Modern Theory of Partial Differential Equations

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Department of Mathematics

Synopsis:

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

Elliptic PDE of Second Order, Lax-Milgram theorem, regularity, maximum principle, harmonic functions.

Requirements:

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

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 (Lax-Milgram 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 self-study 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.

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No time-table has been prepared for this course

The course is a part of the following study plans: