Modern Theory of Partial Differential Equations

Course guarantor:

Matěj Tušek

Lecturer:

Matěj Tušek

Tutor:

Matěj Tušek

Supervisor:

Department of Mathematics

Synopsis:

Sobolev spaces, continuous and compact embedding theorems, trace theorem.

Elliptic PDE of Second Order, existence, uniqueness, regularity, maximum principle, harmonic functions.

Requirements:

Basic knowledge of the theory of distributions, functional analysis, and topological notions.

Syllabus of lectures:

Outline:

1. Sobolev spaces - advanced propoerties, examples.

2. Definition, completeness, examples.

3. Continuous and compact embedding theorems.

4. Trace theorem - details.

5. Weak solution (importance).

6. Elliptic PDE of Second Order.

7. Methods for existence and uniqueness of weak solutions.

8. Regularity of weak solutions.

9. Relation to the calculus of variations, Poincaré inequality.

10. Maximum principle and comparison principle for classical and weak solutions.

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] L. C. Evans: Partial Differential Equations, 2nd ed., American Mathematical Society, Rhode Island, 2010.

[2] G. Leoni: A First Course in Sobolev Spaces, AMS, 2017.

[3] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.

Recommended referemces:

[4] M. H. Protter, H. F. Weinberger: Maximum Principles in Differential Equations, Springer, New York, 1984.

[5] R. A. Adams: Sobolev Spaces, Academic Press, New York, 2003.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans: