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
-Fixed-point theorems
-Monotone operators and the existence theorems of Browder and Minty
-Blow-up phenomena

If time permits, and depending on the wishes of the participants, we can also cover some regularity theory such as the Moser iteration scheme.

Effects of a nonlinear PDE (Mean Curvature Flow): Singularities occur in finite time, blowup permits surgery. [Source: http://www.math.utah.edu/~mayer/math/MCF/dumbbell2_js.html]

Lecture Notes

You can find the complete lecture notes here. All corrections, remarks and suggestions are welcome; please write an email to James Kennedy.

A list of questions you should be able to answer before taking the oral exam can be found here.

General Information

Dates and Rooms

Lecture: Tue. 10-12 O28/2003, Wed. 8-10 HeE60,

Exercise Class: Wed. 16-18 O28/2004

Lecturer: James Kennedy

Class teacher: Adrian Spener

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.

Literature

  • Lawrence C. Evans, Partial di fferential equations, 2nd edition, American Mathematical Society, Providence, R.I., 2010.
  • Ben Schweizer, Partielle Di fferentialgleichungen. Eine anwendungsorientierte
    Einfuhrung, Springer Spektrum
    , Berlin, 2013.
  • Michael Struwe, Variational methods. Applications to nonlinear partial diff erential
    equations and Hamiltonian systems
    , 4th edition, Springer, Berlin, 2008.