Calculus of Variations and Geometric Measure Theory
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The Allen-Cahn equations and multiphase mean curvature flow

Tim Laux

created by pluda on 12 Mar 2021

17 mar 2021 -- 17:00   [open in google calendar]

Dipartimento di Matematica di Pisa (online)

Abstract.

The connection between the Allen-Cahn equation and mean curvature flow has been explored for decades. Still, little is known in the multiphase case, which is an important model for grain growth in polycrystals. In the first part of the talk, I will present a simple convergence proof of the Allen-Cahn equation to a weak solution to mean curvature flow, which generalizes to this multiphase case. The key ingredient is a phase-field approximation of the tilt-excess which allows to pass to the limit in nonlinearities in the equations. In the second part, I will present a recent weak-strong uniqueness result for multiphase mean curvature flow, which applies to precisely those solutions constructed in the first part of the talk. Here, the main ingredient is the new notion of gradient-flow calibrations, which generalizes the well-known concept from minimal surfaces to our dynamic situation. This is based on joint works with Julian Fischer, Sebastian Hensel, and Theresa Simon.

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