Basic knowledge of Distribution Theory, Sobolev Spaces and elliptic PDEs. Solution of exercises on elliptic PDEs.
Course Prerequisites
In general topics in Functional Analysis ( Hilbert and Banach spaces, (weak topology and dual space, L^p spaces) and Measure Theory (positive and real measures, absolute continuity of function and measures). Fundamental results are always recovered whenever needed.
Teaching Methods
Lectures and problem solving sessions, strongly taylored together. Lectures aims at presenting and explaining fundamental concepts and results, accompained with reference examples. Problem solving sessions aims at developing abilities in both reasoning and performing computations.
Assessment Methods
The exam consists of an oral examination which requires a good knowledge of all the topics (definition and theorems, with proofs). In general it is required to answer a question for each of the macro-subjects of the course. Moreover, it is required to provide a rigorous formulation and the solution of an elliptic PDE.
Texts
BASIC REFERENCES. 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. ADVANCED REFERENCES F. Treves: "Topological Vector Spaces, Distributions and Kernels". Academic Press, New York, 1967
Contents
FUNCTIONAL SPACES. Dual spaces and Reisz-Markov representation theorems. Finite and locally finite Radon measures. The metric space L^1_loc. Weak compactness and weak convergence. Continuous and compact embeddings. DISTRIBUTIONS. Definition and topology. Embeddings and convergence. Derivatives, translations and difference quotients. Order of a distribution. Radon measures. Principal Value distribution. Support and distributions with compact support. The space E'. Convolutions. Fundamental solutions for the laplacian in R^n. 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 and Korn inequality.
Course Language
Italian
More information
Students can download notes and podcasts of the lectures, moreover consulting office hours will be available also online.
