The thresholding scheme for mean curvature flow and de Giorgi's ideas\n for minimizing movements

2019/10/24 by Tim Laux, Félix Otto, Laux, Tim +1 · 5 citations
Computer Science · Medicine · #Advanced Mathematical Modeling in Engineering #Advanced Neuroimaging Techniques and Applications #Analysis of PDEs (math.AP) #FOS: Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1910.11442

openalex publication_date 2019/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the thresholding scheme and explore its connection to De Giorgi's\nideas on gradient flows in metric spaces; here applied to mean curvature flow\nas the steepest descent of the interfacial area.\n The basis of our analysis is the observation by Esedoglu and the second\nauthor that thresholding can be interpreted as a minimizing movements scheme\nfor an energy that approximates the interfacial area.\n De Giorgi's framework provides an optimal energy dissipation relation for the\nscheme in which we pass to the limit to derive a dissipation-based weak\nformulation of mean curvature flow.\n Although applicable in the general setting of arbitrary networks, here we\nrestrict ourselves to the case of a single interface, which allows for a\ncompact, self-contained presentation.\n

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• Obstacle Mean Curvature Flow: Efficient Approximation and Convergence Analysis
• Equivalence of weak solution concepts for mean curvature flow
• Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds
• Convex-concave splitting for the Allen-Cahn equation leads to ε²-slow movement of interfaces

Related

• Analysis of thresholding for codimension two motion by mean curvature: a gradient-flow approach
• Convergence of the thresholding scheme for multi-phase mean-curvature\n flow
• Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds
• De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions
• Brakke's inequality for the thresholding scheme