Hensel, Sebastian and Laux, Tim (2025) A new varifold solution concept for mean curvature flow: Convergence of the Allen–Cahn equation and weak-strong uniqueness. JOURNAL OF DIFFERENTIAL GEOMETRY, 130 (1). pp. 209-268. ISSN 0022-040X, 1945-743X
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We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) 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, 417-461, (1993)], any limit point of solutions to the Allen-Cahn 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 non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | IMPLICIT TIME DISCRETIZATION; GAMMA-CONVERGENCE; GRADIENT FLOWS; 1ST VARIATION; LEVEL SETS; MOTION; CONJECTURE; SPACES |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Tim Laux |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 18 Jun 2026 06:04 |
| Last Modified: | 18 Jun 2026 06:04 |
| URI: | https://pred.uni-regensburg.de/id/eprint/67153 |
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