Vorlesung: Viscosity Solutions for nonlinear PDEs 2 (SoSe 2015)

Content of the lecture

Time, room (Zeit und Ort): Tuesday 16-18Uhr in A 027 (First meeting: April 14).

Exercises (Übungen): There are NO exercises!

Synopsis (Kurzbeschreibung): This course is a continuation of my lecture 'Viscosity Solutions for nonlinear PDEs' in the past semester (WiSe 2014/15). It treats the regularity theory of viscosity solutions for linear and nonlinear PDEs (whereas the first course treated the definition, uniqueness (Comparison Principles), and existence (Perron's Method) of viscosity solutions).
Keywords: Viscosity solutions, fully nonlinear elliptic PDEs, Alexandroff estimate and maximum principle (ABP estimate), Harnack inequality, W^{2,p}-regularity, Hölder regularity, Gradient Hölder regularity.

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): This course is a continuation of 'Viscosity Solutions for nonlinear PDEs' (WiSe 2014/15). 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.

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 Caffarelli and Cabré mentioned below. (Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird größtenteils auf folgendes Buch basieren:)

[CC] L. A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, AMS (Colloquium Publications), 1995.

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.

[H-L] Q. Han und F. Lin, Elliptic Partial Differential Equations: Second Edition, AMS (Courant Lecture Notes), 2011.

[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).

[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 longer list of books.

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


Letzte Änderung: 21 July 2015 (no more updates).

Thomas Østergaard Sørensen

Curriculum Vitae