Problem 1.

(a) Let $\Omega \subset \mathbb{R}^{n}$ be open, $\alpha=\left(\alpha_{1}, \ldots, \alpha_{n}\right) \in \mathbb{N}{0}^{n}$ a multi-index and $|\alpha|=\sum{k=1}^{n} \alpha_{k}$. Let
$u \in L_{\text {loc }}^{1}(\Omega) .$ Given $1 0$, but $u \notin W^{1,1}(\mathbb{R})$, i. e. $u$ does not have a weak derivative in $L^{1}(\mathbb{R})$.

Proof .

(a) Let $u \in L_{\text {loc }}^{1}(\Omega)$. Given $1<p \leq \infty$, let $1 \leq q<\infty$ such that $\frac{1}{p}+\frac{1}{q}=1$. Suppose $D^{\alpha} u$ exists as weak derivative in $L^{p}(\Omega)$. Let $\varphi \in C_{c}^{\infty}(\Omega)$ be arbitrary. Then,
$$\left|\int_{\Omega} u D^{\alpha} \varphi d x\right|=\left|(-1)^{|\alpha|} \int_{\Omega}\left(D^{\alpha} u\right) \varphi d x\right| \leq\left|D^{\alpha} u\right|_{L^{p}(\Omega)}|\varphi|_{L^{q}(\Omega)}$$
by Hölder’s inequality which proves the first claim with constant $C=\left|D^{\alpha} u\right|_{L^{p}(\Omega)}$. Conversely, suppose
$$\forall \varphi \in C_{c}^{\infty}(\Omega): \quad\left|\int_{\Omega} u D^{\alpha} \varphi d x\right| \leq C|\varphi|_{L^{q}(\Omega)}$$
Then, since $C_{c}^{\infty}(\Omega)$ is dense in $L^{q}(\Omega)$ for $q<\infty$, the map
$$f: \varphi \mapsto(-1)^{|\alpha|} \int_{\Omega} u D^{\alpha} \varphi d x$$
defines a continuous linear functional $f \in\left(L^{q}(\Omega)\right)^{}$. Since $\left(L^{q}(\Omega)\right)^{}$ for $1 \leq q<\infty$ is isometrically isomorphic to $L^{p}(\Omega)$, there exists $g \in L^{p}(\Omega)$ such that
$$\forall \varphi \in L^{q}(\Omega): \quad f(\varphi)=\int_{\Omega} g \varphi d x$$
By definition of $f$ it follows that $g \in L^{p}(\Omega)$ is the weak derivative $D^{\alpha} u$ of $u$.
(b) Let $u=\chi_{j 0,[1}$ and $\varphi \in C_{c}^{\infty}(\mathbb{R})$. Then
$$\left|\int_{\mathbb{R}} u \varphi^{\prime} d x\right|=\left|\int_{0}^{1} \varphi^{\prime} d x\right|=|\varphi(1)-\varphi(0)| \leq 2|\varphi|_{L^{\infty}(\mathbb{R})} .$$
The function $u$ restricted to $\mathbb{R} \backslash{0,1}$ is differentiable with vanishing derivative. In particular, if $u$ had a weak derivative $u^{\prime} \in L_{\text {loc }}^{1}(\mathbb{R})$, then $u^{\prime}=0$ almost everywhere. A contradiction arises for test functions $\varphi \in C_{c}^{\infty}(\mathbb{R})$ with $\varphi(0) \neq \varphi(1)$ via
$$0=\int_{\mathbb{R}} u^{\prime} \varphi d x=-\int_{\mathbb{R}} u \varphi^{\prime} d x=-\int_{0}^{1} \varphi^{\prime} d x=\varphi(0)-\varphi(1) .$$

Problem 2.

2.5. Green’s function for the half-space Show that Green’s function for the upper half-space, i. e. the unbounded domain $\mathbb{R}{+}^{n}:=\left{\left(x{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}: x_{n}>0\right} \subset \mathbb{R}^{n}$
exists by computing $G(x, y)$ explicitly.

Proof .

Given $x=\left(x_{1}, \ldots, x_{n-1}, x_{n}\right) \in \mathbb{R}{+}^{n}$, let $\bar{x}=\left(x{1}, \ldots, x_{n-1},-x_{n}\right)$ denote its reflection in
the plane $\partial \mathbb{R}{+}^{n} .$ Let $\Phi: \mathbb{R}^{n} \backslash{0} \rightarrow \mathbb{R}$ be the fundamental solution of Laplace’s equation as given on the problem set. Then the function $$\phi^{x}(y):=\Phi(y-\bar{x})=\Phi\left(y{1}-x_{1}, \ldots, y_{n-1}-x_{n-1}, y_{n}+x_{n}\right)$$
satisfies
\left{\begin{aligned} \Delta \phi^{x} &=0 & & \text { in } \mathbb{R}{+}^{n} \ \phi^{x}(y) &=\Phi(y-x) & & \text { for } y \in \partial \mathbb{R}{+}^{n} \end{aligned}\right.
because $y-\bar{x} \neq 0$ for every $y \in \mathbb{R}{+}^{n}$ and since by symmetry of $\Phi$ $$\forall y \in \partial \mathbb{R}{+}^{n}: \quad \Phi(y-x)=\Phi(\overline{y-x})=\Phi(\bar{y}-\bar{x})=\Phi(y-\bar{x})=\phi^{x}(y)$$
Hence, Green’s function for the upper half-space is
\begin{aligned} G(x, y) &=\Phi(y-x)-\phi^{x}(y)=\Phi(y-x)-\Phi(y-\bar{x}) \ &=\left{\begin{array}{ll} -\frac{1}{2 \pi}(\log |y-x|-\log |y-\bar{x}|), & (n=2) \ \frac{1}{n(n-2)\left|B_{1}\right|}\left(|y-x|^{2-n}-|y-\bar{x}|^{2-n}\right), & (n \neq 2) \end{array}\right. \end{aligned}
Remark. Since the domain $\mathbb{R}{+}^{n}$ is unbounded, the representation formula (as given on the problem set) for solutions of the equation $-\Delta u=f$ in $\mathbb{R}{+}^{n}$ with boundary data $\left.u\right|{\partial \mathbb{R}{+}^{n}}=g$ has to be checked separately.

Problem 3.

Let $1 \leq p \leq \infty$. Consider the open set $\Omega=]-1,1\left[{ }^{2} \backslash([0,1[\times{0})\right.$.
(a) Prove that there does not exist an extension operator $E: W^{1, p}(\Omega) \rightarrow W^{1, p}\left(\mathbb{R}^{2}\right)$.
(b) Where and why does the argument that you have seen in class fail in this situation?

Proof .

(a) Let $\Omega=]-1,1\left[^{2} \backslash([0,1[\times{0})\right.$ and let $u: \Omega \rightarrow \mathbb{R}$ be given by
$$u\left(x_{1}, x_{2}\right):=\left{\begin{array}{ll} x_{1} & \text { if } x_{1}>0 \text { and } x_{2}>0 \ 0 & \text { otherwise. } \end{array}\right.$$
As shown in Problem $5.1, u \in W^{1, \infty}(\Omega) .$ Since $\Omega$ is bounded, $u \in W^{1, p}(\Omega)$ for any $1 \leq p \leq \infty$. Suppose, there exists an extension operator $E: W^{1, p}(\Omega) \rightarrow W^{1, p}\left(\mathbb{R}^{2}\right)$ such that $\left.(E u)\right|{\Omega}=u$ almost everywhere in $\Omega .$ Let $\left.Q:=\right]-1,1\left[^{2}\right.$ and $v:=\left.(E u)\right|{Q}$. Then $E u \in W^{1, p}\left(\mathbb{R}^{n}\right)$ implies $v \in W^{1, p}(Q)$. Consequently, as shown in Problem 5.5, $\left(x_{2} \mapsto v\left(x_{1}, x_{2}\right)\right) \in W^{1, p}(]-1,[)1$ for almost every $\left.x_{1} \in\right]-1,1[.$ Moreover, since $[0,1[\times{0}$
has measure zero, $v\left(x_{1}, x_{2}\right)=u\left(x_{1}, x_{2}\right)$ for almost every $\left(x_{1}, x_{2}\right) \in Q$.

Hence, there exists some fixed $\left.x_{1} \in\right] \frac{1}{2}, 1\left[\right.$ such that $\left(g: x_{2} \mapsto v\left(x_{1}, x_{2}\right)\right) \in W^{1, p}(]-1,[)1$ and such that $g\left(x_{2}\right)=u\left(x_{1}, x_{2}\right)$ for almost every $\left.x_{2} \in\right]-1,1[.$ By Sobolev’s embedding in dimension one, $g$ and hence $x_{2} \mapsto u\left(x_{1}, x_{2}\right)$ has a representative in $C^{0}(]-1,[)1$. However, since we chose $x_{1}>\frac{1}{2}$, this contradicts discontinuity of
$$x_{2} \mapsto u\left(x_{1}, x_{2}\right)=\left{\begin{array}{ll} x_{1} & \text { for } x_{2}>0 \ 0 & \text { for } x_{2}<0 \end{array}\right.$$
(b) The issue is that $\Omega$ is not a topological manifold with boundary. In particular, every point $x \in[0,1] \times{0}$ belongs to the topological boundary of $\Omega$ but doesn’t admit an open neighbourhood $U$ such that $U \cap \Omega$ is even only homeomorphic to $Q_{+}$ (compare with the definition of open set with $C^{k}$ boundary given in class).

Problem 4.

As a warm-up to the following problem, please have a look at the proof for the Sobolev-Gagliardo-Nirenberg inequality
Adapt that argument to show that there is an embedding $H^{1}\left(\mathbb{R}^{2}\right) \hookrightarrow L^{4}\left(\mathbb{R}^{2}\right)$ and prove the estimate
$$\forall u \in H^{1}\left(\mathbb{R}^{2}\right): \quad|u|_{L^{4}\left(\mathbb{R}^{2}\right)}^{4} \leq 4|u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2}|\nabla u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2}$$

Proof .

Sobolev’s embedding (in the case $n=2=p$ ) states that the space $H^{1}\left(\mathbb{R}^{2}\right)$ embeds into $L^{q}\left(\mathbb{R}^{2}\right)$ for any $2 \leq q<\infty$, in particular for $q=4$. The Sobolev inequality states
$$\exists C<\infty \quad \forall u \in H^{1}\left(\mathbb{R}^{2}\right): \quad|u|_{L^{4}\left(\mathbb{R}^{2}\right)} \leq C|u|_{H^{1}\left(\mathbb{R}^{n}\right)}$$
In this special case, we claim that the following inequality also holds.
$$\forall u \in H^{1}\left(\mathbb{R}^{2}\right): \quad|u|_{L^{4}\left(\mathbb{R}^{2}\right)}^{4} \leq 4|u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2}|\nabla u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2}$$
Since $C_{c}^{\infty}\left(\mathbb{R}^{2}\right)$ is dense in $H^{1}\left(\mathbb{R}^{2}\right)$, it suffices to prove the inequality for $u \in C_{c}^{\infty}\left(\mathbb{R}^{2}\right)$. Let $u \in C_{c}^{\infty}\left(\mathbb{R}^{2}\right)$ and $\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$. Then,
\begin{aligned} \left|u^{2}\left(x_{1}, x_{2}\right)\right| &=\left|\int_{-\infty}^{x_{1}} \frac{\partial u^{2}}{\partial x_{1}}\left(s, x_{2}\right) d s\right|=\left|\int_{-\infty}^{x_{1}} 2 u\left(s, x_{2}\right) \frac{\partial u}{\partial x_{1}}\left(s, x_{2}\right) d s\right| \ & \leq 2 \int_{\mathbb{R}}\left|u\left(s, x_{2}\right) | \nabla u\left(s, x_{2}\right)\right| d s \end{aligned}

Analogously,
$$\left|u^{2}\left(x_{1}, x_{2}\right)\right| \leq 2 \int_{\mathbb{R}}\left|u\left(x_{1}, t\right) | \nabla u\left(x_{1}, t\right)\right| d t$$
Hence, by Fubini’s theorem and the Cauchy-Schwarz inequality
\begin{aligned} |u|_{L^{4}\left(\mathbb{R}^{2}\right)}^{4} &=\int_{\mathbb{R}} \int_{\mathbb{R}}\left|u\left(x_{1}, x_{2}\right)\right|^{4} d x_{1} d x_{2}=\int_{\mathbb{R}} \int_{\mathbb{R}}\left|u^{2}\left(x_{1}, x_{2}\right) | u^{2}\left(x_{1}, x_{2}\right)\right| d x_{1} d x_{2} \ & \leq 2 \int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left|u\left(s, x_{2}\right) | \nabla u\left(s, x_{2}\right)\right| d s\right) \int_{\mathbb{R}}\left|u^{2}\left(x_{1}, x_{2}\right)\right| d x_{1} d x_{2} \ & \leq 4 \int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left|u\left(s, x_{2}\right) | \nabla u\left(s, x_{2}\right)\right| d s\right) d x_{2} \int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left|u\left(x_{1}, t\right) | \nabla u\left(x_{1}, t\right)\right| d t\right) d x_{1} \ &=4\left(\int_{\mathbb{R}^{2}}|u||\nabla u| d x\right)^{2} \leq 4|u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2}|\nabla u|_{L^{2}\left(\mathbb{R}^{2}\right)}^{2} \end{aligned}

Functional Analysis II Spring 2021

LecturerProf. Dr. Alessandro CarlottoCourse AssistantRiccardo CaniatoTeaching AssistantsFilippo GaiaBian WuLecturesMon10-12 / HG G 5 – Livestream
Thu14-16/ HG G 5 – LivestreamExercise classesMon9-10Office hoursMon16-17.30First lecture22.02.2021Course Catalogue401-3462-00L 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^pLp 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.

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).

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).

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.

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).