MULLINS-SEKERKA AS THE WASSERSTEIN FLOW OF THE PERIMETER

Antonin Chambolle (1) , Tim Laux (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 HCM - Hausdorff Center for Mathematics

We prove the convergence of an implicit time discretization for the Mullins-Sekerka equation proposed in [F. Otto, Arch. Rational Mech. Anal. 141 (1998) 63-103]. Our simple argument shows that the limit satisfies the equation in a distributional sense as well as an optimal energy-dissipation relation. The proof combines simple arguments from optimal transport, gradient flows & minimizing movements, and basic geometric measure theory.

Mots clés: free boundary problems; Mullins- Sekerka; Gradient flows; Wasserstein distance; sets of finite perimeter

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Antonin Chambolle, Tim Laux. MULLINS-SEKERKA AS THE WASSERSTEIN FLOW OF THE PERIMETER. Proceedings of the American Mathematical Society, 2021, 149, pp.2943--2956. ⟨10.1090/proc/15401⟩. ⟨hal-02306665v2⟩