Partial Differential Equations - M1 - 8EC

Prerequisites

Bachelor-level knowledge of

  • Real and Complex Analysis
  • (the theory of) Ordinary Differential equations
  • some basic concepts of Functional Analysis and Measure Theory; more specifically:
    • Hilbert and Banach spaces, dual spaces, and convergence in these spaces
    • Linear operators: definition, basic properties
    • Lebesgue integral and L p -spaces
    • Convergence criteria for L p -functions: Monotone convergence theorem, dominated convergence theorem, Fatou's lemma

Of advantage is knowledge on compact operators, but it is not a strict prerequisite


The necessary background on Functional Analysis can be found in any of the following books:

  • H.W. Alt, Linear Functional Analysis: An Application-Oriented Introduction, Springer, 2016 Chapters 2-5
  • H. Brezis: Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011. Chapters 1-5
  • B. Rynne, M. Youngson, Linear Functional Analysis, 2ed., Springer 2008, Chapters 1-6.

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.

Rules about Homework/Exam
The course counts for 8 EC. The learning goals will be assessed as follows:
• hand-in assignments (30%)
• written exam (70%)
The hand-in assignments still count as part of the grade after retake. To pass the course, at least a 5 is required in the exam.

Lecture notes/Literature
Lecture notes will be developed and made available during the course. They contain all the material
discussed in class.
The course is based on sections from Parts I and II of the book:

  • L.C. Evans, Partial Differential Equations. American Mathematical Society, 1998.

Lecturers
Martina Chirilus-Bruckner (Leiden University) m.chirilus-bruckner@math.leidenuniv.nl
Stefanie Sonner (Radboud University), stefanie.sonner@ru.nl