Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

2022/12/22 by Tim Laux, Kerrek Stinson, Laux, Tim +3 · 3 citations
Mathematics · #35A02 #35D30 #53E30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2212.11939

openalex publication_date 2022/12/22 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

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• Error estimates for a surface finite element method for anisotropic mean curvature flow
• Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness
• The vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions

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