Weighted Energy-Dissipation principle for gradient flows in metric spaces

Abstract

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 \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}ϕ(u(t)) \right) \dd t, \] 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 $\varphi$. 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.