The capillary $L_p$-Minkowski problem

TL;DR

This paper introduces a capillary $L_p$-Minkowski problem in the Euclidean half-space, extending classical Minkowski problems with Robin boundary conditions, and solves it for smooth cases via a Monge-Ampère equation.

Contribution

It formulates a new capillary $L_p$-Minkowski problem for $p eq 1$ and provides a solution in the smooth category using PDE techniques.

Findings

Formulated the capillary $L_p$-Minkowski problem for $p eq 1$.

Reduced the problem to a Monge-Ampère equation with Robin boundary conditions.

Solved the problem in the smooth case.

Abstract

This paper is a continuation of our recent work [54] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\geq 1$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [43]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge-Amp\`ere equation with a Robin boundary condition on the unit spherical cap.