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.

Further Reading

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.

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

• 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.

• There you can ask questions or clarifications, discuss with the assistants and see which other questions other students are asking. In the topic "Welcome! Please read this first" is more information.
• 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.

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

The exam is oral (30 min), covering the content of the course and of the exercise sheets.