MATH-502 / 5 crédits

Enseignant: Michelat Alexis Paul Benjamin

Langue: Anglais

Remark: Cours donné en alternance tous les deux ans

## Summary

The goal of this course is to give an introduction to the theory of distributions and cover the fundamental results of Sobolev spaces including fractional spaces that appear in the interpolation theory. Those notions are central to the study of partial differential equations (PDE).

## Content

Part 1: Toplogy and functional spaces. Fundamental theorems on Banach spaces, weak topology, weak * topology, reflexive spaces, separable spaces.

Part 2: Distributions. Topological vector spaces, distributions: differentiation, restriction, localisation, convolution, tempered distributions and Fourier transform.

Part 3: Sobolev spaces. Extension operators, Sobolev embedding theorem, Sobolev inequality, Poincaré inequality, dual Sobolev space, Hilbert-Sobolev spaces, fractional derivatives, fractional Sobolev spaces.

## Keywords

Distributions, Sobolev Spaces, Interpolation Spaces

## Required courses

• MATH-200: Analysis III
• MATH-205: Analysis IV
• MATH-303: Measure and integration

## Recommended courses

• MATH-302: Functional analysis I

## Learning Outcomes

By the end of the course, the student must be able to:

• Demonstrate proficiency in statements
• Identify use and role of the assumptions
• Recognize which concepts and results could be used in a given context
• Describe concepts and proofs
• Apply theory to specific examples

## Teaching methods

Lectures + Exercises

## Assessment methods

Oral

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.

## Supervision

 Assistants Yes

## Bibliography

"Théorie des distributions," Laurent Schwartz.

"Analyse fonctionnelle. Théorie et applications," Haïm Brezis.

"Cours d'analyse. Théorie des distributions et analyse de Fourier," Jean-Michel Bony.

"Sobolev Spaces," Robert A. Adams and John J. F. Fournier.

"Elliptic Partial Differential Equations of Second Order," David Gilbarg and Neil S. Trudinger.

"Partial differential equations," Lawrence C. Evans.

"An introduction to Sobolev spaces and interpolation spaces," Luc Tartar.

"An introduction to harmonic analysis," Yitzhak Katznelson.

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Distribution and interpolation spaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Distribution and interpolation spaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Distribution and interpolation spaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

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