Modern Theory of Partial Differential Equations
Code  Completion  Credits  Range  Language 

01PDR  ZK  2  2+0  Czech 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

Sobolev spaces, continuous and compact embedding theorems, trace theorem.
Elliptic PDE of Second Order, LaxMilgram theorem, regularity, maximum principle, harmonic functions.
 Requirements:

Basic knowledge of theory of distributions and functional analysis.
 Syllabus of lectures:

1. Sobolev spaces
1.1 Definition, completeness, examples
1.2 Continuous and compact embedding theorems
1.3 Trace theorem
2. Weak solution (importance, derivation of the weak formulation)
3. Elliptic PDE of Second Order
3.1 Existence and uniqueness of weak solutions (LaxMilgram theorem)
3.2 Regularity of weak solutions
3.3 Relation to the calculus of variations, Poincaré inequality
3.4 Maximum principle for classical and weak solutions
 Syllabus of tutorials:
 Study Objective:

Acquired knowledge: fundamental facts about Sobolev spaces; weak solution and its importance; theorems on existence, uniqueness, and regularity of weak solutions of partial differential equations (PDE) of the second order; maximum principle
Acquired skills: derivation of the weak formulation, understanding the relation to the classical theory, to get ready for selfstudy of other important cases (such as evolution equations)
 Study materials:

Compulsory literature:
[1] Evans L.C.: Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
Optional literature:
[2] Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations, Springer, 1984.
[3] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, Springer, 2001 (reprint).
[4] Adams R.A.: Sobolev Spaces, Academic Press, 1975.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Matematické inženýrství (elective course)