Prescribed L_p curvature problem for convex capillary hypersurface

2025/12/18 by Mei, Xinqun, Wang, Guofang, Weng, Liangjun
Mathematics · #52A39 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Primary: 58J05. Secondary: 52A20

paper · pdf · doi:10.48550/arxiv.2512.16686

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

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.

Citations

• Capillary L_p Minkowski Flows
• The capillary Gauss curvature flow
• Capillary curvature images
• The capillary L_p-Minkowski problem
• Convex capillary hypersurfaces of prescribed curvature problem
• Capillary Christoffel-Minkowski problem
• Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II
• Prescribed L_p quotient curvature problem and related eigenvalue problem
• Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces
• An eigenvalue problem for prescribed curvature equations
• Heintze-Karcher inequality and capillary hypersurfaces in a wedge
• Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
• The Neumann problem of Hessian quotient equations
• L^p Christoffel-Minkowski problem: the case 1< p
• Asymptotic behavior of flows by powers of the Gaussian curvature
• Uniqueness of closed self-similar solutions to the Gauss curvature flow
• A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions
• Global C2‐Estimates for Convex Solutions of Curvature Equations
• Convexity estimates for mean curvature flow and singularities of mean convex surfaces

Related

• Convex capillary hypersurfaces of prescribed curvature problem
• The L_p dual Minkowski problem for capillary hypersurfaces
• Capillary Christoffel-Minkowski problem
• The capillary L_p-Minkowski problem
• The capillary Christoffel-Minkowski problem