Weighted Energy-Dissipation principle for gradient flows in metric spaces

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories I-epsilon[u] = integral(infinity)(0) e(-t/epsilon) (1/2 vertical bar u'vertical bar(2)(t) + 1/epsilon phi(u(t))) dt, featuring the weighted sum of energetic and dissipative terms. As the parameter epsilon is sent to 0, the minimizers u(epsilon) of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional phi. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation. (C) 2018 Elsevier Masson SAS. All rights reserved.