Collapsing and the convex hull property in a soap film capillarity model

肥皂膜中的Plateau边界与Gauss毛细理论

Plateau borders in soap films and Gauss' capillarity theory

本文在Gauss毛细理论框架下，通过测度论改造同伦张成条件，证明紧性、能量表示与极小元正则性，从而严格推导出湿肥皂膜及泡沫中Plateau边界的平衡定律。

Smoothness of collapsed regions in a capillarity model for soap films

We study generalized minimizers in the soap film capillarity model introduced in [arXiv:1807.05200,arXiv:1907.00551]. Collapsed regions of generalized minimizers are shown to be smooth outside of dimensionally small singular sets, which are thus empty in physical dimensions. Since generalized minimizers converge to Plateau's type surfaces in the vanishing volume limit, the fact that collapsed regions cannot exhibit $Y$-points and $T$-points (which are possibly present in the limit Plateau's surfaces) gives the first strong indication that singularities of the limit Plateau's surfaces should al…

通过毛细作用求解平面 Plateau 问题

The planar Plateau's problem via capillarity

本文基于二维毛细作用表述重新审视 Plateau 问题，通过在变分长度极小化方案中引入非局部几何势来研究肥皂膜模型，从而纳入厚度影响并为解释塌缩现象等物理现象提供新见解。

毛细理论中的润湿定理

Wetting theorems in capillarity theory

该研究在毛细理论框架下，证明了肥皂膜模型体积趋于零时Plateau边界一致收敛到Plateau奇点，相关结果基于考虑肥皂膜局部空间分隔特性的新型局部面积极小化器得出。

Plateau's problem as a singular limit of capillarity problems

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection principle for Plateau's problem, points at physical features of soap films that are unaccessible by simply looking at minimal surfaces, and opens several challenging questions.

平面边界的肥皂膜毛细问题的经典解

Classical solutions to the soap film capillarity problem for plane boundaries

本文研究肥皂膜毛细问题，针对平面边界（光滑Jordan曲线或超平面嵌入的闭管状邻域），证明了经典极小元的存在性与唯一性，其边界为两个对称光滑法向图，并具有常平均曲率，且解随体积单调有序，形成空间叶状结构。