Moon hypersurfaces and some related existence results of capillary hypersurfaces without gravity and of rotational symmetry

Fei-Tsen Liang

Vol. 172 (1996), No. 2, 433–460

DOI: 10.2140/pjm.1996.172.433

Abstract

Let Ω_∗(R) be a domain in ℝⁿ bounded by two spherical caps Σ₁ and Σ₂ of respective radii and R, with < R < 1. (cf. Figure 1 for n = 3). We consider the vertical cylinder Z over ∂Ω_∗(R) and seek a hypersurface u_R(x₁,…,x_n) over Ω_∗(R) of constant mean curvature H ≡ 1 which meets Z in the angle π (vertically downward) over Σ₁(R) and the angle 0 (vertically upward) over Σ₂(R); intuitively and essentially, this amounts to seeking a solution to the problem

(1)

ν being outward unit normal.

Mathematical Subject Classification 2000

Primary: 58E12

Secondary: 35J60, 53A10

Milestones

Received: 20 August 1993

Revised: 7 December 1993

Published: 1 February 1996