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.