Embeddedness of liquid-vapour interfaces in stable equilibrium
Embeddedness of liquid-vapour interfaces in stable equilibrium
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DOI：
10.4171/ifb/490
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发表时间：
2021-04
影响因子：
1
通讯作者：
C. Bellettini
中科院分区：
文献类型：
--
作者：
C. Bellettini
我们考虑一个经典的（毛细管）模型的单相液体平衡。液体（例如水）受到体积约束，它不与周围的蒸气（例如空气）混合，它可以与固体载体（例如容器）接触，并且受到分析势场（例如重力）的作用。液体所占据的区域被描述为$\mathbb{R}^3 $中的一组局部有限周长（Caccioppoli集）;在其边界上没有先验的正则性假设。本说明的（双重）范围是提出一组最弱的数学假设，这些假设合理地描述了液-汽界面（毛细管表面）的稳定平衡条件，并从这些假设中推断出该界面是一个光滑嵌入的分析表面。(The可能存在液-固-汽交界面或自由边界，但此处不进行分析。这个结果基本上依赖于最近由Wickramasekera和作者发展的变倍正则性理论，以及确定一个合适的稳定性条件的公式。
We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g. water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g. air), it may come into contact with solid supports (e.g. a container), and is subject to the action of an analytic potential field (e.g. gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in $\mathbb{R}^3$; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by Wickramasekera and the author, and on the identification of a suitable formulation of the stability condition.