Stieltjes Onderwijsweek Global and Variational Methods for ODEs and PDEs
Start: Monday, February 14, 2005, 9.30-10 am: reception in the
(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:
13.30-15.30: exercise class
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
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.
- 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)
- 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)
- 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