Comparison of the Pluricomplex and the Classical Green Functions

摘要

We define the fundamental solution for the Laplacian in R~N as formula (ell) Let Ω be a bounded domain in R~N with Lipschitz boundary, and fix y ∈ Ω. Then Ω is regular for the Dirichlet problem Formula (ell) That is, there is a function h_y(x), continuous on Ω, that solves this problem. Define G(x, y) = p(x-y)+h_y(x). This is the classical Green function for the Laplacian, with pole at y. It is negative And subharmonic in Ω, harmonic in Ω{y}, and tends to zero on aΩ. Near y, it Behaves like p(x-y). Furthermore, it is symmetric, that is G(y, x) = G(x, y).