Vorlesung: Semi-linear Elliptic PDEs (WS 2015/16)

Content of the lecture

Time, room (Zeit und Ort): Wednesday 16-18Uhr in B 045 (First meeting: October 14).

Exercises (Übungen): There are NO exercises!

Synopsis (Kurzbeschreibung): This course studies existence of weak solutions of semi-linear elliptic Partial Differential Equations (PDEs). Existence of solutions of PDEs is not easy to establish, the best strategy is often to first show the existence of solutions in some generalised sense, and then establish regularity (to conclude existence of a classical solution). For equations in divergence form, this leads to the study of weak solutions (and Sobolev spaces) by testing (multiplying and integrating) against smooth functions (as studied for linear elliptic PDEs in the course PDE 2 last semester).

Examples of semi-linear elliptic PDEs are abundant, in particular from Physics, Geometry, and Biology. They in particular describe solitary (or, stationary) waves for nonlinear time-dependent equations from Physics, such as the Klein-Gordon equation and the nonlinear Schrödinger equation (sometimes called 'nonlinear scalar field equations' in these cases). They also appear as stationary states for nonlinear heat equations, or in nonlinear diffusion in population genetics. On the other hand, such equations often appear in problems in Differential Geometry, such as the Yamabe Problem. There are also connections with constant mean curvature and minimal surfaces, as well as to stationary solutions for various geometric flows.

In this course we will study various techniques to prove existence of weak solutions to such equations in bounded domains.
Keywords: Nonlinear functional analysis; Critical Points; variational methods (Minimization Techniques: compact problems, constrained minimization, lack of compactness; Minimax Methods: Palais-Smale sequences, Mountain Pass Theorem, Saddle Point Theorem).

(Depending on interest, a sequel treating non-variational methods (monotone operators; fix point methods) might be planned in the following semester.)

Audience (Hörerkreis): Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master.

Credits: 3 ECTS.

Exam (Prüfung): There will be an oral exam of 30min (Es wird eine mündliche Prüfung von 30min geben).

Prerequisites (Vorkenntnisse): Knowledge of Sobolev spaces (on domains) and the theory of weak solutions of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage. The course will start with a (quick!) review of this material. Students who wish to follow this course, but did not yet follow a course on this material, should (in due time!) contact the Lecturer (Prof. Sørensen) via email to discuss the prerequisites needed. (These are basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned below.)

Language (Sprache): English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Literature: There will be no lecture notes. Here you will find a short description of the content of the lecture (to be updated as we go along). The lecture will mainly follow the book by Badiale and Serra mentioned below. (Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird größtenteils auf folgendes Buch basieren:)

[BS] M. Badiale, E. Serra (2011), Semilinear Elliptic Equations for Beginners, Springer (Universitext), 2011. (Login with your Campus-account.)

Supplementary literatur (Ergänzende Literatur):

Here is a longer list of books.

Office hours (Sprechstunde): Thursday 10:15-11:00 (Room B 408) or by appointment via email.


Letzte Änderung: 15 February 2016 (no more updates).

Thomas Østergaard Sørensen

Curriculum Vitae