The capillary Orlicz-Minkowski problem

Abstract: In this paper, we introduce a Robin boundary analogue of the Orlicz-Minkowski problem, which seeks to find a capillary convex body with a prescribed capillary Orlicz surface area measure in the upper Euclidean half-space. We obtain the volume-normalized smooth solutions to the capillary even Orlicz-Minkowski problem by the continuity method. In addition, we also establish a capillary Orlicz-Brunn-Minkowski inequality and a capillary Orlicz-Minkowski inequality, which can change our solutions to a spherical cap under some conditions.