The distribution of the eigenvalues of differential operators has long fascinated mathematicians. Advances have shed light upon classical problems in this area, and this book presents a fresh approach, largely based upon the results of the authors. The emphasis here is on a topic of central importance in analysis, namely the relationship between i) function spaces on Euclidean n-space and on domains; ii) entropy numbers in quasi-Banach spaces; and iii) the distribution of the eigenvalues of degenerate elliptic (pseudo) differential operators. The treatment is largely self-contained and accessible to non-specialists. Both experts and newcomers alike will welcome this unique exposition.

This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and makes applications to such questions. After the exposition of the abstract theory in the first four chapters, Sobolev spaces are introduced and their main properties established. The remaining seven chapters are largely concerned with second-order elliptic differential operators and related boundary-value problems. Particular attention is paid to the spectrum of the Schrödinger operator. Its original form contains material of lasting importance that is relatively unaffected by advances in the theory since 1987, when the book was first published. The present edition differs from the old by virtue of the correction of minor errors and improvements of various proofs. In addition, it contains Notes at the ends of most chapters, intended to give the reader some idea of recent developments together with additional references that enable more detailed accounts to be accessed.

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Let m and n be positive integers with n 2 and 1 m n&1. We study rearrangement-invariant quasinorms * R and * D on functions f: (0, 1) Ä R such that to each bounded domain 0 in R n , with Lebesgue measure |0|, there corresponds C=C( |0| )>0 for which one has the Sobolev imbedding inequality, involving the nonincreasing rearrangements of u and a certain m th order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which * D need not be rearrangementinvariant,In both cases we are especially interested in when the quasinorms are optimal, in the sense that * R cannot be replaced by an essentially larger quasinorm and * D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Bre zis, and Wainger.

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