Workshop on geometric analysis and spectral theory
May 18-19, 2009
Laboratoire CNRS J.-V. Poncelet, Independent University of Moscow, Moscow, Russia
Nikolai Nadirashvili, Christophe Pittet, Yannick Sire.
"On a class of random walks on groups with infinite number of generators".
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.
"Optimal transportation, weak flows and very
degenerate elliptic PDE's".
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
"Reaction-diffusion equations and generalized transition waves".
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.
"Singular solutions to fully nonlinear elliptic equations".
We discuss regularity of solutions to fully nonlinear elliptic equations
and we show the existence of singular solutions.
"A Weyl type formula relating spectral distribution to the isoperimetric profile".
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.
"Geometry of elliptic equations".
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 and more dimensions".
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.