Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.