Bachelor-level knowledge of Real and Complex Analysis and the theory of Ordinary Differential equations as well as some concepts of Functional Analysis and Measure Theory; more specifically, knowledge of:

  • Hilbert and Banach spaces, dual spaces, and convergence in these spaces
  • Linear operators: definition, basic properties and compact operators
  • Lebesgue integral and Lp -spaces
  • Convergence criteria for Lp-functions: Monotone convergence theorem, dominated convergence theorem, Fatou's lemma

The necessary background on these topics can be found in Chapters 2-5 in the book by H. W. Alt, Linear Functional Analysis: An Application-Oriented Introduction, Springer, 2016.

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 (PDEs). To this end, we introduce methods that are needed to analyze linear PDEs from a functional analytic point of view. Primarily focusing on equations of second order we aim to study the existence, uniqueness and qualitative properties of solutions.
We first discuss how PDEs differ from ordinary differential equations and what natural questions can be considered. We address classes of PDEs that can be solved explicitly using the heat equation, the wave equation and Laplace's equation as guiding examples. Subsequently, we discuss limitations of the classical theory and motivate the notion of weak solutions and the functional analytic approach.
We use the functional analytic approach to establish the basic theory of weak solutions for elliptic boundary value problems. The latter requires Sobolev spaces that we discuss in considerable details. Sobolev spaces allow a weak (re-)formulation of the problem and to establish existence, uniqueness and regularity results. If time permits, at the end of the course we address linear evolution equations.

Martina Chirilus-Bruckner (Leiden University) (first part)
Stefanie Sonner (Radboud University), (second part)