Embeddedness of liquid-vapour interfaces in stable equilibrium

Abstract

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.