Thomas Stieltjes Institute for Mathematics 

Stieltjes Onderwijsweek Global and Variational Methods for ODEs and PDEs


Start: Monday, February 14, 2005, 9.30-10 am: reception in the Lorentz Centrum (Oortgebouw, third floor) in Leiden.
Closing: Friday February 18, 2005 around 5 pm.


The lectures on Monday through Thursday will be given by On Friday there will be a separate program of research seminars.


Concluding session on Friday:


Many ordinary and partial differential equations that arise in applications have a variational structure. This structure is important in many ways: it is often closely linked to the modelling context, it provides a single description of the problem that unites equation, boundary conditions, and any interface conditions, and — relevant for this course — provides important handles for the analysis of solutions.

In this course we will discuss various types of information that may be derived from ODEs and PDEs by exploiting their variational structure. Elliptic variational equations arise as stationarity equations of functionals, and we present a variety of techniques for finding stationary points, thereby proving existence of solutions. An important distinction can be made here between local minimizers and saddle points, both in their relevance to other issues and in the methods for finding them.

An important class of evolution equations with a variational structure is that of gradient flows. In recent years it has become clear that a large class of evolution equations and systems can be formulated as a gradient flow, by a well-chosen combination of the functional that is to be decreased and the dissipation metric that opposes this decrease. Such a formulation provides an important insight in the energy-dissipation structure of a given system.

The goal of this course is to provide a toolbox of widely applicable methods that students can apply to their own problems.


  1. Introduction to variational methods for elliptic problems: Euler equations; functional (Hilbert) spaces (L2, H1, H2), Poincare's lemma; weak topology in Hilbert spaces; weak formulation, intrinsic and natural boundary conditions; Riesz and Lax-Milgram theorems. (Peletier)
  2. Methods for finding stationary points of nonlinear functionals: direct minimization, lower semi-continuity, the Euler-Lagrange equation, differentiability of functionals, Sobolev embeddings, compactness and the Palais-Smale condition, constrained minimization, gradient flows, the Deformation lemma, the Mountain-Pass Theorem, the Linking Theorem, applications to nonlinear elliptic equations. (Van den Berg and Van der Vorst)
  3. The geometry of dissipative evolution equations: gradient flows on Riemannian manifolds, the space of probability measures as a Riemannian manifold, the Wasserstein metric as its induced metric, the (displacement-) convexity of the entropy functional; applications to the porous medium equation. (Otto)


This Stieltjes Onderwijsweek is intended for final-year Masters students and PhD students in mathematics; students in theoretical physics with an interest in the methods of this course are also invited to attend. Participants should have a working understanding of differential equations and a basic knowledge of Lebesgue integration theory and topology.


Participants can register at the Lorentz Center web site.


For further information about this course, please contact Mark Peletier, General information about the Stieltjes Onderwijsweken can be found (in Dutch) at the Stieltjes web site.

Maintained by Mark Peletier.
Last modified: Monday, 27 September 2004