# 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 both in L^p spaces (p in (1,+oo)) and in spaces of bounded and Holder continuous functions.

## Prerequisites

Calculus for functions of several variables. Linear algebra. Topology. Measure theory and integration.

Linear functional analysis.

## Course contents summary

Weak derivatives and Sobolev spaces.

Weak solutions of boundary value elliptic problems. Compact operators.

## Course contents

An overview on the L^p spaces.

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

Some characterization of the Sobolev spaces W^{k,p}. Traces of Sobolev functions. Variational formulation of elliptic boundary value problems. Lax-Milgram lemma. Regularity of weak solutions. Compact operators.

Boundary value problems for elliptic equations in L^p-spaces and in spaces of bounded and Holder continuous functions.

## 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.

Notes by the teacher.

## Teaching methods

Lectures in the classroom.

## Assessment methods and criteria

The examination consists of an oral test which is aimed at evaluating the knowledge of the results seen during the course, their proofs and the skills in using such results to solve simple problems in the fields of the course.