Due to the latest developments in the spread of the coronavirus, lectures and exercise classes are no longer taking place (here is ETH's dedicated page).
Michael Struwe, Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.
Addendum:
different proof for Satz 8.4.2.
Lawrence C. Evans, Partial Differential Equations. Second Edition. Graduate Studies in Mathematics. AMS, 2010.
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, 2011.
David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, 2001.
Online recordings of the lectures from the V week on are posted at the ➡ ETH Video Portal. Use your ETH username and password to access.
Week | Content | References |
---|---|---|
I (17.02 & 20.02) | Laplace's equation, Mean value property of harmonic functions, Maximum principle, Green's function | Evans § 2.2 |
II (24.02 & 27.02) | Sobolev spaces: motivation | Struwe § 7.1-2 Brezis § 8.1-2 |
III (02.03 & 05.03) | Sobolev spaces on an interval | Struwe § 7.2-3 Brezis § 8.2-3 |
IV (09.03 & 12.03) | Sobolev spaces on an interval (continued) and one-dimensional boundary value problems: Theorems 7.3.1, 7.3.2, 7.3.3, 7.3.5, Corollaries 7.3.1-2, Sections 7.4.1-2. |
Struwe § 7.3-4 Brezis § 8.2-4 |
V (16.03 & 19.03) | Sobolev spaces on an interval (continued) and one-dimensional boundary value problems: Variants of the Neumann problem, the Laplace operator on an interval. Self-adjointness of the closure of the Laplace-operator with homogeneous Dirichlet boundary conditions, notion of capacity |
Struwe § 7.4-5, 8.1 |
VI (23.03 & 26.03) | Capacity, Mayers-Serrin approximation theorem.
Further properties of Sobolev functions: Lipschitz functions and \(W^{1,\infty}\), chain rule, change of variables |
Struwe § 8.1-2-3 |
VII (30.03 & 02.04) | Extension theorems for regular domains, approximation with smooth functions, traces and their properties | Struwe § 8.4 |
VIII (20.04 & 23.04) | Sobolev embedding case \(p < n \), compact
embeddings, the limiting case \(p = n \). Hölder and Campanato spaces |
Struwe § 8.5-6 |
IX (27.04 & 30.04) | Poincaré's inequalities, Sobolev embedding case \(p > n \),
Morrey's Dirichlet Growth Theorem,
a.e. differentiability of Sobolev functions.
General Sobolev embeddings |
Struwe § 8.6, 9.1 |
X (04.05 & 07.05) | \(L^2\)–regularity theory (interior and at the boundary) for the Laplace operator | Struwe § 9.1-2-3 |
XI (11.05 & 14.05) | \(L^2\)–regularity theory for the Laplace operator (continued) | Struwe § 9.3 |
XII (18.05 & 21.05) | \(L^2\)–regularity theory for general elliptic operator in divergence form; applications | Struwe § 9.4-5-6 |
XIII (25.05 & 28.05) | Schauder theory: Campanato estimates,
Campanato spaces, a priori estimates in Höolder spaces, existence of solutions via continuity method |
Struwe § 10 |
At the following link, an ➡ Online Forum is available as replacement for the exercise classes.
Upload date | Exercise Sheet | Due date | Comments |
---|---|---|---|
20.02.2020 | Sheet 1 – Solution | 02.03.2020 | – |
27.02.2020 | Sheet 2 – Solution | 09.03.2020 | Added remark and updated solution to 2.3 |
5.03.2020 | Sheet 3 – Solution | 16.03.2020 | – |
12.03.2020 | Sheet 4 – Solution | 23.03.2020 | Corrected 4.3 |
19.03.2020 | Sheet 5 – Solution | 30.03.2020 | – |
26.03.2020 | Sheet 6 – Solution | 06.04.2020 | – |
02.04.2020 | Sheet 7 – Solution | 13.04.2020 | – |
23.04.2020 | Sheet 8 – Solution | 04.05.2020 | Added more explanation to 8.3 (ii) |
30.04.2020 | Sheet 9 – Solution | 11.05.2020 | – |
7.05.2020 | Sheet 10 – Solution | 18.05.2020 | – |
14.05.2020 | Sheet 11 – Solution | 25.05.2020 | – |
22.05.2020 | Sheet 12 – Solution | 01.06.2020 | – |
02.06.2020 | Sheet 13 – Solution | – | Multiple choice questions for self-test |
– | Extra sheet | – | Facultative exercises. More information inside |