# Functional Analysis II Spring 2020

Lecturer
Michael Struwe
Course Assistant
Francesco Palmurella
Teaching Assistants
Federico Franceschini
Peter Wildemann

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).

• Lectures from the V week on are replaced with recordings; on the Topics section precise referencing is given for every lecture.
• Exercise Classes are replaced with an online forum.

See below for more. We will mark in red all the changes induced by the new situation.

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.

If you want to read more on elliptic equations and related topics, here are some friendly monographs (out of a huge literature).
• L. Ambrosio, A. Carlotto and A. Massaccesi, Lectures on elliptic Partial Differential Equations. Edizioni della Normale, 2018
• A. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. Second Edition, Edizioni della Normale, 2012
• E. Giusti, Direct Methods in the Calculus of Variations. World Scientific, 2003
• N. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces. AMS, 1996
• F. Lin, Q. Han, Elliptic Partial Differential Equations. Second Edition. AMS, 2011

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.

WeekContentReferences
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
• An exercise sheet is uploaded every Thursday. The corresponding solution appears one week after the due date.
• You have the possibility to hand in your solution to all or only to some of the exercises that are in the exercise sheets to your teaching assistant and have it graded and commented. You can send your assistant your work per email It can be either typed or scanned/photograped, as long as it is clearly readable. In the same way the assistant will answer you back.
• Handing in, even at different time intervals, is a good way to test your preparation and understanding of the subject. It will not contribute to the final grade.
• If you have troubles solving a particular exercise, do not understand the solution or have any other question, do not refrain from asking or mailing your teching assistant or the course assistant.
• If you spot any typo or you think there may be some mistake, do not hesitate to contact the course assistant. Except for small typos, all changes made after the initial release are marked in blue.

At the following link, an ➡ Online Forum is available as replacement for the exercise classes.

• As for the classes, (online) attendance is not compulsory but recommended. Register using your student mail account and look for the Functional Analysis II thread.

20.02.2020 Sheet 1Solution 02.03.2020
27.02.2020 Sheet 2Solution 09.03.2020 Added remark and updated solution to 2.3
5.03.2020 Sheet 3Solution 16.03.2020
12.03.2020 Sheet 4Solution 23.03.2020 Corrected 4.3
19.03.2020 Sheet 5Solution 30.03.2020
26.03.2020 Sheet 6Solution 06.04.2020
02.04.2020 Sheet 7Solution 13.04.2020
23.04.2020 Sheet 8Solution 04.05.2020 Added more explanation to 8.3 (ii)
30.04.2020 Sheet 9Solution 11.05.2020
7.05.2020 Sheet 10Solution 18.05.2020
14.05.2020 Sheet 11Solution 25.05.2020
22.05.2020 Sheet 12Solution 01.06.2020
02.06.2020 Sheet 13Solution Multiple choice questions for self-test