- 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

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.

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.

Date | Content | Reference 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 |

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.

Assistant | Classroom |
---|---|

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.

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.

Assignment date | Due date | Problem Sets | Extra Hints | Solutions | Comments |
---|---|---|---|---|---|

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):

Day | Date | Time | Location | Assistant |
---|---|---|---|---|

Tuesday | 20.02.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 21.09.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 27.02.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 28.02.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 06.03.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 07.03.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 13.03.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 14.03.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 20.03.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 21.03.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 27.03.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 28.03.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 10.04.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 11.04.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 17.04.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 18.04.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 24.04.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 25.04.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Wednesday | 02.05.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 08.05.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 09.05.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 15.05.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 16.05.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Tuesday | 22.05.2018 | 14:00–15:00 | HG FO 27.9 | Alessandro Pigati |

Wednesday | 23.05.2018 | 13:00–15:00 | HG G 28 | Xavier Fernandez-Real Girona |

Tuesday | 29.05.2018 | 14:00–15:00 | HG F 28.3 (update) | Mario Schulz |

Wednesday | 30.05.2018 | 13:00–15:00 | HG F 28.3 | Mario Schulz |