Moon hypersurfaces and some related existence results of capillary hypersurfaces without gravity and of rotational symmetry (Q1919910)

The author shows that if \({{n-1} \over n}< R< 1\), \(n\geq 2\), there is a ``moon domain'' \(\Omega_* (R)\in \mathbb{R}^n\) bounded by two spherical caps, \(\sigma_1\) of radius \({{n-1} \over n}\) and \(\Sigma_2\) of radius \(R\), and satisfying the condition \(|\Sigma_2|- |\Sigma_1 |= n|\Omega_* |\), and she proves the existence of a solution \(u(x)\), in a variational sense, of the singular ``capillary'' problem \[ \text{div } Tu=n \quad \text{in }\Omega_*, \qquad Tu= {{Du} \over {\sqrt {1+|Du|^2}}}, \] such that \(\nu\cdot Tu=-1\) on \(\Sigma_1\) and \(\nu\cdot Tu= +1\) on \(\Sigma_2\), \(\nu\)= unit exterior normal. The solution is uniquely determined up to an additive constant. For \(n=2\) this result had been proved by the reviewer; \(u(x)\) then has the physical significance of a capillary surface over \(\Omega_*\) in the absence of gravity, which meets vertical walls over \(\Sigma_1\) and \(\Sigma_2\) in the respective angles \(\pi\) and 0. That surface can be used when \(n=2\) as a majorant for gradient, leading to a conceptually simplified proof of a gradient bound, due originally to Giusti and the reviewer, for any graph \(u(x)\) of constant mean curvature that covers a sufficiently large disk. It was used in a very effective way by the author [Indiana Univ. Math. J. 41, No. 3, 569-604 (1992; Zbl 0837.53049)] in characterizing the largest disks over which such an a priori bound is valid. When \(n>2\) it is not known whether \(u(x)\) can be used as a majorant in those senses. However, from the author's procedure for constructing the surface as a generalized BV minimizer for the corresponding energy functional, in a sense introduced by M. Miranda, conditions can be inferred for which no such gradient bound can hold. The existence proof is technically much more difficult than in the two-dimensional case, as the boundaries of the extremal sets when \(n=2\) can be shown to consist of circular arcs of known radius, whereas for \(n>2\) they are known only to be surfaces of given mean curvature. The author overcomes the difficulties by making essential use of the symmetries in the boundary configuration, and showing that minimizing configurations must have corresponding symmetry.