The $L_p$ dual Minkowski problem for capillary hypersurfaces
Ya Gao
TL;DR
The paper addresses the capillary $L_p$ dual Minkowski problem in the Euclidean half-space, focusing on the case $p>q$. It reduces the geometric problem to a Monge–Ampère type equation with Robin boundary on the spherical cap $C_{\theta}$ and formulates data in terms of the capillary support function $h$. A full suite of a priori estimates ($C^0$, gradient, and $C^2$) are established, and the continuity method yields existence and uniqueness of a smooth solution for every positive smooth data $f$ on $C_{\theta}$ when $\theta \in (0, \pi/2)$. This extends capillary Minkowski theory to the dual $L_p$ setting and provides a rigorous PDE framework with sharp boundary control for capillary convex bodies.
Abstract
In this paper, we consider the $L_p$ dual Minkowski problem for capillary hypersurfaces for $p>q$, which aims to find a capillary convex body with a prescribed capillary $(p,q)$-the dual curvature measure in the Euclidean half-space. We reduce it to a Monge-Ampère type equation with a Robin boundary condition on the unit spherical cap, and prove that there exists a unique smooth solution that solves this problem provided $θ\in (0,\fracπ{2})$.