# ADVANCED ANALYSIS 1

## Learning outcomes of the course unit

The course gives an overview of Sobolev spaces with applications to the study of elliptic, second order, linear partial differential equations.

## Prerequisites

Calculus and advanced calculus. Linear algebra. Topology. Measure theory and integration.

Linear functional analysis.

## Course contents summary

1. Sobolev spaces.

Distributional and weak derivatives. Sobolev spaces W^{k,p}.

Sobolev spaces as Banach spaces. Duals of Sobolev spaces.

Approximation of Sobolev functions. The space W_0^{k.p}.

Calculus for Sobolev functions. Differential quotient of Sobolev functions.

Sobolev functions of one variable.

Extension theorems and traces.

Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Morrey inequality.

Embeddings of higher order Sobolev spaces.

Weak compactness and Rellich-Kondrachov theorem. Poincare' inequalities.

2. Spectral theory of compact, selfadjoint operators.

Compact operators. Fredholm theory. Spectral representation of compact, selfadjoint operators

3. Second order linear elliptic PDEs: L^2 theory.

Weak, strong and classical solutions. Dirichlet problems. Variational formulation.

Lax-Milgram theorem. Existence and uniqueness of weak solutions. H^2 regularity of weak solutions.

## Recommended readings

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Verlag 2011.

L.C. Evans, Partial differential equations, 2nd Edition, American Mathematical Society 2010.

D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, 2nd Edition, Springer Verlag 1983.

## Teaching methods

Lectures

## Assessment methods and criteria

The exam consists of an oral part which is aimed at evaluating the knowledge of the abstract results seen during the course, their proofs and the skills in using such results.