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

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114-172, 2022. https:// doi.org/10.1137/21M1424925) 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 90.. 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 e 1 2 for general contact angles alpha is an element of (0, pi). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order e; again for general contact angles alpha is an element of (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 et al. (SIAM J Math Anal 52(6):6222-6233, 2020. https:// doi.org/10.1137/20M1322182), 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 et al. (arXiv:2003.05478v2).

Automated summary

In this study, we extend the recent rigorous convergence result of the Allen-Cahn equation towards evolution by mean curvature flow with constant contact angle. We manage to remove the perturbative assumption on the contact angle being close to 90 degrees, and obtain more general convergence rate results. Our proof deviates from the popular strategies and instead relies on a relative entropy technique.