Let X denote a bounded, open,
and connected subset of Rn+1(n ≥ 1) which we consider to represent the interior of a
liquid drop (when n = 2). The principal result of this paper will be to show that
under suitable conditions X is an axially symmetric drop in the sense that there is a
vertical line (axis) such that any nonempty intersection of X with a horizontal
hyperplane is an open disk whose center lies on the axis. Condition 1: X adheres to a
horizontal hyperplane, Σ (i.e., X ∩ Σ = Φ but X∩ Σ≠Φ), with the mean curvature,
H, of the liquid-air interface, Ω, a differentiable function of the vertical coordinate
and the angle of contact, α, of Ω with Σ a constant along ∂Ω, 0 ≤ α ≤ π,
(Theorem 1. 1). Condition 2: X adheres to Σ with the mean curvature a smooth
function of height and the contact region of X with Σ a disk (special case of
Theorem 1. 2).
