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Mullins-Sekerka as the Wasserstein flow of the perimeter

Chambolle, Antonin; Laux, Tim (2021), Mullins-Sekerka as the Wasserstein flow of the perimeter, Proceedings of the American Mathematical Society, 149, 7, p. 2943-2956. 10.1090/proc/15401

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
Proceedings of the American Mathematical Society
Volume
149
Number
7
Publisher
American Mathematical Society
Pages
2943-2956
Publication identifier
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Author(s)
Chambolle, Antonin cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laux, Tim
Hausdorff Center for Mathematics [HCM]
Abstract (EN)
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.
Subjects / Keywords
Gradient flows, Wasserstein distance, sets of finite perimeter, Mullins-Sekerka, free boundary problems, Hele-Shaw cell