The Anisotropic Capillary $L_p$-Minkowski Problem

TL;DR

This paper develops a new geometric framework for anisotropic capillary convex bodies, introduces an anisotropic capillary $L_p$-Minkowski problem, and provides solutions for $p \, \geq \, 1$ based on variation formulas and surface area measures.

Contribution

It introduces the anisotropic capillary $L_p$-Minkowski problem and establishes a theoretical foundation for anisotropic capillary convex bodies and their associated measures.

Findings

Defined the anisotropic $ ext{capillary } p$-sum for hypersurfaces.

Computed variations of anisotropic capillary $k$-th quermassintegrals.

Formulated and solved the anisotropic capillary $L_p$-Minkowski problem for $p \geq 1$.

Abstract

This paper introduces the \textit{anisotropic $\omega_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{R}_+^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $\Sigma $ with anisotropic $\omega_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $\omega_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape $\mathcal{C}_{\omega_{0}}$. This motivates us to propose and solve the anisotropic capillary $L_{p}$-Minkowski problem for $p\geq1$.