Vorlesung: Viscosity Solutions for nonlinear PDEs (WS 2014/15)

Content of the lecture

News (03.11.2014): The Lecture on Tuesday Dec 23th is moved to Wednesday Dec 17th 12--14 in C 113.

News (03.11.2014): The Lecture on Tuesday Nov 25th is moved to Wednesday Nov 19th 12--14 in C 113.

Time, room (Zeit und Ort): Tuesday 16-18Uhr in B 045 (First meeting: Oct 07)

Exercises (Übungen): There are NO exercises!

Synopsis (Kurzbeschreibung): This course treats the viscosity solution theory for linear and nonlinear Partial Differential Equations (PDEs). Existence of solutions of PDEs is not easy to establish, the best strategy is 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 in the course PDE 2 last semester).
For general nonlinear PDEs, and equations in non-divergence form, this approach does (often) not work. However, one can define a new type of generalised solutions (called viscosity solutions) by testing the solution in a whole new sense (inspired by the Maximum Principle for harmonic functions).
Keywords: Viscosity solutions, fully nonlinear elliptic PDEs, Hamilton-Jacobi(-Bellman-Isaacs) equations, Maximum Principles and Comparison Principles (for uniqueness), Perron's Method (for existence), stability (for continuity in the data), regularity (if time permits).

Audience (Hörerkreis): Master students of Mathematics (WP 17.2, 18.1, 18.2), 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): Analysis I-III, Linear Algebra I-II .
Knowledge from PDE 1 (harmonic functions, Laplace and Poisson equations, elliptic equations) and PDE 2 (weak solutions, uniformly elliptic PDEs in divergence form) is an advantage, but not needed.

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 Lecture Notes by Koike mentioned below . (Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird größtenteils auf folgende Skript von Koike basieren:)

[K] S. Koike (Department of Mathematics, Saitama University, Japan), A Beginner’s Guide to the Theory of Viscosity Solutions, 2nd edition (version: June 28, 2012).

Supplementary literatur (Ergänzende Literatur):

[De] R. Denk (Universität Konstanz), Skript zur Vorlesung 'Nichtlineare partielle Differentialgleichungen', Sommersemester 2014 (Last version: 22.07.2014).

[Dr] F. Dragoni (University of Bristol), Introduction to Viscosity Solutions for Nonlinear PDEs, Imperial College London, Autumn 2009.

[L] Q. Liu (University of Pittsburgh), An introduction to viscosity solution theory, (Version: April 4, 2013).

[P] K. Payne (Universita dì Milano), Fully nonlinear equations and viscosity solutions, Bibliography Part 2 for the course: Nonlinear Partial Differential Equations, Academic year 2011-2012.

[Y] X. Yu (University of Alberta, Edmonton, Canada), Lecture 11: Viscosity solutions, from Math 527: Intermediate Partial Differential Equations (Fall 2008).

Here is a list of books.

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


Letzte Änderung: 17 December 2014 (no more updates).

Thomas Østergaard Sørensen

Curriculum Vitae