The Anisotropic Capillary $L_p$-Minkowski Problem

Abstract

This paper introduces the \textit{anisotropic $\ddot{I}{‰}_0$-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 $\hat{I}\pounds$ with anisotropic $\ddot{I}{‰}_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $\ddot{I}{‰}_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape ${\mathcal{C}}_{\ddot{I}{‰}_0}$. This motivates us to propose and solve the anisotropic capillary $L_p$-Minkowski problem for $p\ge 1$.

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