Two Variational Techniques for the Approximation of Curves of Maximal Slope

Abstract

This report generalizes several results on approximations of curves of maximal slope in the book by Ambrosio, Gigli and Savare (Gradient Flows, 2005), allowing general approximations of the functional and of the metric. The conditions guaranteeing the convergence of the approximations are closely related to the conditions of -convergence. Complete convergence results are obtained for -convex functionals together with general results indicating a possible procedure for even more general problems.

The theory presented provides convergence results as well as explicit a-priori error estimates for gradient flows of functionals which may be non-convex as well as non-differentiable.