A new varifold solution concept for mean curvature flow: Convergence of the AllenCahn equation and weakstrong uniqueness
Abstract
We propose a new weak solution concept for (twophase) mean curvature flow which enjoys both (unconditional) existence and (weakstrong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417461, (1993)], any limit point of solutions to the AllenCahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478]  and hence any classical solution to mean curvature flow  is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally nonunique. Finally, we propose an extension of the solution concept to the multiphase case which is at least guaranteed to satisfy a weakstrong uniqueness principle.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.04233
 Bibcode:
 2021arXiv210904233H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 53E10 (primary);
 49Q20;
 35K57;
 35Q49;
 28A75
 EPrint:
 38 pages, added additional references