Quantization of the Energy for the inhomogeneous Allen-Cahn mean curvature

TL;DR

This paper studies the energy distribution and convergence properties of varifolds associated with the Allen--Cahn phase transition, demonstrating energy equidistribution and convergence to integer rectifiable varifolds with controlled mean curvature.

Contribution

It establishes energy equidistribution between Dirichlet and potential energies and proves convergence to integer rectifiable varifolds with mean curvature in L^{q_0}, extending Allard's theorem.

Findings

Energy is evenly distributed between Dirichlet and potential components.

Varifolds converge to integer rectifiable varifolds with bounded mean curvature.

Extension of Allard's convergence theorem to Allen--Cahn varifolds.

Abstract

We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ 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 $L^{q_0}, q_0 > n$. The latter is a diffused version of Allard's convergence theorem for integer rectifiable varifolds.