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. Using the support function, we formulate the problem as a Hessian quotient equation on a spherical cap with a Robin boundary condition. For $p\ge 1$, we establish existence results under suitable assumptions on the prescribed data, together with uniqueness in the corresponding parameter ranges. At the critical exponent, we obtain an associated eigenvalue problem with uniqueness up to dilation. In the subcritical range, existence is established for capillary even data. As applications, we obtain existence results for the capillary $L_p$ Christoffel--Minkowski problem and prescribed $L_p$ quotient curvature problems, providing a framework for studying these equations in the presence of a capillary boundary.

arXiv comment55 pages