OVERVIEW

The course will introduce the students to Sobolev spaces and then we will investigate variational aspects of elliptic differential equations.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide an introduction on Sobolev spaces, and to provide variational interpretation of some partial differential equations, also studying the regularity of the solutions. As an application we will provide simple results of existence of regular solutions.

AIMS AND LEARNING OUTCOMES

To give fundamental basis around weakly differentiable functions (AC functions, Sobolev functions and their characterization). To interpret weak solution to PDEs as minima of variational problems and vice-versa through Euler-Lagrange equations; to deduce existence and regularity using this principle. The double objective is to give the students fundamental tools and results in PDEs and general principles of Calculus of Variations.

Learning outcomes: Comprehension of concept and proofs shown at lecture. To be able to provide proof of results similar to those seen at lectures, to be able to come up with example and counterexamples and to solve exercises related to the course.

PREREQUISITES

Analisi matematica 1,  2 e 3, Analisi Funzionale 1. Preferrably also Analisi Funzionale 2 and Equazioni differenziali 1.

TEACHING METHODS

Lectures at the blackboard

SYLLABUS/CONTENT

Space W^{1,p} on the interval and definition of weak derivative; density of smooth functions in W^{1,p}. Generalization of W^{1,p} on R^n: equivalent definitions through integration by parts formula and closure of smooth functions (possibly AC characterization on lines). Localization on an open set and extension operator from an open and regular set. Compact embeddings and Poincaré-Wirtinger inequality. Coarea formula, Isoperimetric inequality, Sobolev inequality.

Differential equations in weak form and relation to variational principles through Euler-Lagrange equations (mainly elliptic equations). Existence via Hopf-Lax. Caccioppoli estimates and Schauder H^2 regularity: bootstrap. (possibly L^p regularity and explicit formulas for Laplace and Poisson equations. Regularity in the case of non-costant regular coefficients (frezing)). Outline of Holder regularity à la De Giorgi in the case of L^{\infty} coefficients.

RECOMMENDED READING/BIBLIOGRAPHY

H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations

L. Ambrosio, A. Massaccesi, A. Carlotto - Lectures on Elliptic Partial Differential Equations

TEACHERS AND EXAM BOARD

Exam Board

SIMONE DI MARINO (President)

MATTEO SANTACESARIA

FLAVIANA IURLANO (President Substitute)

ANDREA BRUNO CARBONARO (Substitute)

FILIPPO DE MARI CASARETO DAL VERME (Substitute)

EDOARDO MAININI (Substitute)

LESSONS

LESSONS START

February 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam. At the request of the students part of the exam could be done with a mini-seminar about something close to the material of the course.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

ASSESSMENT METHODS

In the exam some aspect on the theory will be considered, as well as some exercises related to the course. This allows to test the knowledge and the grasp of the course by the students, as well as their capacity in applying theoretical results in order to solve exercises.

At the request of the students part of the exam could be done with a mini-seminar about something close to the material of the course.

Exam schedule