Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result

Article type: Research Article

Authors: Moser, Maximilian^{; *}

Affiliations: Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria

Correspondence: [*] Corresponding author. E-mail: maximilian.moser@ist.ac.at.

Abstract: We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N⩾2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂Ω. The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [0,T] for some time T>0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).

Keywords: Sharp interface limit, mean curvature flow, contact angle, Allen–Cahn equation, vector-valued Allen–Cahn equation

DOI: 10.3233/ASY-221775

Journal: Asymptotic Analysis, vol. 131, no. 3-4, pp. 297-383, 2023