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

Lectures.

Oral exam

Main properties of Banach (weak topology and dual spaces) and L^p spaces.

FUNCTIONAL SPACES. Dual spaces and Reisz representation theorems. Finite and locally finite Radon measures. Inductive limit topology. Weak compactness and weak convergence.

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

SOBOLEV SPACES. Definition, norms and scalar products, separability and reflexivity. Friedrich's Theorem. Chain rule and truncation. Characterization by translation. Extensions. Meyers-Serrin Theorem. Continuous Embeddings: Sobolev-Gagliardo-Nirenberg and Morrey Theorem. Lipschitz and absolutely continuous functions. Compact embedding. Dual spaces. The space H^{-1}. Poincarè and Poincarè-Wirtinger inequalities. Traces in L^p. Green's formulas.

ELLIPTIC EQUATIONS. Lax-Milgram Theorem. Elliptic equation with bounded coefficients with Dirichlet, Neumann and mixed boundary conditions. The space L^2(div). H^2 regularity for the Dirichlet problem (Niremberg). Maximum principle (Stamapacchia). Eigenvalues of the Laplacian. Linearized elasticity.

H. Brezis: "Functional Analysis, Sobolev Spaces and Partial Differential Equations". Springer, New York, 2011.
L.C. Evans: "Partial Differential Equations", Americal Mathematical Society, Providence, 1998.
G. Leoni: "A First Course in Sobolev Spaces". Americal Mathematical Society, Providence, 2009.
F. Treves: "Topological Vector Spaces, Distributions and Kernels". Academic Press, New York, 1967