[Fall 2020]

Prerequisites

A basic knowledge of real and complex analysis and/or calculus. A background in ordinary differential equations and/or dynamical systems. Several concepts from functional analysis will be used, but they will introduced and discussed according to the needs of the participants.

Aim of the course

The aim of the course is to gain an understanding of some of the basic techniques that underpin modern research in the field of partial differential equations. To this end, we provide an introduction to linear and nonlinear partial differential equations from a functional analytic and topological point of view. Understanding the question of existence and uniqueness/multiplicity of solutions is the primary goal. To achieve this we explore both functional analytic (mostly linear theory) and topological methods (nonlinear theory). The first half of the course (R. van der Vorst) focusses on linear and nonlinear elliptic partial differential equations. We start with linear theory:

  • Existence of weak solutions using Hilbert space methods;
  • Sobolev spaces;
  • Regularity of weak solutions;
  • H\"older and $L^p$-theory;

and we continue with the application of linear theory to nonlinear equations:

  • Nonlinear equations via variational principles;
  • The mountain pass theorem;
  • The Ljusternick-Schnirelmann category and multiplicity;
  • Morse theory (if time permits).

 In the second part of the course (H.J. Hupkes) we study semigroups, based on Evans and Engel\&Nagel. We discuss several different notions of continuity (uniform, strong and maybe even analytic) and prove the Hille-Yosida generation theorem. A great deal of time will be spent on examples to illustrate how the abstract results can be applied to concrete (parabolic) PDEs.

Lecturers

Part I, prof. dr. R.C.A.M. van der Vorst (VU). Part II, dr. H.J. Hupkes (UL).