On De Giorgi's lemma for variational interpolants in metric and Banach spaces

TL;DR

This paper extends De Giorgi's lemma for variational interpolants from metric spaces to Banach spaces, establishing conditions under which the energy-dissipation inequality becomes an equality, thereby advancing the theory of gradient flows.

Contribution

It generalizes De Giorgi's lemma to Banach spaces and identifies conditions for the energy-dissipation inequality to hold as an equality in this broader context.

Findings

Extended De Giorgi's lemma to Banach spaces.

Established conditions for energy-dissipation equality.

Provided examples demonstrating the sharpness of results.

Abstract

Variational interpolants are an indispensable tool for the construction of gradient-flow solutions via the Minimizing Movement Scheme. The De Giorgi lemma provides the associated discrete energy-dissipation inequality. It was originally developed for metric gradient systems. Drawing from this theory we study the case of generalized gradient systems in Banach spaces, where a refined theory allows us to extend the validity of the discrete energy-dissipation inequality and to establish it as an equality. For the latter we have to impose the condition of radial differentiability of the dissipation potential. Several examples are discussed to show how sharp the results are.