LEARNING OUTCOMES

The aim of the course is to provide an introduction on Sobolev spaces, and to provide variational interpretation of some partial differential equations, also studying the regularity of the solutions. As an application we will provide simple results of existence of regular solutions.

AIMS AND LEARNING OUTCOMES

To give fundamental basis around weakly differentiable functions (AC functions, Sobolev functions and their characterization). To interpret weak solution to PDEs as minima of variational problems and vice-versa through Euler-Lagrange equations; to deduce existence and regularity using this principle. The double objective is to give the students fundamental tools and results in PDEs and general principles of Calculus of Variations.

Learning outcomes: Comprehension of concept and proofs shown at lecture. To be able to provide proof of results similar to those seen at lectures, to be able to come up with example and counterexamples and to solve exercises related to the course.

PREREQUISITES

Analisi matematica 1,  2 e 3, Analisi Funzionale 1. Preferrably also Analisi Funzionale 2 and Equazioni differenziali 1.

TEACHING METHODS

Lectures at the blackboard

SYLLABUS/CONTENT

Space W^{1,p} on the interval and definition of weak derivative; relations with AC functions and density of smooth functions in W^{1,p}. Generalization of W^{1,p} on R^n: equivalent definitions through integration by parts formula and closure of smooth functions, and AC characterization on lines. Localization on an open set and extension operator from an open and regular set. Compact embeddings and Poincaré-Wirtinger inequality.

Differential equations in weak form and relation to variational principles through Euler-Lagrange equations (mainly elliptic equations). Existence via Hopf-Lax. Caccioppoli estimates and Shauder H^2 regularity: bootstrap. L^p regularity and explicit formulas for Laplace and Poisson equations. Regularity in the case of non-costant coefficients (frezing). Holder regularity à la De Giorgi in the case of L^{\infty} coefficients.