A VARIATIONAL APPROACH TO GRADIENT FLOWS IN METRIC SPACES

Abstract. In this note we report on a new variational principle for Gradient Flows in metric spaces. This new variational formulation consists in a functional defined on entire trajectories whose minimizers converge, in the case in which the energy is geodesically convex, to curves of maximal slope. The key point in the proof is a reformulation of the problem in terms of a dynamic programming principle combined with suitable a priori estimates on the minimizers. The abstract result is applicable to a large class of evolution PDEs, including Fokker Plack equation, drift diffusion and Heat flows in metric-measure spaces. 1