The Anisotropic Capillary $L_p$-Minkowski Problem

This paper introduces the $\mathit{\text{anisotropic}}{\omega }_0\mathit{\text{-capillary}}p\mathit{\text{-sum}}$ of two hypersurfaces in ${\mathbb{R}}_{+}^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $\mathrm{\Sigma }$ 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 $p\ge 1$.