# ADVANCED ANALYSIS 2

## Learning outcomes of the course unit

The course aims at providing a first introductory and yet significant knowledge on the problems and the (mainly abstract) methods of PDE's.

The course aims at providing a first introductory and yet significant knowledge on the problems and the methods of the Calculus of Variations.

## Prerequisites

The contents of mandatory courses of a standard italian course of Mathematics

Basic knowledge of Analysis and Measure Theory, and Sobolev Spaces

## Course contents summary

Aim is first to give a panoramic about the PDE's (=partial differential equations), second to study in a more detailed way II order hyperbolic equations with abstract methods.

The course is focused on some classical topics of the Calculus of Variations. It is divided in two parts. The first one deals with a thorough study of firts and second order necessary and sufficient minimality conditions for one-dimensional problems. Among the various examples, a complete treatment of the brachistochrone problem will be given. The second part is devoted to the DIrect Methods of the Calculus of Variations: after proving a general existence theorem for integral functionals in N-dimensions, we will expose De Giorgi's regularity theory.

## Course contents

A detailed program in format docx. or .pdf may be asked via email to alberto.arosio@unipr.it - as well as any clarification about this course

## Recommended readings

No specific reference textbooks. The topics of the course are chosen by the instructor and may be found in several different classical and modern books on the topic.

No specific reference textbooks. The topics of the course are chosen by the instructor and may be found in several different classical books on the Calculus of Variations.

## Teaching methods

Frontal lesson, by means of slides (= transparencies) and traditional blackboard. Motivations, applications, examples and counterexamples will be provided. If possible, Heuristics will have space.

The course will consist in lectures delivered at the blackboard. The topics will be complemented with motivations, instructive exercises and applications. Emphasis will be placed on the mathematical rigor of the presentation, which will be as much detailed and self-contained as possible.

## Assessment methods and criteria

The final exam will consist in an oral exam, aiming to evaluate the learning

ability of the scholar and the quality and rigor of the exposition.

The final exam will consist in an oral interview, aiming at evaluating the learning ability of the student and the quality and rigor of his/her exposition.