Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum

On the expansion of convex hypersurfaces by symmetric functions of their principal radii of curvature

SIGLETIB Hannover: RO 5389(36) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

W^{2,1}- regularity for solutions of the Monge-Ampere equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is W2,1loc. This is obtained by showing higher integrability a-priori estimates for D2u, namely D2u 08LlogkL for any k 08N