Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain
Abstract
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 Lipschitz-continuous 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.
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BibTeX
@misc{eto2026convergence,
title = {{Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain}},
author = {Tokuhiro Eto and Jiwoong Jang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.00094},
eprint = {2607.00094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00094},
}
RIS
TY - UNPB
TI - Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain
AU - Tokuhiro Eto
AU - Jiwoong Jang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2607.00094
UR - https://arxiv.org/abs/2607.00094
AB - 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 Lipschitz-continuous 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.
ER -