Nov 2019 - March 2020

Teachers: Sara Maad Sasane, Niels Christian Overgaard, Erik Wahlén

Syllabus

Official syllabus for the course.

Literature

Main book: J. Mawhin & M. Willem, Critical Point Theory and Hamiltonian Systems.

Complementary book: M. Struwe, Variational Methods

Schedule

Updated 19 February.

We meet on Wednesdays at 10.15-12.00 in MH:332A the coming weeks (26 Feb, 4 Mar, 11 Mar). The last week we will meet on Monday, 10.15-12.00 in the same room (16 Mar).

Lectures

Nov 8 (Niels Christian): Introduction. See lecture notes and exercise sheet.
Nov 15 (Erik): Introduction. 1.1-1.4 in Mawhin & Willem (we got to the point of introducing the Sobolev space $LaTeX: W^{1,p}$ ; the rest is saved for the next lecture). Suggested exercises: 1-7 in chapter 1 (in particular 2, 4, 7). Follow-up question to exercise 4: what about weakly lower semi-continuous functions? Some notes on Sobolev spaces (from the Specialised Course on Integration Theory, 2015).
Nov 22. No lecture.
Nov 29 (Erik): 1.4-1.5 in Mawhin & Willem. Study 1.6 on your own. Suggested exercises: 8, 9 in chapter 1 and additional exercises.
Dec 6 (Erik): 1.7, "2.0" (pp. 28-30) and 2.3 in M & W.
Dec 13 (Erik): 2.1, 2.2, 2.4 in M & W, I.6 (pp. 58-60) in Struwe. Suggested exercises: 1-8 in chapter 2 and additional exercise. Note: Exercise 4 should say $LaTeX: F : \mathbf{R}^n \to \mathbf{R}$ .
Dec 20: No lecture.
Jan 20 (Erik): 3.1, 3.2, 3.4, 3.5 (we didn't go through the proofs in sections 3.4 and 3.5) in M & W. Suggested exercises: 1-5 in chapter 3.
Jan 27 (Sara): Introduction of Ch. 4 in M & W. 4.1. Suggested exercises: 1-6 of Ch. 4 in M & W. Lecture notes.
Feb 3 (Sara): Section 4.2 in M & W. Suggested exercise: 7 of Ch. 4 in M & W. Lecture notes.
Feb 10 (Sara) Section 4.5 in M & W. Lecture notes. The gap in Lemma 4.5 has been filled by Olof. See the attached file. You can also see Ekeland's original paper from 1979, from which this result is taken. I also mentioned another minimax theorem: the saddle point theorem (Section 4.3 in M&W). The details and an application of this (e.g. the one given in Section 4.4.) could be a topic for a project. Suggested exercises: 10 of Ch. 4 in M & W.
Feb 19 (Erik & Sara) Information about projects (see below).
Feb 26 (Sara) Section 6.1 and 6.2 of M & W, and some material from Section 5.1 and 5.4. Lecture notes. Suggested exercises: 1 and 2 of Chapter 6 in M&W.
Mar 4 (Sara) Section 5.1 and 5.4. The aim was to compute (or estimate) the S^1 index for some concrete sets in a Banach space, with the aim of using Theorem 6.1 to obtain infinitely many critical points of a functional. Exercises: Fill in the gaps in the proofs of the lecture. Lecture notes. You can also do exercises 4-8 in M&W.
Mar 11 Presentation of projects: Douglas, George, Tien, Adam
Mar 16 Presentation of projects: Alexia, Daniele, Jonathan
Mar 20 Presentation of projects: Charis, Olof, Mårten

Projects

Each presentation should be approximately 20 minutes. This means that you don't have time to do all the details, but should concentrate on conveying the main ideas. You can use a computer or do it on the blackboard, as you prefer.

The dates for the presentations are Wednesday, March 11, 10.15-12, Monday, March 16, 10.15-12 and Friday, March 20, 13.15-15. All in 332A.

Topics for the presentations:

1. Constraints and Lagrange multipliers. Struwe Chapter I.2. Adam.

2. Application of minimax on semilinear elliptic PDE. Struwe Chapter II.6. Jonathan.

3. Saddle point theorem with application. Mawhin-Willem Chapter 4.3-4.4. (requires knowledge about topological degree). Charis.

4. Concentration-compactness. Struwe Chapter I.4. Daniele.

5. Periodic solutions with prescribed energy. Mawhin-Willem Chapter 3.5 and Struwe I.6. George.

6. Symmetry and compactness and applications to solitary waves. Willem (Minimax theorems) 1.5-1.6. Tien.

7. Choquard’s nonlinear equation and symmetric rearrangements. Paper by Lieb Lieb & Loss (Analysis). Olof.

8. Variational inequalities and obstacle problems. Kinderlehrer & Stampacchia (Introduction to variational inequalities and their applications) Introduction and Chapter II, or Friedman (Variational inequalities and free boundary problems) chapter 1.1-1.2. Douglas.

9. Applications of Krasnoselskii genus to a nonlinear system of ODEs. Struwe II.5. Alexia.

10. Viscosity solutions. Katzourakis 7. Mårten.