Description:

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

Contents:

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.

Recommended literature:

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.

Keywords:

partial differential equations, Sobolev spaces, elliptic regularity, maximum principle.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar