The Christoffel problem by fundamental solution of the Laplace equation

2019/11/24 by Qirui Li, Li, Qi-Rui, Dongrui Wan +3 · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1911.10507

openalex publication_date 2019/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere Sⁿ. Necessary and sufficient conditions have been found by Firey and Berg, using the Green function of the Laplacian on the sphere. Expressing the Christoffel problem as the Laplace equation on the entire space Rⁿ⁺¹, we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equations. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the L_p extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p≥ 2.

Cited by

• The capillary Christoffel-Minkowski problem
• A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem
• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• The Christoffel problem for the disk area measure

Related

• Smooth solutions to the Christoffel problem in ℍⁿ⁺¹
• A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem
• The L_p-dual Christoffel-Minkowski problem for the case p≥ q
• L^p Christoffel-Minkowski problem: the case 1< p
• The Christoffel problem and two analogs of the Minkowski problem in Riemannian space