Lecture Description: 

Differential equations are a fundamental tool to model processes in science and technology. In this lecture we will first consider fixed-point theorems and use them to show solvability of nonlinear differential equations. Building on linear elliptic theory and Lax-Milgram's lemma, we will prove Browder--Minty's theorem, a general existence result for monotone operators. The generalization to pseudomonotone operators will allow us to treat the stationary Navier--Stokes equations and other examples. In the last part of the course, we will turn to nonsmooth problems and study maximal monotone operators.

Prerequisites: Analysis I-III, Linear Algebra

Literature: 

Michael Růžička: Nichtlineare Funktionalanalysis,

Tomas Roubicek: Nonlinear partial differential equations with applications

Etienne Emmrich: Gewöhnliche und Operatorgleichungen

Haim Brezis: Operateurs Maximaux Monotone

Eberhard Zeidler: Nonlinear functional analysis and its applications (IIA,IIB)

Michel Chipot: Elliptic equations: An introductory course

Gajewski, Gröger, Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen 

There are weekly homework sheets that should be worked on in groups of two or three people. The exercise sheets should be handed in on Wednesdays before the exercise starts at 12p.m. 

My Office Hours are on Monday 10-12 in my Office A9/K012. You can also contact me or ask for a date via mail: lasarzik@wias-berlin.de/robert.lasarzik@fu-berlin.de. 

Please do not hesitate to contact me, if there is anything to discuss.