*Submitted Paper*

**Inserted:** 15 jan 2018

**Last Updated:** 15 jan 2018

**Year:** 2018

**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}\phi(u(t)) \right) d t, \]
featuring the weighted sum of energetic and dissipative terms. As the parameter $\varepsilon$ is sent to $0$, the minimizers $u_\varepsilon$ 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.

**Download:**