A new varifold solution concept for mean curvature flow: Convergence of the Allen–Cahn equation and weak-strong uniqueness

Mean curvature flow is one of the most fundamental geometric evolution equations and appears in many surface-tension driven problems. Although the equation has an instantaneous smoothing effect, generically, singularities appear in finite time. One is led to consider weak solutions which persist through these singular events. Folklore says that mean curvature flow is a gradient flow with the caveat that the underlying metric is completely degenerate. In this talk, after discussing known weak notions of solution, I will present a new concept which has its roots in the theory of gradient flows and relies on basic geometric measure theory. I will show that these solutions arise naturally in the sharp-interface limit of the Allen-Cahn equation and in addition satisfy a weak-strong uniqueness principle. The latter property is a fundamental difference to well-known Brakke solutions, which a priori may disappear at any given time and are therefore fatally non-unique.

This is joint work with Sebastian Hensel (U Bonn).