Capillary curvature images

TL;DR

This paper addresses the capillary $L_p$-Minkowski problem for specific parameter ranges, introducing an iterative approach that extends solutions from the classical case and uses monotonicity properties of specialized operators.

Contribution

It presents a novel iterative scheme for solving the capillary $L_p$-Minkowski problem for $-n < p < 1$, expanding the understanding of capillary curvature images.

Findings

Established existence of solutions for the problem in the specified range.

Developed a new class of capillary curvature image operators.

Proved monotonicity properties leading to solution fixed points.

Abstract

In this paper, we solve the even capillary $L_p$-Minkowski problem for the range $-n < p < 1$ and $\theta \in (0,\frac{\pi}{2})$. Our approach is based on an iterative scheme that builds on the solution to the capillary Minkowski problem (i.e., the case $p = 1$) and leverages the monotonicity of a class of functionals under a family of capillary curvature image operators. These operators are constructed so that their fixed points, whenever they exist, correspond precisely to solutions of the capillary $L_p$-Minkowski problem.