to the successful execution of the plans for the seminar and to the enjoyment of the particip