Capillary John ellipsoid theorem with applications to capillary curvature problems

Abstract

In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_{+}}$. This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining $C^{0}$ estimates for solutions to some capillary curvature problems (including the capillary $L_{p}$ Christoffel-Minkowski problem and the capillary $L_{p}$ curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary $L_{p}$ dual Minkowski problem. By deriving a gradient estimate, refining a $C^{2}$ estimate, and combining these with the non-collapsing estimate, we establish existence in the case $1<p\leq q\leq 3$ and improve upon the existing existence result for the case $p > q$ in $\overline{\mathbb{R}^3_{+}}$.