Vorlesung: Semi-linear Elliptic PDEs 2 (SoSe 2016)

News (01.07.2016): Information (sign up!) about the exam is online.

Content of the lecture

Time, room (Zeit und Ort): Tuesday 16-18 Uhr in B 040 (First meeting: April 12).

Exercises (Übungen): There are NO exercises.

Synopsis (Kurzbeschreibung): This course is a continuation of my lecture Semi-linear Elliptic PDEs in the past semester.

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 in SoSe2015).

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 continue the study of various techniques to prove existence of weak solutions to such equations in bounded and unbounded domains.
Keywords: Variational methods (Minimization Techniques: constrained minimization (on spheres and Nehari manifolds), lack of compactness; Minimax Methods: Saddle Point Theorem).
Non-variational methods (Fixpoint theory, Method of lower and upper solutions).

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 (also on domains) and the theory of weak solutions of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage.
(This is basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned below.)
The course will start with a (quick!) review of the material covered last semester. Students who wish to follow this course, but did not follow the course last semester, 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, 4.1-4.3, and 4.4.1 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 in part follow the book by Badiale and Serra mentioned below. Further literature to come. (Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird zum Teil 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 (to be updated).

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


Letzte Änderung: 08 July 2016 (no more updates).

Thomas Østergaard Sørensen

Curriculum Vitae