Nonlinear Partial Differential Equations
The theory of partial differential equations is one of the largest and most active areas of modern mathematics, drawing on tools and results from many other areas of mathematics such as functional analysis, operator theory, topology and harmonic analysis, and having applications in practically all areas of the natural sciences.
The principal goal of this course is to give an overview of the most important methods and phenomena particular to the nonlinear theory, mostly revolving around the existence and non-existence of solutions. The focus will be on elliptic and parabolic problems. In opposition to the linear case, there is no 'systematic' existence theory, rather a collection of commonly used approaches based on various analytical and topological techniques; a rough list of topics to be covered is thus:
-A (very incomplete) classification of the most common types of PDEs
-Generalised solutions (including distribution and Sobolev space theory as needed)
-Direct methods in the calculus of variations
-Minimax methods and mountain pass theorems
-Monotone operators and the existence theorems of Browder and Minty
If time permits, and depending on the wishes of the participants, we can also cover some regularity theory such as the Moser iteration scheme.
Prerequisites and Exam
This is a master level course consisting of two lectures and one tutorial per week (4+2 SWS). The course prerequisites are a good grounding in analysis and in particular knowledge of at least the basics of functional analysis. Previous experience of PDEs (such as the course "Partielle Differentialgleichungen" in Summer Semester 2015) is advantageous but not essential. The language of instruction can be English or German, depending on the wishes of the participants.
The exam will probably be oral.
To take this course you need to register at Moodle.
- Lawrence C. Evans, Partial differential equations, 2nd edition, American Mathematical Society, Providence, R.I., 2010.
- Ben Schweizer, Partielle Differentialgleichungen. Eine anwendungsorientierte
Einfuhrung, Springer Spektrum, Berlin, 2013.
- Michael Struwe, Variational methods. Applications to nonlinear partial differential
equations and Hamiltonian systems, 4th edition, Springer, Berlin, 2008.