article published yes Helmut Abels author Maximilian Moser author a60047a9-da77-11eb-85b4-c4dc385ebb8c JuFi department This paper is concerned with the sharp interface limit for the Allen--Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Ω⊂\R2. We assume that a diffuse interface already has developed and that it is in contact with the boundary ∂Ω. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant α-contact angle. For α close to 90° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen--Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument. Society for Industrial and Applied Mathematics2022 eng Applied MathematicsComputational MathematicsAnalysis 0036-1410 1095-7154 2105.08434 00076276800000410.1137/21m1424925 541114-172 H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172. Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1, Society for Industrial and Applied Mathematics, 2022, pp. 114–72, doi:<a href="https://doi.org/10.1137/21m1424925">10.1137/21m1424925</a>. Abels H, Moser M. 2022. Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. SIAM Journal on Mathematical Analysis. 54(1), 114–172. Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2022. <a href="https://doi.org/10.1137/21m1424925">https://doi.org/10.1137/21m1424925</a>. H. Abels and M. Moser, “Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 114–172, 2022. Abels H, Moser M. Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(1):114-172. doi:<a href="https://doi.org/10.1137/21m1424925">10.1137/21m1424925</a> Abels, H., &#38; Moser, M. (2022). Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/21m1424925">https://doi.org/10.1137/21m1424925</a> 123052023-01-16T10:07:00Z2024-10-09T21:03:58Z