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

TL;DR

This paper addresses the prescribed Lp curvature problem for convex capillary hypersurfaces in Euclidean half-space, establishing existence and uniqueness of smooth solutions and applying results to related geometric problems.

Contribution

It reduces the problem to a Hessian quotient equation with Robin boundary conditions and proves the existence and uniqueness of smooth convex solutions, advancing the understanding of capillary hypersurfaces.

Findings

Established existence and uniqueness of smooth convex solutions.

Solved the capillary Lp Christoffel Minkowski problem in the smooth category.

Derived results for the prescribed Lp curvature and eigenvalue problems.

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.