  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.
