<p>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.</p>.

<p>Linear and non-linear elliptic boundary problems are a fundamental subject in analysis and the spaces of weakly differentiable functions (also called Sobolev spaces) are an essential tool for analysing the regularity of its solutions.</p><p> </p><p>The complete theory of Sobolev spaces is covered whilst also explaining how abstract convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. Other kinds of functional spaces are also included, useful for treating variational problems such as the minimal surface problem.</p><p> </p><p>Almost every result comes with a complete and detailed proof. In some cases, more than one proof is provided in order to highlight different aspects of the result. A range of exercises of varying levels of difficulty concludes each chapter with hints to solutions for many of them.</p><p> </p><p>It is hoped that this book will 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 Schwartz spaces. </p><p>