# Weak Convergence Methods for Nonlinear Partial Differential Equations (PDE 2)

## Termine

• Vorlesung (4 SWS): Mo 8:30-10:00, MI 00.07.011, und Mi 14:15-15:45, MI 03.08.011
• Übung (2 SWS): Mo 14:15-15:45, MI HS 3

## Inhalt

The weak convergence method is a powerful tool for studying partial differential equations. E.g., one may try to prove existence of solutions for certain nonlinear equations by considering approximate problems which are much more easily solved. The main task is then to pass from the sequence of approximate solutions to some limit which in turn hopefully defines a solution of the original problem. In general, however, only weak convergence can be guaranteed. This is the source of major difficulties as nonlinear operations do not commute with weak convergence and hence it is by no means evident that the limiting function will solve the appropriate equation. The same kind of problems arise from the opposite point of view e.g. in multiscale models where the equations naturally contain a small parameter. Here one is led to investigate the asymptotics of the corresponding sequence of solutions. The main goal is then to identify an effective equation, which typically is much easier to understand, and to show that the limiting ("coarse grained") object can be found by solving this equation.

The main topics covered in this course include:
• Convergence properties of (nonlinear functionals of) sequences of functions in Lebesgue- and Sobolev spaces (key words: weak convergence , Young measures, quasiconvexity, compensated compactness),
• Applications to the theory of hyperbolic conservation laws, nonlinear elasticity theory and homogenization
• Vector valued problems in the Calculus of Variations and their relaxation

Prerequisites: Some familiarity with Sobolev spaces. (If you have read Chapter 4 of my PDE 1 notes available on the PDE 1 website, you'll be fine.)

## Übungsblätter

Es werden wöchentlich Übungsaufgaben ausgegeben, deren Lösungen dann in der darauffolgenden Woche in den Tutorien besprochen werden.

## Skript

Leicht zeitversetzt zum Fortgang der VL wird es ein Online-Skript geben. Bitte teilen Sie mir Tipp- (und andere) Fehler, die Sie bemerken, mit.

Aktuelle Version: July_25.pdf

## Aktuelles

• Das vollständige Skript ist nun im Netz (s.o.)
• Terminänderung: Der Freitagstermin wird ab dieser Woche (3.5.) auf Mo 8:30-10:00, MI 00.07.011, verlegt.

## Scheinkriterium

Scheinkriterium ist das Bestehen der mündlichen Prüfung nach der Vorlesung.