Distribution and interpolation spaces

MATH-502 / 5 credits

Teacher: Michelat Alexis Paul Benjamin

Language: English

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

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

Moodle Link

In the programs

  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Distribution and interpolation spaces
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Distribution and interpolation spaces
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Distribution and interpolation spaces
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional

Reference week

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