Prescribed $L_{p}$ curvature problem for convex capillary hypersurface

Abstract

We study the prescribed Lp curvature problem for convex capillary hypersurfaces in the Euclidean half-space. By reducing the problem to finding a convex solution of a Hessian quotient type equation with a Robin boundary condition on a spherical cap, we establish the existence and uniqueness of smooth admissible (in fact, strictly convex) solutions. As applications, we solve the capillary Lp Christoffel Minkowski problem in the smooth category, and we also obtain corresponding results for the prescribed Lp curvature problem and the related eigenvalue problem for convex capillary hypersurfaces in the Euclidean half-space.