# VARIATIONAL MEYHODS IN ANALYSIS

## Learning outcomes of the course unit

The course aims at introducing to some of the fundamentals topics and techniques of the Calculus of Variations and to their applications to other branches of Mathematical Analysis. Classes will be aimed at stimulating a deeper understanding of the ideas and methods of mathematical thinking.

The main objectives are to allow the student to:

1) acquire a solid knowledge of the fundamental concepts of the Calculus of Variations and Elliptic Partial Differential Equations and develop an understanding of the language, techniques and contents of a wide spectrum of modern mathematical theories;

2) understand advanced texts and research articles in Mathematics;

3) to apply the theoretical advanced tools learned during the course to solve problems and to analyze new mathematical models; ; develop that propensity to research required to undertake a possible PhD path;

4) evaluate the consistency and correctness of mathematical demonstrations and reasoning; analyze and propose appropriate resolution strategies to solve a given problem;

5) improve their ability to communicate their knowledge in a clear, precise and formally rigorous way;

6) develop a flexible mentality that allows her/him to easily adapt to face new problems.

## Prerequisites

Measure Theory and basic Functional Analysis.

## Course contents summary

-Direct Methods in the Calculus of Variations and their application to Partial Differential Equations (PDE).

-Introduction to Regularity Theory: Campanato's approach, De Giorgi-Nash Theorem and their application to the study of the regularity of minimizers of some integral functionals of the Calculus of Variations.

-Introduction to Critical Point Theory: the Mountain Pass Theorem and its applications to PDE.

-Introduction to Gamma-convergence, examples and applications.

-Evolutions problems:the Minimizing Movements Method.

Complements (as time permits):

- Vectorial Calculus of Variations.

-Examples of functionals defined on spaces of discontinuous functions.

## Recommended readings

-"Partial Differential Equations" by L. C. Evans

-"An introduction to Gamma-convergence" byi G. Dal Maso

-"Gamma-convergence for beginners" by A. Braides

## Teaching methods

Lectures at the blackboard

## Assessment methods and criteria

Oral examination.