Distribution and interpolation spaces
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
Learning Prerequisites
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 |
Resources
Bibliography
"Théorie des distributions," Laurent Schwartz.
"Analyse fonctionnelle. Théorie et applications," Haïm Brezis.
"Functional analysis, Sobolev spaces and partial differential equations," 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.
Ressources en bibliothèque
- Elliptic Partial Differential Equations of Second Order / Gilbarg
- Théorie des distributions / Schwartz
- Functional analysis, Sobolev spaces and partial differential equations / Brezis
- Cours d'analyse / Bony
- Partial differential equations / Evans
- Analyse fonctionnelle / Brezis
- An introduction to Sobolev spaces and interpolation space / Tartar
- An introduction to harmonic analysis / Katznelson
- Sobolev Spaces / Adams
Moodle Link
In the programs
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Distribution and interpolation spaces
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Distribution and interpolation spaces
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Distribution and interpolation spaces
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Distribution and interpolation spaces
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
Reference week
| Mo | Tu | We | Th | Fr | |
| 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 |
Légendes:
Lecture
Exercise, TP
Project, other