# ADVANCED ANALYSIS 1

## Learning outcomes of the course unit

At the end of the course students should know the basic theory of Sobolev spaces with respect to the Lebesgue measure, of compact operators in normed spaces, and of the variational formulation of elliptic PDEs.

Through exercises solved in the classroom students should understand how to apply his/her theoretic knowledges to solve explicit problems.

Students should be able to evaluate the correctness of the results obtained by himself/herself or by other people.

Students should be able to communicate in a clear and precise way the mathematical contents of the course. Lectures in the classroom and discussion with the teacher will help to be able to use the appropriate scientific language.

Students will be able to deepen their knowledge on the subjects of the course, starting from the basic knowledge given by the course itself. They will be able to consult autonomously specialized monographs, even on related subjects not directly treated in the lectures.

## Prerequisites

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

Basic theory of linear functional analysis.

## Course contents summary

The course gives an overview of Sobolev spaces and on compact operators, with applications to the study of elliptic, second order, linear partial differential equations with boundary conditions.

## Course contents

An overview on the L^p spaces and on convolutions.

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

Some characterization of the Sobolev spaces W^{k,p}, and some of their properties: embeddings, product of two Sobolev functions, superposition, local invariance under diffeomorphisms. Traces of Sobolev functions. The spaces W^{1,p}_0 and Poincaré inequalities.

Variational formulation of elliptic boundary value problems. Lax-Milgram lemma. Regularity of weak solutions. Boundary value problems for elliptic equations in L^p-spaces and in spaces of bounded and Holder continuous functions (hints). Compact operators. Spectrum and resolvent set of linear operators. The spectral theorem for compact self-adjoint linear operators in Hilbert spaces. Applications to elliptic differential operators.

## 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.

L.C. Evans, R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press 1992

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

## Teaching methods

Online lectures held through the Teams platform. A part of the teaching hours will be devoted to the discussion of exercises.

## 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.