Sagun Chanillo: Borderline Sobolev Inequalities on Symmetric Spaces with Applications

The lecture was held within the framework of the Hausdorff Trimester Program: Evolution of Interfaces.

Abstract:
Bourgain and Brezis proved the following remarkable estimate: Consider the equation, −∆u = f where f : R^n → R^n is a vector field with zero divergence. Further u : R^n → R^n and the Laplacian acts componentwise. Then one has ||∇u|| n/n−1 ≤ c(n)||f||1. We show that the same estimate is valid for non-compact globally symmetric spaces. Next we show that the original Bourgain-Brezis inequality can be applied to obtain Strichartz inequalities for wave and Schrodinger equations and also we can obtain new estimates for the Maxwell equations and 2D, incompressible Navier-Stokes flow. These results have been obtained jointly with Jean van Schaftingen and Po-lam Yung.