Functional Spaces for the Theory of Elliptic Partial Differential Equations 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. Korns Inequality in Lp Appendix on Regularity. 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. 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. 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. 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.