Gamma-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

TL;DR

This paper develops a theoretical framework connecting minimizing movement schemes and gradient flows in metric spaces, accommodating relaxed minimization steps and Gamma-convergence of functionals.

Contribution

It introduces new abstract results linking Gamma-convergence with gradient flows and extends the scheme to include relaxed minimization steps.

Findings

Established conditions for convergence of relaxed minimizing movement schemes

Extended the theory to general metric spaces

Provided new relaxation results for the scheme

Abstract

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.