Martial Agueh, Carnegie Mellon University
"Some Applications of Displacement Convexity to PDEs and to Geometric Inequalities"
Based on a notion of convexity - displacement convexity - along geodesics in the space of probability densities equipped with the Wasserstein metric, we establish a basic inequality relating the total - internal, potential and interactive- energies of two arbitrary states of a system, their Wasserstein distance and their energy production function. This inequality is remarkably encompassing as it implies many known geometric inequalities, while allowing a direct and unified way for computing their best constants and extremals. It also leads to exponential rates of decay to equilibria for solutions of parabolic equations such as the Fokker-Planck, the porous medium, the fast diffusion, the parabolic p-Laplacian, the McKea-Vlasov type equations, and some doubly degenerate diffusion equations.
TUESDAY, September 28, 2004