A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

2025/12/09 by Colesanti, A., Focardi, M., Guan, P. +1
Mathematics · #35J15 #52A20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.2512.08670

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

We study the mixed Christoffel problem for C^2,+ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \citeGuan-Ma-2003 to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a C^2,+ convex body.

Citations

• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• The Christoffel problem for the disk area measure
• Mixed Christoffel-Minkowski problems for bodies of revolution
• Minkowski Problems for Geometric Measures
• The Christoffel problem by fundamental solution of the Laplace equation

Cited by

• The Christoffel problem for the disk area measure

Related

• The Christoffel problem by fundamental solution of the Laplace equation
• Smooth solutions to the Christoffel problem in ℍⁿ⁺¹
• The L_p-dual Christoffel-Minkowski problem for the case p≥ q
• A constant rank theorem for partial convex solutions of partial differential equations
• Capillary Christoffel-Minkowski problem