Thomas Stieltjes Institute for Mathematics  Stieltjes Onderwijsweek Global and Variational Methods for ODEs and PDEs

DATES

LECTURERS

• Jan Bouwe van den Berg
• Felix Otto
• Mark Peletier
• Rob van der Vorst

SCHEDULE

10.00-11.00: lecture
11.00-11.30: coffee
11.30-12.30: lecture
12.30-13.30: lunch
13.30-15.30: exercise class
15.30-16.00: tea
16.00-17.00: exercise class

10.00-11.00 Jos Maubach, Numerical methods for variational formulations
11.30-12.30 Brenny van Groesen, Variational principles for extreme waves
13.30-14.30 Giuseppe Savare, Differential calculus and Gradient flows in spaces of probability measures
15.00-16.00 Joost Hulshof, Strongly indefinite variational systems

CONTENT

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.

PROGRAMME

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)

PREREQUISITES

REGISTRATION

FURTHER INFORMATION