Modern Theory of Partial Differential Equations

Garant předmětu:

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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:

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.

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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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: