# Abstracts for Rivière-Fabes Symposium on Analysis and PDE

April 23-25, 2004

### H. Brezis

Lecture 1 - New estimates for the Laplacian, the div-curl, and related elliptic systems.

Abstract 1: I will present a recent joint work with J. Bourgain concerning new estimates for integrals on loops, for the Laplacian , for the div-curl system, and more general first order elliptic systems in L^1 .

### Alex Kiselev

Lecture 1 - Spectrum and dynamics of Schrëodinger operators with decaying potentials.

Abstract 1: We review recent progress in spectral and scattering theory of Schrëodinger operators. In particular, we will discuss sharp results on the rate of decay of potential needed for asymptotic completeness of (modified) wave operators in dimension one. The counterexample which shows sharpness of the result involves the construction of potentials which lead to imbeeded singular continuous spectrum. The inspiration for this contruction goes back to the classical Wigner von Neumann example of positive imbedded eigenvalue for a Schrëodinger operator with potential decaying at a Coulomb rate.

### Alexander Nagel

Lecture 1 - Regularity of the Kohn-Laplacian in decoupled domains

Abstract 1: Abstract: We obtain optimal estimates for solutions of the Kohn-Laplacian on decoupled domains, where the eigenvalues of the Levi form can degenerate at different rates. In domains with comparable eigenvalues, it is known that the relevant singular integral operators are variants of the standard classical Calderon-Zygmund operators. In contrast, for decoupled domains one is led to the study of operators which are more related to product theory and flag kernels.

### S.R.S Varadhan

Lecture 1 - Homogenization of Random Hamilton-Jacobi-Bellman equations and applications to large Deviations in a quenched Random Environment.

Abstract 1: The problem of establishing a quenched large deviation principle for a diffusion in a random environment is a special case of the following larger class of problems. Under suitable scaling, the Hamilton-Jacobi-Bellman type equation that describes the optimal value of a controlled diffusion, in a random environment, i.e with a random cost function, is to be replaced by a first order Hamilton-Jacobi equation. We will review the literature and discuss some new results.

### Sijue Wu

Lecture 1 - Recent Progress in Mathematical Analysis of Vortex Sheets

Abstract 1: The vortex sheet problem serves as a prototype for the evolution of the vorticity in fluid flows. One can think for example of the wake of an airfoil as a typical problem of this type. This problem can be described by the incompressible Euler equation, where the initial vorticity is ideally a finite Radon measure supported on a curve. The issue is to determine the specific nature of the evolution of this curve--the vortex sheet, after the singularity formation time. We answer this question through results on the regularity of the vortex sheet, and the existence and nonexistence of solutions to the initial value problem.