Convergence of the Allen-Cahn equation with a zero Neumann boundary condition on non-convex domains

Abstract

We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke's mean curvature flow with a generalized right angle condition on the boundary of the domain.

arXiv comment31 pages and 2 figures

References in corpus (5)

• A general regularity theory for weak mean curvature flow
• A second derivative Hölder estimate for weak mean curvature flow
• Convergence of the Allen-Cahn equation with Neumann boundary conditions
• Existence and regularity of mean curvature flow with transport term in higher dimensions
• A singular perturbation limit of diffused interface energy with a fixed contact angle condition

Cited by in corpus (3)

• Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to $90$°
• Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with $90$°-Contact Angle in Higher Dimensions
• Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension