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

Abstract

We address the prescribed $L_p$ curvature problem for convex capillary hypersurfaces in the Euclidean half-space. By reducing it to a convex solution of a Hessian quotient equation on a spherical cap with a Robin boundary condition, we establish the existence and uniqueness of smooth admissible, and indeed strictly convex, solutions. In particular, we solve the capillary $L_p$ Christoffel--Minkowski problem for $p\geq 1$ in the smooth category, providing a natural Robin boundary counterpart of the classical $L_p$ Christoffel--Minkowski problem of Hu--Ma--Shen [25] and Guan--Xia [24]. We further obtain analogous existence and uniqueness results for the prescribed $L_p$ curvature problem and the associated eigenvalue problem for convex capillary hypersurfaces in the Euclidean half-space. These results form a capstone to our series of works [48,49,51].