2022/2023

2022/2023

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (01/03/2023 - 09/06/2023)

6

56 lesson hours

Italian

ORAL TEST

NEGRI MATTEO (titolare) - 6 ECTS

Properties of Banach spaces (weak topology and dual space) and L^p spaces from the course on Functional Analysis.

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

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.

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.

Lectures.

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

The exam consists of an oral examination which requires a good knowledge of all the course topics (definition and theorems, with proofs) and the solution of an elliptic PDE.