# Fiches de cours

## Sobolev spaces and elliptic equations

Nguyên Hoài-Minh

English

#### Summary

This is an introductory course on "Linear Elliptic Partial Differential Equations".

#### Content

1, Harmonic functions. Mean value properties. Fundamentai solutions. Green's identities, Maximum principles. Caccioppoli's inequality.
2. Sobolev spaces. Soiobev's inequality, Poincare's inequality, Reillich-Kondrachov's inequality. Trace theorems.
3. Dirichlet problems. Existence and uniqueness of weak solutions. Lax-Milgram's theorem and compactness arguments. Maximum's principle. A connection with variational method.
4, Neumann problems. Existence and uniqueness of weak solutions. Lax-Milgram's theorem and comptactness arguments. A connection with variational method.
5. Mixed boundary problems, An example.
6. Separation of variables. Solving Laplace's equations in a ball and in a circular. Three spheres inequality.
7. Laplace equation in unbounded domains.

#### Learning Prerequisites

##### Required courses

The students are strongly recommended to have sufficiently knowledge on real analysis, theory of integrations. Having taken a functional analysis course will be an advantage.

##### Important concepts to start the course

By the end of the course, the student must be able to:
. Apply basic theory to solve several problems in sciences
. Analyze partial dilforential equations

#### Teaching methods

The course is given during the first 7 weeks with 5 hours ex-cathedra and 3 hours of exercises.

#### Assessment methods

Written exam

Dans le cas de l'art. 3 al. 5 du Réglement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand