# Fiches de cours

## Introduction to partial differential equations

Nobile Fabio

English

#### Summary

This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.

#### Content

• Laplace equation; mean value property; maximum principle; fundamental solution; Dirichlet problem; Poisson integral and Newtonian potential; regularity theory in Holder spaces;
• General second order linear elliptic equations; maximum principle; a priori bounds;
• Sobolev spaces; weak derivatives and their properties; density results; extension results; traces; imbedding theorems; Poincaré inequalities;
• Weak solutions of general elliptic equations;  Lax Milgram theorem; existence and uniqueness results; regularity theory in Sobolev spaces; compactness results and non coercive problems;

#### Learning Prerequisites

Analysis I-IV

##### Recommended courses

Measure and Integration; Functional Analysis I

#### Learning Outcomes

By the end of the course, the student must be able to:
• Classify different types of PDEs
• Define different notions of solutions
• Analyze the properties of solutions of PDEs
• Prove existence and regularity results of solutions of elliptic PDEs

#### Transversal skills

• Use a work methodology appropriate to the task.
• Demonstrate a capacity for creativity.
• Demonstrate the capacity for critical thinking

#### Teaching methods

Ex cathedra lectures, exercises in classroom

#### Assessment methods

Oral exam

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

#### Resources

No

##### Bibliography

• David Gilbarg, Niel S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2nd edition, 2001.
• Lawrence C. Evans. Partial Differential Equations, AMS-Graduate Studies in Mathematics, 2nd edition, 2010.
• Haïm Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011
• Fritz John, Partial Differential Equations, Springer-Verlag, 4th edition, 1982

#### Prerequisite for

• Master courses on theory of PDEs: Equations aux dérivées partielles d'évolution, Calculus of variations, Optimal Transport, Dispersive PDEs, Théorie du calcul stochastique;
• Bachelor / Master courses on numerical approximation of PDEs: Numerical Approximation of PDEs; Numerical methods for conservation laws; Computational finance; Numerical integration of stochastic differential equations.

### Dans les plans d'études

• Semestre
Automne
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Introduction to partial differential equations
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

### Semaine de référence

LuMaMeJeVe
8-9 MAB111
9-10
10-11 MAB111
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

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