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; ~ Lectures on ELLIPTIC BOUNDARY VALUE PROBLEMS by SHMUEL AGMON Professor of Mathematics The Hebrew University of Jerusalem ~ ... • ...• '.:.II!I Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. D. VAN NOSTRAND COMPANY, INC. TORONTO PRINCETON, NEW JERSEY NEW YORK Univer~itv ('f HI~"l.;:"''Tfon LONDON Dtirary Marsh R I!:e C(i1.IVtr'S/f~ fI()l.Isl-~YlI 1ex4sSummer Irlsfriu te.. '~r 'Ad vtlf"1C!.~d Graduate :S1i-1de;;;-f $) Ie; 68,] r..W/{/IOI'YI D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey (Principal office) 24 West 40 Street, New York 18, New York 012 PREFACE D. VAN NOSTRAND COMPANY, LTD. 358, Kensington High Street, London, W.14, England D. VAN NOSTRAND COMPANY (Canada), LTD. 25 Hollinger Road, Toronto 16, Canada © COPYRIGHT 1965, BY D. VAN NOSTRAND COMPANY, D. INC. Published simultaneously in Canada by VAN NOSTRAND COMPANY (Canada), LTD. No reproduction in any form of this book, in whole or in part (except for brief quotation in critical articles or reviews), may be made without written authorization from the publishers. ,. I I r I '.!. .' J.' .. \ . ,,, '1..,.' V -/ ' " \.,,' ·\I 1 } 1•• , / This book reproduces with few corrections notes of lectures given at the Summer Institute for Advanced Graduate Students held at the William Rice University from July 1, 1963, to August 24, 1963. The Summer Institute was sponsored by the National Science Foundation and was directed by Professor Jim Douglas, Jr., of Rice University. The subject matter of these lectures is elliptic boundary valued problems. In recent years considerable advances have been made in .. developing a general theory for such problems. It is the purpose of these lectures to present some selected topics of this theory. We . consider elliptic problems only in the framework of the L 2 theory. This approach is particularly simple and elegant. The hard core of the theory is certain fundamental L 2 differential inequalities. The discussion of most topics, with the exception of that of eigen value problems, follows more or less along well-known lines. The' treatment of eigenvalue problems is perhaps less standard and differs iD some important details from that given in the literature. This approach yiel ds a very general form of the theorem on the asymptotic distribution of eigenvalues of elliptic operators. Only a few references are given throughout the text. The literature . on elliptic differential equations is very extensive. A comprehensive bibliography on elliptic and other differential problems is to be found in the book by J. L. Lions, Equations diffe'rentielles ope'rationelles, ~• . Springer-Verlag, 1961. 'li:i' These lectures were prepared for publication by Professor B. Frank ~ Jones, Jr., with the assistance of Dr. George W.Batten, Jr. I am " greatly indebted to them both. Professor Jones also took upon him \ self the trouble of inserting explanatory and complementary material t'-. in several places. I am particularly grateful to him. I would also ~ like to thank Professor Jim Douglas for his active interest in the \) publication of these lectures. ~ PRINTED IN THE 'UNITED STATES OF AMERICA' ~ Jerusalem Shmuel Agmon ;c ~ CONTENTS 1'0 Section 00, rq;u"'dx = (-l)iexI U ID"'uI2dx]'~ J uD"'q;dx, lal·~ m. U Taking this last relation on its own