Edmunds D.E., Evans W.D. Elliptic Differential Operators and Spectral Analysis

Springer, 2018. — 324 p. — (Springer Monographs in Mathematics). — ISBN: 978-3-030-02124-5.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.Preliminaries
The Laplace Operator
Second-Order Elliptic Equations
The Classical Dirichlet Problem for Second-Order Elliptic Operators
Elliptic Operators of Arbitrary Order
Operators and Quadratic Forms in Hilbert Space
Realisations of Second-Order Linear Elliptic Operators
The Lp Approach to the Laplace Operator
The p-Laplacian
The Rellich Inequality
More Properties of Sobolev Embeddings
The Dirac Operator