Convergence rates for the Allen-Cahn equation with boundary contact energy: The non-perturbative regime

TL;DR

This paper proves convergence rates for the Allen-Cahn equation with boundary contact energy towards mean curvature flow, removing previous restrictions on contact angle proximity to ninety degrees, and introduces a novel approach based on relative entropy.

Contribution

It extends convergence results to general contact angles without perturbative assumptions and achieves sharper rates for specific boundary energy densities using a relative entropy method.

Findings

Established sub-optimal convergence rate of order ε^{1/2} for general contact angles.

Achieved a sharp convergence rate of order ε for specific boundary energy densities.

Removed the perturbative assumption on contact angle being close to ninety degrees.

Abstract

We extend the recent rigorous convergence result of Abels and the second author (arXiv preprint 2105.08434) concerning convergence rates for solutions of the Allen-Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to ninety degree. We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order $\smash{\varepsilon^{\frac{1}{2}}}$ for general contact angles $\alpha \in (0,\pi)$. For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order $\varepsilon$; again for general contact angles $\alpha \in (0,\pi)$. Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen-Cahn operator. Instead, we follow the recent approach by Fischer, Laux and Simon (SIAM J. Math. Anal. 52, 2020), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer, Laux, Simon and the first author (arXiv preprint 2003.05478).