Universität Wien

250310 VO Selected topics in partial differential equations (2007S)

Selected topics in partial differential equations

6.00 ECTS (3.00 SWS), SPL 25 - Mathematik

Details

Language: German

Lecturers

Classes (iCal) - next class is marked with N

  • Thursday 01.03. 14:00 - 15:00 Seminarraum
  • Tuesday 06.03. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 08.03. 14:00 - 15:00 Seminarraum
  • Tuesday 13.03. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 15.03. 14:00 - 15:00 Seminarraum
  • Tuesday 20.03. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 22.03. 14:00 - 15:00 Seminarraum
  • Tuesday 27.03. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 29.03. 14:00 - 15:00 Seminarraum
  • Tuesday 17.04. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 19.04. 14:00 - 15:00 Seminarraum
  • Tuesday 24.04. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 26.04. 14:00 - 15:00 Seminarraum
  • Thursday 03.05. 14:00 - 15:00 Seminarraum
  • Tuesday 08.05. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 10.05. 14:00 - 15:00 Seminarraum
  • Tuesday 15.05. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Tuesday 22.05. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 24.05. 14:00 - 15:00 Seminarraum
  • Thursday 31.05. 14:00 - 15:00 Seminarraum
  • Tuesday 05.06. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Tuesday 12.06. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 14.06. 14:00 - 15:00 Seminarraum
  • Tuesday 19.06. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 21.06. 14:00 - 15:00 Seminarraum
  • Tuesday 26.06. 11:00 - 13:00 (ehem. Seminarraum A 1.01)
  • Thursday 28.06. 14:00 - 15:00 Seminarraum

Information

Aims, contents and method of the course

The Obstacle Problem: deeper insight:

(based on the materials of Fermi lectures, read by Prof. L. Caffarelli at Scoula Normale de Pisa, 1998)

The optimal regularity of the solution; optimal gradient bound; global solutions;
a detailed proof of the regularity of free boundary - employing the famous
Alt-Caffarely-Friedman Monotonicity formula; the structure of singular points.

Parabolic Obstacle Problem:

Parabolic variational inequalities; the strong maximum principle; the Stefan problem; properties of free boundary.

Assessment and permitted materials

Minimum requirements and assessment criteria

The prime goal is to arm students with some very strong techniques, illustrating them on examples form the frontier of active research filed. These techniques are applicable not only in the field of free boundary problems, but also in many other areas of PDEs. This course design had two goals: to be a dynamic continuation of the previous one and to be self contained, providing
easy understanding also for students that did not attend the previous part, which was an introduction to Free boundary problems.

Examination topics

Basic functional analysis, function spaces, Green¿s formula and boundary value problems, second order elliptic PDE from variational inequality; the projection theorem, existence results, stability, comparison and maximum principles, Harnack inequality, Liouville theorem, Alt-Caffarelli-Friedman monotonicity formula, DeGiorgi oscilation lemma, DeGiorgi-Nash-Moser interior Harnack inequality, Littman Stampacia Weinberger theorem on the behavior of fundamental solution.

Reading list

L. A. Caffarelli "The Obstacle Problem",
A. Friedman "Variational Principles and Free Boundary Problems",
additional handouts.

Association in the course directory

Last modified: Mo 07.09.2020 15:40