article published yes Sebastian Hensel author 4D23B7DA-F248-11E8-B48F-1D18A9856A870000-0001-7252-8072 Maximilian Moser author a60047a9-da77-11eb-85b4-c4dc385ebb8c JuFi department Bridging Scales in Random Materials project 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 ε12 for general contact angles α∈(0,π). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order ε; again for general contact angles α∈(0,π). 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). https://research-explorer.ista.ac.at/download/12079/12320/2022_Calculus_Hensel.pdf application/pdfno Springer Nature2022 eng 0944-2669 1432-0835 00084424730000810.1007/s00526-022-02307-3 616 S. Hensel, M. Moser, Calculus of Variations and Partial Differential Equations 61 (2022). Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6, 201, Springer Nature, 2022, doi:<a href="https://doi.org/10.1007/s00526-022-02307-3">10.1007/s00526-022-02307-3</a>. Hensel S, Moser M. 2022. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. Calculus of Variations and Partial Differential Equations. 61(6), 201. Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00526-022-02307-3">https://doi.org/10.1007/s00526-022-02307-3</a>. Hensel S, Moser M. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. 2022;61(6). doi:<a href="https://doi.org/10.1007/s00526-022-02307-3">10.1007/s00526-022-02307-3</a> S. Hensel and M. Moser, “Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6. Springer Nature, 2022. Hensel, S., &#38; Moser, M. (2022). Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href="https://doi.org/10.1007/s00526-022-02307-3">https://doi.org/10.1007/s00526-022-02307-3</a> 120792022-09-11T22:01:54Z2025-04-14T07:53:59Z