Capillary John ellipsoid theorem with applications to capillary curvature problems

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 $1<p\le q\le 3$ and improve upon the existing existence result for the case $p>q$ in $\overline{{\mathbb{R}}_{+}^3}$.