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

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.

💡 Research Summary

The paper investigates the discrete energy‑dissipation inequality associated with variational interpolants—also known as De Giorgi interpolants—within the framework of the Minimizing Movement Scheme (MMS). While the classical De Giorgi lemma was originally formulated for metric gradient flows, the authors extend its validity to generalized gradient systems in Banach spaces and, under suitable hypotheses, upgrade the inequality to an exact identity.

The work is organized around two parallel settings. First, a generalized metric gradient system (gMGS) is defined as a quadruple ((M,E,D,\psi)) where ((M,D)) is a complete metric space, (E) is a proper lower‑semicontinuous functional, and (\psi) is a strictly convex, (C^{1}) dissipation potential on (