The capillary Minkowski problem

Abstract

In this article, we introduce a capillary Minkowski problem, which asks for the existence of a strictly convex capillary hypersurface $\Sigma \subset \overline{R_{+}^{n+1}}$ supported on $\overline{R_{+}^{n+1}}$ with a prescribed Gauss-Kronecker curvature on a spherical cap $C_{\theta }$. We reduce it to a Monge-Ampère type equation with a Robin boundary value problem and then obtain a necessary and sufficient condition for solving this problem provided $\theta \in (0,\frac{\pi }{2}]$. The restriction $\theta \in (0,\frac{\pi }{2}]$ comes from the difficult part of the proof, $C^2$-estimation. We manage to prove $C^2$-estimates by using this restriction and leave the problem open if $\theta >\frac{\pi }{2}$. This is a natural Robin boundary version of the classical Minkowski problem.