Basic knowledge of Distribution Theory, Sobolev Spaces and elliptic PDEs.

Lectures.

Oral examination.

Main properties of Banach and L^p spaces.

DISTRIBUTIONS. Definition and topology. Embeddings and convergence. Radon measures. Derivatives, translations and difference quotients. Order of a distribution. Support and distributions with compact support. The space E'. Convolutions. Tempered distributions and Fourier transform in S'. Fundamental solutions for the laplacian and the heat equation.

SOBOLEV SPACES. Definition, norms and scalar products, separability and reflexivity. Friedrich's Theorem. Chain rule and truncation. Characterization by translation. Extention by reflexion. Meyers-Serrin Theorem. Continuous Embeddings. Sobolev-Gagliardo-Nirenberg and Morrey's Theorem. Functions of class Lip, AC, UC and BV. Compact embedding: Ascoli-Arzela', Riesz-Frechet-Kolmogorov and Dunford-Pettis Theorems. Compactness in L^p, W^{1,p} and in M. Dual spaces. The space H^{-1}. Poincare' and Poincare'-Wirtinger inequalities. Traces in L^p.

ELLIPTIC EQUATIONS. Lax-Milgram Theorem. Laplacian with Dirichlet and Neumann boundary conditions. The space L^2(div). H^2 regularity for the Dirichlet problem (Niremberg). Maximum principle. (Stamapacchia).

S. Kesavan: "Topics in functional analysis and applications". John Wiley & Sons, New York, 1989.

H. Brezis: "Functional Analysis, Sobolev Spaces and Partial Differential Equations". Springer, New York, 2011.