Convex capillary hypersurfaces of prescribed curvature problem

TL;DR

The paper addresses the problem of prescribing the $k$-th Weingarten curvature on strictly convex capillary hypersurfaces in the upper half-space, recasting it as a Hessian quotient equation with Robin boundary on a spherical cap. It proves existence for capillary-even data when $\theta\in(0,\frac{\pi}{2}]$, with uniqueness for $f$ close to $1$, and shows that a natural necessary condition is not sufficient in general for $1\le k\le n-1$. The key methodological contributions include a capillary-adapted a priori estimate framework (C^0, C^1, C^2) built via a capillary support function and a Chou-Wang-type lemma, and a degree-theoretic approach combined with the inverse function theorem. The results extend capillary Minkowski-type problems to the broad family $1\le k\le n-1$, providing a robust PDE-geometry bridge for capillary curvature problems with Robin boundary data.

Abstract

In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed $k$-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40].

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof