Distributions, Sobolev Spaces, Elliptic Equations
Many problems in mathematical physics reduce to elliptic differential equations of second order and their boundary value problems, and also to the spectral theory of such operators. Central role are played by the Dirichlet problem, the Neumann problem and the eigenvalue and eigenfunction problem. The main aim of the book is to develop the L2 theory of the listed problems on bounded domains with smooth boundaries in Euclidean space. This is done in a reader-friendly way at a moderate level of difficulty, aiming at graduate students and their teachers.
Among the topics covered, we find the classical theory of the Laplace-Poisson equation, the theory of distributions, the theory of Sobolev spaces on domains, abstract spectral theory in Hilbert and Banach spaces and compact embeddings. One of the principal assets of the book is a very good, friendly and accessible introduction to various aspects of function space theory and their applications in the theory of partial differential equations. The book is richly furnished with interesting exercises and a thorough set of notes is attached to each chapter. This is an ideal book for both students and teachers of a modern graduate course on partial differential equations.