Modern Theory of Partial Differential Equations
Code  Completion  Credits  Range 

01PDRMI  ZK  3  3P+0C 
 Garant předmětu:
 Lecturer:
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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.
 Syllabus of tutorials:
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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 timetable has been prepared for this course
 The course is a part of the following study plans: