The Christoffel problem by the fundamental solution of the Laplace equation

Qi Rui Li; Dongrui Wan; Xu Jia Wang

doi:10.1007/s11425-019-1674-1

The Christoffel problem by the fundamental solution of the Laplace equation

Qi Rui Li, Dongrui Wan, Xu Jia Wang^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

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 (1967) and Berg (1969), by 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 equation. 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.

Original language	English
Pages (from-to)	1599-1612
Number of pages	14
Journal	Science China Mathematics
Volume	64
Issue number	7
DOIs	https://doi.org/10.1007/s11425-019-1674-1
Publication status	Published - Jul 2021

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title = "The Christoffel problem by the fundamental solution of the Laplace equation",

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 (1967) and Berg (1969), by 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 equation. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p ⩾ 2.",

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AU - Wan, Dongrui

AU - Wang, Xu Jia

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N2 - 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 (1967) and Berg (1969), by 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 equation. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p ⩾ 2.

AB - 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 (1967) and Berg (1969), by 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 equation. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p ⩾ 2.

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JO - Science China Mathematics

JF - Science China Mathematics

IS - 7

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The Christoffel problem by the fundamental solution of the Laplace equation

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