Workshop on geometric analysis and spectral theory

May 18-19, 2009

Laboratoire CNRS J.-V. Poncelet, Independent University of Moscow, Moscow, Russia

Organizers**Nikolai Nadirashvili, Christophe Pittet, Yannick Sire.**

This is a joint work with Ch. Pittet. We consider a class of r.w. on groups which are inductive limits of finite groups and establish a Weyl type relation between probability of return to the origin and L^2-isoperimetric profile of the corresponding Laplacian, similar to the one proved recently in the work of A. Bendikov, Ch. Pittet and R. Sauer.

Starting from a model of traffic congestion, we introduce a minimal-flow--like variational problem whose solution is characterized by a very degenerate elliptic PDE. We precisely investigate the relations between these two problems, which can be done by considering some weak notion of flow for a related ODE. We also prove regularity results for the degenerate elliptic PDE, which enables us in some cases to apply the DiPerna-Lions theory.

This talk is concerned with various generalizations of the usual notions of waves, fronts and propagation mean speed in a general setting. These new notions involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. General intrinsic properties, some monotonicity properties and some uniqueness results for almost planar fronts have been obtained with H. Berestycki. In a second part (with H. Berestycki and H. Matano), we will see how to use these notions to describe the propagation of almost-planar fronts around obstacles for bistable reaction-diffusion equations.

We discuss regularity of solutions to fully nonlinear elliptic equations and we show the existence of singular solutions.

On non-compact amenable Lie groups, a simple formula relates the spectral distribution of the Laplace operator with the isoperimetric profile. This is recent work with A. Bendikov and R. Sauer, based on previous works by Coulhon and Grigor'yan, and Gromov and Shubin.

I will describe several results related to a conjecture of De Giorgi on the flatness of level sets of elliptic equations. I will give some hints on how this conjecture is related to minimal surface theory and I will describe some flatness results for non local equations on one hand and on elliptic equations on riemannian manifolds on the other hand.

Zero-pressure gas dynamics in one spatial dimension is a nice and simple model which can be explicitly integrated in two ways, very different conceptually but leading to the same answer: a variant of the famous Hopf convex hull construction and a projection construction inspired by a (reverse) analogy with Arnold's geometric formulation of the incompressible Euler dynamics. Moreover, both representations have been independently rediscovered, with different motivations, and published at least three times each! I will recall this confusing story and discuss the tantalizing multidimensional case.