Comparison of the Pluricomplex and the Classical Green Functions on Convex Domains of Finite Type

摘要

Let D be a bounded domain with Lipschitz boundary in R~n, and let y be a fixed point in D. Then there is a solution H_y(x) to the Dirichlet problem {Δu(x) = 0 in D, u(x) = -η(x-y) on partial derivD, where η(x) = {logx if N = 2, -x~(2-N) if N ≥ 3. The function G_D(x,y) = η(x-y) + h_y(x) is called the classical (negative) Green function for the Laplacian, with pole at y. It is harmonic in D{y} and tends to zero on the boundary; furthermore, it is symmetric.