Calculus of Variations and Geometric Measure Theory
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R. Rossi - G. Savaré - A. Segatti - U. Stefanelli

Weighted Energy-Dissipation principle for gradient flows in metric spaces

created by rossi on 15 Jan 2018

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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.


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