VOL. 85 | 2020 The thresholding scheme for mean curvature flow and de Giorgi's ideas for minimizing movements
Tim Laux, Felix Otto

Editor(s) Yoshikazu Giga, Nao Hamamuki, Hideo Kubo, Hirotoshi Kuroda, Tohru Ozawa

Adv. Stud. Pure Math., 2020: 63-93 (2020) DOI: 10.2969/aspm/08510063


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


Published: 1 January 2020
First available in Project Euclid: 29 December 2020

Digital Object Identifier: 10.2969/aspm/08510063

Primary: 35A15
Secondary: 53E10 , 65M12 , 74N20

Keywords: diffusion generated motion , gradient flows , Mean curvature flow , minimizing movements , thresholding

Rights: Copyright © 2020 Mathematical Society of Japan


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