Elliptic Differential Operators and Spectral Analysis

Description

This book deals with elliptic differential equations, providing the analytic background necessary for the treatment of associated spectral questions, and covering important topics previously scattered throughout the literature. Starting with the basics of elliptic operators and their naturally associated function spaces, the authors then proceed to cover various related topics of current and continuing importance. Particular attention is given to the characterisation of self-adjoint extensions of symmetric operators acting in a Hilbert space and, for elliptic operators, the realisation of such extensions in terms of boundary conditions. A good deal of material not previously available in book form, such as the treatment of the Schauder estimates, is included. Requiring only basic knowledge of measure theory and functional analysis, the book is accessible to graduate students and will be of interest to all researchers in partial differential equations. The reader will value its self-contained, thorough and unified presentation of the modern theory of elliptic operators. Front Matter ....Pages i-xiii Preliminaries (David E. Edmunds, W. Desmond Evans)....Pages 1-33 The Laplace Operator (David E. Edmunds, W. Desmond Evans)....Pages 35-56 Second-Order Elliptic Equations (David E. Edmunds, W. Desmond Evans)....Pages 57-63 The Classical Dirichlet Problem for Second-Order Elliptic Operators (David E. Edmunds, W. Desmond Evans)....Pages 65-81 Elliptic Operators of Arbitrary Order (David E. Edmunds, W. Desmond Evans)....Pages 83-113 Operators and Quadratic Forms in Hilbert Space (David E. Edmunds, W. Desmond Evans)....Pages 115-158 Realisations of Second-Order Linear Elliptic Operators (David E. Edmunds, W. Desmond Evans)....Pages 159-202 The \(L_{p}\) Approach to the Laplace Operator (David E. Edmunds, W. Desmond Evans)....Pages 203-211 The \(p-\)Laplacian (David E. Edmunds, W. Desmond Evans)....Pages 213-233 The Rellich Inequality (David E. Edmunds, W. Desmond Evans)....Pages 235-247 More Properties of Sobolev Embeddings (David E. Edmunds, W. Desmond Evans)....Pages 249-280 The Dirac Operator (David E. Edmunds, W. Desmond Evans)....Pages 281-301 Back Matter ....Pages 303-322