# Functional Analysis II Spring 2021

Lecturer
Prof. Dr. Alessandro Carlotto
Course Assistant
Riccardo Caniato
Teaching Assistants
Filippo Gaia, Bian Wu
Lectures
Mon10-12 / HG G 5 - Livestream
Thu14-16/ HG G 5 - Livestream
Exercise classes
Mon9-10
Office hours
Mon16-17.30
First lecture
22.02.2021
Course Catalogue
401-3462-00L Functional Analysis II

### Prerequisites

Deep understanding of the topics covered in the course Functional Analysis I and a solid background in measure theory, Lebesgue integration and $$L^p$$ spaces.

### Content

Sobolev spaces; weak solutions of elliptic boundary value problems; basic results in elliptic regularity theory (including Schauder estimates); maximum principles.

## Literature

### Primary references

Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2019/20.

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018.

### Extra references

David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.

Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003.

## Diary of the lectures

The live streaming of the lectures is available here. After each lecture, the recording is published here.

For further information, check the following information about lecture recording.

Date Content Notes References Extras
1 22.02. Introduction of the course. A model problem: the elastic membrane with fixed boundary. The Euler-Lagrange equation associated to a functional. A general roadmap to elliptic regularity. Notes - L01
2 25.02. Distributional and weak derivatives, examples and basic facts. Definition of Sobolev spaces. The Poincaré inequality. Completeness, separability and reflexivity. Notes - L02

Struwe: section 7.2

Brezis: section 9.1
3 01.03. The notion of weak solution for elliptic problems with Dirichlet boundary conditions and an existence result via Riesz theorem.Absolute continuity of functions in $$W^{1,p}(I)$$, weak vs. pointwise derivative. Notes - L03

Struwe: section 7.1

Brezis: section 8.1 and 8.3
Absolute continuity of functions in $$W^{1,p}(I)$$, weak vs. pointwise derivative.

Struwe: section 7.3 till Satz 7.3.2

Brezis: section 8.2 till Proposition 8.3
4 04.03. Three equivalent characterisations of $$W^{1,p}(I)$$ for $$p>1$$. The Sobolev extension operator. Density of test functions in $$W^{1,p}(I)$$ for $$1\leq p<\infty$$, special cases and related comments. Notes - L04

Struwe: section 7.3 till Satz 7.3.5

Brezis: section 8.2 till Theorem 8.7
5 08.03. The one-dimensional Sobolev embedding theorem. The "undergrad student dream" corollary. The product rule and $$W^{1,p}(I)$$ as a Banach algebra. Notes - L05

Struwe: section 7.3 (last part)

Brezis: section 8.2 (last part)
6 11.03. Discussion of some examples of second-order ODEs with either Dirichlet or Neumann boundary conditions. Notes - L06

Struwe: section 7.4

Brezis: section 8.4
Sobolev functions of N variables: A criterion for the coincidence of pointwise and weak derivative (null capacity).

Struwe: section 8.1

Brezis: section 9.1
7 15.03. Examples of singular functions in $$W^{1,p}$$. The Meyers-Serrin approximation theorem, comments on the boundary behaviour. Equivalent characterisations of $$W^{1,p}$$ for $$1 < p\le\infty$$. Notes - L07

Struwe: section 8.2 and 8.3 till Korollar 8.3.1

Brezis: section 9.1 till Proposition 9.4
8 18.03. Lipschitz versus $$W^{1,\infty}$$. Calculus rules for Sobolev functions: sums, products and compositions (chain rule). Notes - L08

Struwe: section 8.3 (last part)

Brezis: section 9.1 (last part)
9 22.03. Extension operators for $$W^{1,p}(\Omega)$$ for $$\Omega$$ a relatively compact domain of class $$C^1$$. Two approximation theorems for $$W^{1,p}$$ functions, on bounded and unbounded domains. Notes - L09

Struwe: section 8.4 till Satz 8.4.2

Brezis: section 9.2
10 25.03. Imposing boundary values for elliptic problems: trace operators and their properties. The canonical splitting of $$H^1(\Omega)$$; two equivalent characterisations of $$H^1_0(\Omega)$$. Notes - L10

Struwe section 8.4 (last part)

Brezis: complements to chapter 9
11 29.03. A panorama on the Sobolev embedding theorems. The Sobolev-Gagliardo-Nirenberg inequality, link with the isoperimetric inequality in $$\R^n$$. The Sobolev embedding theorem for $$p< n$$. Notes - L11

Struwe: section 8.6

Brezis: section 9.3

Isoperimetric inequality

Co-area formula
12 01.04. The easy Sobolev embedding theorems for $$W^{1,n}(\Omega)$$, comments on $$BMO(\Omega)$$ and exponential integrability. Notes - L12 Struwe: section 8.6.1
Review on spaces of Hölder-continuous functions, completeness; the embedding $$C^{0,\alpha}\subset C^{0,\beta}$$ is compact for $$\alpha>\beta$$.
13 12.04. Campanato spaces and integral characterisation of Hölder continuity. The Poincaré-Wirtinger inequality. The Sobolev embedding for $$p>n$$ and associated compactness results. Notes - L13

Struwe: section 8.6.2

Brezis: section 9.3
Domains of type A
14 15.04. Pointwise differentiability of functions in $$W^{1,p}$$ for $$p>n$$. Notes - L14 Struwe: section 8.6.3 Nowhere differentiable Sobolev functions
Higher-order Sobolev embedding theorems and their applications to regularity of weak solutions. Struwe: section 8.6.4
15 19.04. Interior regularity for solutions of the Poisson equation: $$H^1$$ and formal $$H^2$$ estimates, and rigorous counterpart via Nirenberg's method (difference quotients). Notes - L15

Struwe: section 9.1 and section 9.2

Brezis: section 9.5 and section 9.6
16 22.04. Higher Sobolev estimates for weak solutions of the Poisson equation via an inductive scheme. Notes - L16

Struwe: section 9.2

Brezis: section 9.6
The general notions of ellipticity for operators in divergence form, and corresponding $$H^{k+2}$$ interior estimates.

Struwe: section 9.4.4

Brezis: section 9.5 and section 9.6
17 26.04. Sobolev estimates and boundary regularity for weak solutions of the Poisson equation on a half-space. The case of curved boundary: flattening via diffeomorphisms and the modified equation. Notes - L17

Struwe: section 9.3

Brezis: section 9.6
18 29.04. Transformation of functionals and operators under diffeomorphisms, the Laplace-Beltrami operator. Sobolev estimates and boundary regularity for weak solutions of elliptic equations. Notes - L18

Struwe: section 9.4.1 and section 9.4.2

Brezis: section 9.6
19 03.05. Global $$H^{k+2}$$ estimates for weak solutions of elliptic partial differential equations. Notes - L19

Struwe: section 9.4.3 and section 9.4.4

Brezis: section 9.6
A min-max characterization for the Dirichlet eigevalues of the laplacian
The Dirichlet spectrum of the Laplace operator and the min-max characterization of its eigenvalues.

Struwe: section 9.5

Brezis: section 9.8
20 06.05. The Weyl law for the Laplacian. Can one hear the shape of a drum? Notes - L20 Notes in the extras Weyl law for the Laplacian
Schauder theory: basic heuristics and motivations. Struwe: section 10.1
21 10.05. Campanato estimates for solutions of homogeneous elliptic problems with constant coefficients (both in the interior and in the boundary case). Notes - L21

Struwe: section 10.2 till Lemma 10.2.1

22 17.05. Estimates for solutions of inhomogeneous problems (with constant coefficients). Notes - L22 Struwe: section 10.2 (last part)
Morrey spaces and their equivalence to Campanato spaces for $$\nu< n$$. Struwe: section 10.3
23 20.05. Local $$C^{2,\alpha}$$ Schauder estimates (interior and boundary cases). Global $$C^{2,\alpha}$$ Schauder estimates. Related results: elliptic problems on compact Riemannian manifolds; operators in non-divergence form. Notes - L23 Struwe: section 10.4
24 27.05. Solvability in $$C^{2,\alpha}$$ for elliptic boundary value problems. The case of the Laplacian and the method of continuity. Notes - L24 Struwe: Section 10.5
25 31.05. The weak maximum principle for elliptic operators and applications to the inhomogeneous case. Connections with the maximum modulus principle and the Schwarz Lemma for holomorphic functions. Notes - L25 Notes in the extras Weak maximum principle for elliptic operators
26 03.06. The strong maximum principle via the Hopf boundary point lemma. Elliptic barriers. The method of sub- and super-solutions, examples of geometric relevance. Notes - L26 Notes in the extras Strong maximum principle for elliptic operators

## Final exam

The final assessment will be an oral exam lasting 30 minutes. The rules for the exam are available here. Some useful advice to prepare for the exam can be found here. If you wish to self-check your preparation, here you can find some sample questions. The dates and the modality for the final exam will be provided as soon as possible on this page.

## Forum

In order to easily interact, we set up a forum for our course at the link Functional Analysis II (Spring 2021) - Forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc. Use it!

## Exercise classes

Please register and enroll for a teaching assistant in myStudies. The enrollment is needed to attend the exercise class and to hand in your homework. Due to the current measures concerning the undergoing COVID-19 pandemic, all the exercise classes are held online. Below you find the link to the Zoom meetings of each exercise class (accessible with the password we sent you by email).

Assistant Online room
Filippo Gaia ETH Zoom
Bian Wu ETH Zoom

Here is the diary of the exercise classes. The online exercise classes are recorded, the videos are accessible at the links below (with the password that we sent you by email) and the notes are available in polybox - Functional Analysis II (with the same password).

Date Content Recordings
1 22.02. General information about the exercise classes. Review on the direct method of calculus of variations. Examples of applications (in particular, discussion about the fourth problem given in the winter session exam). Filippo Gaia / Bian Wu
2 01.03. Review about distributional derivatives, weak derivatives and Sobolev spaces. Some examples involving the computation of distributional derivatives. Filippo Gaia / Bian Wu
3 08.03. A review on harmonic functions: mean value property, Liouville theorem and Harnack inequality. Filippo Gaia / Bian Wu
4 15.03. Some existence and regularity results for second order linear ODEs in divergence form. The Cantor function: pointwise derivative a.e. vs distributional derivative. Filippo Gaia / Bian Wu
5 22.03. Discussion of some exercises given in the online quiz. In particular, solution of problems 4.8, 4.9, 4.12 and 4.15. Filippo Gaia / Bian Wu
6 29.03. Non-existence of trace operators in $$L^p$$ (problem 5.4). The weak gradient of the positive and the negative part of a Sobolev function (problem 5.6). Filippo Gaia / Bian Wu
7 12.04. Review on Sobolev embedding theorems. Non-compactness of the embedding $$W^{1,p}(\mathbb{R}^n)\hookrightarrow L^p(\mathbb{R}^n)$$, for any $$p\in[1,+\infty]$$ (problem 6.4). Compactness of the embedding $$W_0^{1,p}(\Omega)\hookrightarrow L^p(\Omega)$$, for every open set $$\Omega\subset\mathbb{R}^n$$ with finite measure having $$C^1$$ boundary and for any $$p\in(1,n]$$ (problem 6.5). Filippo Gaia / Bian Wu
8 19.04. Relation between $$H_0^1(\Omega)$$ and functions in $$H^1(\mathbb{R}^n)$$ vanishing on $$\mathbb{R}^n\smallsetminus\Omega$$, depending on the regularity of $$\partial\Omega$$ (problem 6.2). Existence of nowhere differentiable functions in $$W^{1,p}(\Omega)$$, for $$n\ge 2$$ and $$p\in[1,n]$$. Filippo Gaia / Bian Wu
9 26.04. Review about higher order Sobolev embedding theorems and discussion about the case $$W^{n,1}\hookrightarrow L^{\infty}$$, through the solution of problem 7.4. Different Poincarè inequalities, comments on problem 7.5. Filippo Gaia / Bian Wu
10 03.05. Rieview of some boundary regularity results for uniformly elliptic operators and discussion of problems 9.2 and 9.3. Filippo Gaia / Bian Wu
11 10.05. Uniform ellipticity and monotonicity of the eigenvalues of the Laplacian with respect to the domain (discussion about problems 10.4 and 10.5). Explicit computation of the Dirichlet spectrum of the Laplacian on rectangles (problem 10.10). Filippo Gaia / Bian Wu
12 17.05. Wrap up session about Schauder estimates and discussion about problem 11.1. Filippo Gaia / Bian Wu
13 31.05. Discussion about elliptic regularity up to the boundary on cubes (problem 12.1), extensions of Hölder continuous functions (problem 12.10) and regularity for solutions of the minimal surface equation (problem 12.13). Filippo Gaia / Bian Wu

## Problem sets

Every Thursday, at 4pm, a new problem set is uploaded here. You have seven days to solve the problems and hand in your solutions via the platform SAMUpTool (the precise deadline is the following Thursday, no later than 8pm). Your work will be carefully graded and given back to you after a few days. During exercise classes on Monday some of the problems are discussed. Hints for all problems of any given problem set will be posted on Monday evenings.

Every problem is marked by one of the following symbols.

Computation
Get your hands dirty and calculate.
Bookkeeping
Apply what you learn in basic situations.
Comprehension
Construct examples and give full proofs.
Previous exam
Exercise given in an old exam.
Hard problem
Challenging problems are denoted by one up to three diamonds. It is recommended that you start working on these problems only after you have reviewed the weekly material and carefully solved all other exercises in the assignment.
Assignment dateDue dateProblem setSolution
Thu 25.02. Thu 04.03.Problem set 1 - HintsSolutions 1
Thu 04.03. Thu 11.03.Problem set 2 - HintsSolutions 2
Thu 11.03. Thu 18.03.Problem set 3 - HintsSolutions 3
Thu 18.03. Thu 25.03.Problem set 4 - Online quizSolutions 4
Thu 25.03. Thu 01.04.Problem set 5 - HintsSolutions 5
Thu 01.04. Thu 15.04.Problem set 6 - HintsSolutions 6
Thu 15.04. Thu 22.04.Problem set 7 - HintsSolutions 7
Thu 22.04. Thu 29.04.Problem set 8 - Online quizSolutions 8
Thu 29.04. Thu 06.05.Problem set 9 - HintsSolutions 9
Thu 06.05. Thu 13.05.Problem set 10 - Online quizSolutions 10
Thu 13.05. Thu 27.05.Problem set 11 - HintsSolutions 11
Thu 27.05. Thu 03.06.Problem set 12 - Online quizSolutions 12 - Extra

## Office hours

You are free to come and ask questions. The office hours are held via Zoom. The schedule is as follows (up to possible short-term changes, please check for updates).

DateTimeLocationAssistant
Mon 01.03. 16-17.30 ETH Zoom Riccardo Caniato
Mon 08.03. 16-17.30 ETH Zoom Filippo Gaia
Mon 15.03. 16-17.30 ETH Zoom Bian Wu
Mon 22.03. 16-17.30 ETH Zoom Riccardo Caniato
Mon 29.03. 16-17.30 ETH Zoom Filippo Gaia
Mon 12.04. 16-17.30 ETH Zoom Bian Wu
Wed 21.04. 16-17.30 ETH Zoom Riccardo Caniato
Mon 26.04. 16-17.30 ETH Zoom Filippo Gaia
Mon 03.05. 16-17.30 ETH Zoom Bian Wu
Mon 10.05. 16-17.30 ETH Zoom Riccardo Caniato
Mon 17.05. 16-17.30 ETH Zoom Filippo Gaia
Mon 31.05. 16-17.30 ETH Zoom Riccardo Caniato