Modern theory of partial differential equations

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

1. Sobolev spaces.

2. Definition, completeness, examples.

3. Continuous and compact embedding theorems.

4. Trace theorem.

5. Weak solution (importance, derivation of the weak formulation).

6. Elliptic PDE of Second Order.

7. Existence and uniqueness of weak solutions (Lax-Milgram theorem).

8. Regularity of weak solutions.

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

10. Maximum principle for classical and weak solutions.

Requirements:

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

Syllabus of lectures:

1. Sobolev spaces.

2. Definition, completeness, examples.

3. Continuous and compact embedding theorems.

4. Trace theorem.

5. Weak solution (importance, derivation of the weak formulation).

6. Elliptic PDE of Second Order.

7. Existence and uniqueness of weak solutions (Lax-Milgram theorem).

8. Regularity of weak solutions.

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

10. Maximum principle for classical and weak solutions.

Syllabus of tutorials:

Study Objective:

Study materials:

Povinná literatura

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

Doporučená literatura

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

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

4. R. A. Adams: Sobolev Spaces, Academic Press, New York, 2003.

5. G. Leoni: A first course in Sobolev spaces, AMS, 2017.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství (compulsory elective course)