The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the...The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).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 R^(n+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.展开更多
基金supported by the One-Thousand-Young-Talents Program of Chinasupported by National Natural Science Foundation of China(Grant No.11871345)supported by Australian Research Council(Grant Nos.DP170100929 and DP200101084)。
文摘The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).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 R^(n+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.
