Functional Spaces for the Theory of Elliptic Partial Differential Equations, 2012 Universitext Series

The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions.

This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem.

The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space.

There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.

Preliminaries on ellipticity.- Notions from Topology and Functional Analysis.- Sobolev Spaces and Embedding Theorems.- Traces of Functions on Sobolev Spaces.- Fractional Sobolev Spaces.- Elliptic PDE: Variational Techniques.- Distributions with measures as derivatives.- Korn's Inequality in L^p.- Appendix on Regularity.