Embeddedness of liquid-vapour interfaces in stable equilibrium

TL;DR

This paper establishes that under minimal assumptions, the liquid-vapour interface in stable equilibrium is a smoothly embedded analytic surface, based on advanced varifold regularity theory.

Contribution

It introduces the weakest possible assumptions for stable equilibrium of capillary surfaces and proves their smoothness and analyticity.

Findings

The liquid-vapour interface is a smooth analytic surface under minimal stability assumptions.

The approach relies on recent varifold regularity theory by Wickramasekera and the author.

The model accommodates contact with solid supports and external potential fields.

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.