Published Paper

Inserted: 27 nov 2024

Journal: Mathematische Annalen
Volume: 390
Year: 2024
Doi: https://doi.org/10.1007/s00208-024-02909-6

Abstract:

We consider the varifold associated to the Allen--Cahn phase transition problem in ℝn+1(or n+1-dimensional Riemannian manifolds with bounded curvature) with integral Lq0 bounds on the Allen--Cahn mean curvature (first variation of the Allen--Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in Lq0,q0>n. The latter is a diffused version of Allard's convergence theorem for integer rectifiable varifolds.