## Advanced Partial Differential Equations, Winter 2017

Place and Time
 Type: Time: Place: Start: Lecture (VO) 3 hrs. 9:45-11:1513:30-14:14 HS02SR11 3.10. Proseminar 1 hr. 12:30-13:15 SR11 4.10.
This course gives an introduction to the functional analytic treatment of partial differential equations.
Proseminar
The following problems should be prepared:
• Problem Set 1 (due 11.10): [Teschl] 10.18, 10.19, 10.21, 10.23
• Problem Set 2 (due 18.10): [Teschl] 10.24, 13.1, 13.3, 13.5
• Problem Set 3 (due 25.10): [Teschl] 13.7, 13.8, 13.9 [Evans] 5.10 Problem 3
• Problem Set 4 (due 8.11): [Evans] 5.10 Problem 4, 7, 8 (Instructions for 4: The absolutely continuous functions are the antiderivatives of integrable functions (see Theorem 11.49 in my notes). Now have a look at the hint for Problem 13.2 in my notes.)
• Problem Set 5 (due 15.11): [Teschl] 13.13, 13.16
• Problem Set 6 (due 22.11): [Teschl] 13.17 [Evans] 5.10 Problem 20, 21
• Problem Set 7 (due 29.11): [Evans] 6.6 Problem 2, 4 (Hint for 4: Find a solution of the homogenous problem)
• Problem Set 8 (due 6.12): [Evans] 6.6 Problem 5, 6
• Problem Set 9 (due 6.12): [Teschl] 7.3, 7.4, 7.5
• Problem Set 10 (due 10.1): [Teschl] 7.6, 7.7, 7.8, 7.9
• Problem Set 11 (due 17.1): [Teschl] 7.10, 7.11, 7.12 Please download the latest version of the notes!
• Problem Set 12 (due 24.1): [Evans] 7.6 Problem 15, 8.7 Problems 2,3
• Problem Set 13 (due 31.1): [Evans] 8.7 Problem 1 [Teschl] Present the proof of Theorem 5.2 (geometric Hahn-Banach) and Corollary 5.4; show how to obtain Lemma 4.36
Content
Approximation in Lp and mollification [Teschl, Section 10.4]. Sobolev spaces [Teschl, Chapter 13], [Evans, Section 5.1-5.8]. Second-order elliptic PDE [Evans, Section 6.1-6.3.1]. Operator semigroups [Teschl, Chapter 7], [Evans, Section 7.4]. Calculus of Variations [Evans, Sections 8.1.1-8.2.3] To be continued....
Target audience
Module "Advanced Partial Differential Equations" in the Master's programme in Mathematics
Assessment
The course assessment for the lecture (VO) will be via an oral examination at the end of the course. The course assessment for the introductory seminar (PS) will be via participation (solving/presenting assigned problems) during the seminar.
Literature

Some textbooks

1. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
2. L.C. Evans, Partial Differential Equations, 2nd ed., Amer. Mat. Soc., 2010.
3. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.
4. G. Grubb, Distributions and Operators, Springer, New York, 2009.
5. G. Teschl, Topics in Real and Functional Analysis, Amer. Math. Soc., Providence, to appear.
Looking forward to seeing you, Gerald Teschl