# Abstracts for Rivière-Fabes Symposium on Analysis and PDE

April 8-10, 2005

### Luca Capogna

Lecture 1 - Mean curvature flow and the isoperimetric problem in the Heisenberg group

Abstract 1: I will describe recent results (joint with Mario Bonk, U. Michigan) concerning a notion of mean curvature flow in the Heisenberg group. We derive an equation for the flow, characterize self-similar solution and prove basic existence and comparison results. We also study the induced flow on the Legendrian foliation of the manifolds and indicate how it relates to the isoperimetric problem in the Heisenberg group.

### Svatlana jitomirskaya

Lecture 1 - The ten martini problem

Abstract 1: In these talks I will describe recent advances to the theory of homogenization of first- and second-order partial differential equations set in general stationary ergodic environments.

### Vladimir Maz'ya

Lecture 1 - Unsolved mysteries of solutions to PDEs near the boundary

Abstract 1: Throughout its long history, specialists in the theory of partial differential equations gained a deep insight into the boundary behaviour of solutions.Yet despite the apparent progress in this area achieved during the last century, there are fundamental unsolved problems and surprising paradoxes related to solvability, spectral, and asymptotic properties of boundary value problems in domains with irregular boundaries. I shall formulate some challenging questions arising naturally when one deals with unrestricted, polyhedral, Lipschitz graph, fractal and convex domains.

### Natasa Pavlovic

Lecture 1 - Dyadic models for the equations of fluid motion

Abstract 1: In this talk we shall introduce a scalar dyadic model for the Euler and the Navier-Stokes equations in three dimensions and will discuss some of the results that were obtained for these models. For the dyadic Euler equations we prove finite time blow-up, while in the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in case when the degree of dissipation is sufficiently small (joint work with Nets Katz). These results can be generalized to analogous results for a vector dyadic model (joint work with Susan Friedlander). Also time permitting, we shall present some results for the actual Navier-Stokes equations that are inspired by observing similar phenomena present in dyadic models.

### Steve Wainger

Lecture 1 - Some discrete operators arising in Harmonic Analysis

Abstract 1: A great deal of attention has been given to integral operators acting on functions defined on R(d) where the integration is over a submanifold of R(d) of positive co-dimension. In these talks we will discuss operators acting on functios defined on Z(d)-points in R(d) with integral coordinates. Integration is replaced by summation and the sum is not over all of Z(d), but rather over certain arithmetic subsets of Z(d).

### Yu Yuan

Lecture 1 - Global solutions to special Lagrangian equations

Abstract 1: We survey some recent results on global solutions to special Lagrangian equations. These elliptic equations arise in calibrated geometry and have applications to string theory. The global results have close relation to the regularity theory of the special Lagrangian equations. In the minimal surface equation case, one has the classical Bernstein theorem.