Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

This paper studies a Chambolle-type minimizing movement scheme for mean curvature flow with prescribed contact angle in a smooth bounded domain.

The scheme is based on the capillary functional and the geodesic signed distance relative to the container, and yields a time-discrete level-set approximation.

The main result asserts that, for every $C^1$ boundary function prescribing a strictly nondegenerate contact angle, the approximate solutions converge locally uniformly to the unique viscosity solution of the corresponding level-set mean curvature equation with oblique derivative boundary condition.

This improves a previous convergence theorem, where the container was assumed to be convex and a curvature-type condition relating the tangential derivative of the prescribed contact-angle function to the principal curvatures of the container boundary was imposed.

The main new ingredient is a uniform Lipschitz estimate for the solutions of the variational problems defining the scheme.

This estimate is derived by applying a Bernstein-type argument to a suitable weighted gradient, rather than to the gradient itself, which rules out boundary maxima without relying on the previous curvature-type condition.