# Functional Analysis II Spring 2018

Lecturer
Prof. Dr. Alessandro Carlotto
Course Assistant
Mario Schulz
Teaching Assistants
Alessandro Pigati
Xavier Fernandez-Real Girona
Lectures
Monday 10–12 / HG G 5
Thursday 13–15 / HG G 5
Exercise classes
Monday 09–10
Office hours
Tuesday 14–15
Wednesday 13–15
First Lecture
19.02.2018
First Exercise class
19.02.2018
Course Catalogue
401-3462-00L Functional Analysis II
Prerequisite
Functional Analysis I

### Primary references

Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.

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

### 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.

DateContentReference 1 (Struwe)Reference 2 (Brezis)Extras
19.02.2018 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.
–  –  –
22.02.2018 Solvability of the Poisson equation: the Poincaré inequality and two existence results for weak solutions,
either via Riesz theorem or by Fredholm alternative.
§ 7.1 § 8.1,  § 8.3  –
26.02.2018 Distributional and weak derivatives, examples and basic facts.
Definition of Sobolev spaces. Completeness, separability and reflexivity.
§ 7.2 § 9.1  –
01.03.2018 Absolute continuity of functions in $$W^{1,p}(I)$$, weak vs. pointwise derivative.
(Dual) characterizations of $$W^{1,p}(I)$$ for $$p>1$$. The Sobolev extension operator.
§ 7.3
(till Satz 7.3.4)
§ 8.2
(till Thm 8.6)
–
05.03.2018 Density of test functions in $$W^{1,p}(I)$$ for all $$1\leq p < \infty$$.
The one-dimensional Sobolev embedding theorem. $$W^{1,p}$$ as a Banach algebra.
§ 7.3 § 8.2  –
08.03.2018 Discussion of some examples of second-order ODEs with either Dirichlet or Neumann boundary conditions.
Perturbation methods and convergence of approximation schemes.
§ 7.4 § 8.4  –
12.03.2018 Sobolev functions of $$N$$ variables, examples of singular functions in $$W^{1,p}$$. A criterion for the coincidence of pointwise and weak derivative (null capacity). The Meyers-Serrin approximation theorem, and comments on boundary behaviour. § 8.1,  § 8.2 § 9.1  –
15.03.2018 Equivalent characterizations of $$W^{1,p}$$. Lipschitz vs. $$W^{1,\infty}$$.
Calculus rules for Sobolev functions: sums, products and compositions (chain rule).
§ 8.3 § 9.1  –
19.03.2018 Extension of functions in $$W^{1,p}(\Omega)$$ for $$\Omega$$ a relatively compact domain of class $$C^1$$.
Two approximation theorems for $$W^{1,p}$$ functions, on bounded or unbounded domains.
§ 8.4
(till Satz 8.4.2)
§ 9.2  –
22.03.2018 Imposing boundary values for elliptic problems: trace operators and their properties.
Two characterizations of $$H^1_0(\Omega)$$ and the canonical splitting of $$H^1(\Omega)$$.
§ 8.4 complements to § 9  –
26.03.2018 The Sobolev-Gagliardo-Nirenberg inequality, link with the isoperimetric inequality in $$\mathbb{R}^n$$.
The Sobolev embedding theorem for $$p < n$$, comments on the borderline case $$p = n$$.
§ 8.5 § 9.3  –
29.03.2018 A panorama on the Sobolev embedding theorems, exponential integrability of elements in $$W^{1,n}(\Omega)$$. Hölder continuity, completeness of the associated functional spaces, compact embedding of $$C^{0,\alpha}(\Omega)\subset C^{0,\beta}(\Omega)$$ for $$\alpha>\beta$$. § 8.6.1  –  –
09.04.2018 Morrey-Campanato spaces, integral characterization of Hölder continuity.
The Sobolev embedding theorem for $$p>n$$ and associated compactness results.
§ 8.6.2 § 9.3  –
12.04.2018 Pointwise differentiability of functions in $$W^{1,p}(\mathbb{R}^n)$$ for any $$p>n$$.
Higher-order Sobolev embedding theorems and their applications to regularity of weak solutions.
§ 8.6.3-4,  § 9.1 § 9.3,  § 9.5  –
16.04.2018 Interior regularity for solutions of the Poisson equation: formal $$H^1$$ and $$H^2$$ estimates, and their rigorous counterpart via Nirenberg's method (difference quotients). § 9.2
except § 9.2.1
§ 9.6  –
19.04.2018 Higher Sobolev estimates for weak solutions of the Poisson equation.
Ellipticity for operators in divergence form, and corresponding $$H^{k+2}$$ interior estimates.
§ 9.2,  § 9.5 § 9.6  –
23.04.2018 Sobolev estimates and boundary regularity for weak solutions of the Poisson equation with flat boundary.
The case of curved boundary: flattening via diffeomorphisms and the modified equation.
§ 9.3
(till Lem 9.3.4)
§ 9.6  –
26.04.2018 Sobolev estimates and boundary regularity for weak solutions of elliptic equations.
Global $$H^{k+2}$$ estimates for weak solutions of elliptic equations.
§ 9.3 § 9.6  –
30.04.2018 The closure of $$-\Delta\colon C^2_0\subset L^2\to L^2$$.
The Dirichlet spectrum of the Laplace operator and the min-max characterization of its eigenvalues.
§ 9.4 § 9.8  –
03.05.2018 Introduction to Schauder theory, context and motivations.
Campanato estimates for solutions of homogeneous problems (with constant coefficients).
§ 10.1,  § 10.2  –  –
07.05.2018 Campanato estimates for solutions of homogeneous problems (with constant coefficients): the boundary case.
Estimates for solutions of inhomogeneous problems (with constant coefficients).
§ 10.2  –  –
14.05.2018 Morrey spaces and their equivalence to Campanato spaces for $$\nu < n$$.
Local $$C^{1,\alpha}$$ Schauder estimates (interior and boundary cases).
§ 10.3,  § 10.4
(till Lem 10.4.3)
–  –
17.05.2018 Local $$C^{2,\alpha}$$ Schauder estimates (interior and boundary cases).
Global $$C^{2,\alpha}$$ Schauder estimates.
§ 10.4  –  –
24.05.2018 Solvability in $$C^{2,\alpha}$$ for elliptic boundary value problems.
The case of the Laplacian and the method of continuity.
§ 10.5  –  –
28.05.2018 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.
–  – weak maximum principle
31.05.2018 The strong maximum principle via the Hopf boundary point lemma. Elliptic barriers.
The method of sub- and super-solutions, examples of geometric relevance.
–  – strong maximum principle
Dates
6–9 August 2018
Place
HG G 26.1
 Rules of the exam: Rules.pdf Advice for the exam: Advice.pdf Sample Questions: Samplequestions.pdf

Please register in myStudies and use echo.ethz.ch to enroll for an exercise class.

AssistantClassroom
Xavier Fernandez-Real Girona HG G 26.3
Alessandro Pigati HG E 33.3
Mario Schulz HG F 26.5

Every Thursday, a new problem set is uploaded here. You have one week time to solve the problems. The following Monday during exercise class the assistants show the right approaches for the more involved problems. If you still have difficulties understanding or solving certain tasks, please prepare your questions and join the office hours on Tuesday or Wednesday.

#### Homework collection and delivery:

You may hand in your homework for grading to the assistant of the exercise class you enrolled for in myStudies. Either leave it in the respective letterbox in HG F28 or – if you wrote your homework in LaTeX – you may send the pdf-file to your assistant by email. The deadline is on Thursdays 4 pm. The graded homework sheets are delivered in class on Thursdays by the instructor.

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.
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.
19.02.2018 review
22.02.2018 01.03.2018 Problem Set 1 Hints 1 Solution 1 updated 28.02.18
01.03.2018 08.03.2018 Problem Set 2 Hints 2 Solution 2
08.03.2018 15.03.2018 Problem Set 3 Hints 3 Solution 3
15.03.2018 22.03.2018 Problem Set 4 Solution 4 Online quiz: echo.ethz.ch
22.03.2018 29.03.2018 Problem Set 5 Hints 5 Solution 5
29.03.2018 12.04.2018 Problem Set 6 Hints 6 Solution 6
12.04.2018 19.04.2018 Problem Set 7 Hints 7 Solution 7
19.04.2018 26.04.2018 Problem Set 8 Solution 8 Online quiz: echo.ethz.ch update: 8.12. convergence of a subsequence
26.04.2018 03.05.2018 Problem Set 9 Hints 9 Solution 9 updated 30.04.18
03.05.2018 11.05.2018 Problem Set 10 Solution 10 Online quiz: echo.ethz.ch
11.05.2018 24.05.2018 Problem Set 11 Hints 11 Solution 11
25.05.2018 31.05.2018 Problem Set 12 Solution 12 Online quiz: echo.ethz.ch

You are free to come and ask questions. The location changes depending on which assistant is on duty. The schedule is as follows (up to possible short-term changes, please check for updates):

DayDateTimeLocationAssistant
Tuesday20.02.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday21.09.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday27.02.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday28.02.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday06.03.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday07.03.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday13.03.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday14.03.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday20.03.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday21.03.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday27.03.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday28.03.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday10.04.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday11.04.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday17.04.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday18.04.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday24.04.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday25.04.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Wednesday02.05.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday08.05.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday09.05.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday15.05.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday16.05.201813:00–15:00 HG F 28.3 Mario Schulz
Tuesday22.05.201814:00–15:00 HG FO 27.9 Alessandro Pigati
Wednesday23.05.201813:00–15:00 HG G 28 Xavier Fernandez-Real Girona
Tuesday29.05.201814:00–15:00 HG F 28.3 (update) Mario Schulz
Wednesday30.05.201813:00–15:00 HG F 28.3 Mario Schulz