The capillary Christoffel-Minkowski problem

TL;DR

This paper introduces a new capillary area measure for convex bodies in half-space and formulates a Christoffel-Minkowski problem, proving existence and uniqueness of solutions under certain conditions.

Contribution

It defines a novel capillary area measure and formulates a related geometric problem, providing existence and uniqueness results for its solutions.

Findings

Defined a $k$-th capillary area measure for convex bodies in half-space.

Formulated a Christoffel-Minkowski problem for these bodies.

Proved existence and uniqueness of smooth solutions under natural conditions.

Abstract

In this article, we introduce a $k$-th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed $k$-th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition.