光滑有界域中接触角平均曲率流的极小化移动格式的收敛性

中文摘要 AI 辅助

本文研究光滑有界域中具有给定接触角的平均曲率流的Chambolle型极小化移动格式。该格式基于毛细管泛函和相对于容器的测地符号距离，并产生时间离散水平集近似。主要结果断言：对于每个严格非退化接触角的$C^1$边界函数，近似解局部一致收敛到具有斜导数边界条件的对应水平集平均曲率方程的唯一粘性解。这改进了先前的收敛定理，其中假设容器是凸的，并施加了将给定接触角函数的切向导数与容器边界的主曲率相关联的曲率型条件。主要新内容是定义该格式的变分问题解的一致Lipschitz估计。该估计是通过将Bernstein型论证应用于适当的加权梯度（而非梯度本身）推导得出的，从而在不依赖先前曲率型条件的情况下排除了边界最大值。

英文摘要

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.

发表机构

• Université Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、中央里昂学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）
• Department of Mathematics, University of Maryland-College Park（马里兰大学帕克分校数学系）

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补充信息

备注

27 pages; the acknowledgment by T. E. has been added