Remarks on the Metric Induced by the Robin Function Ⅱ

摘要

Let D be a C~(∞)-smoothly bounded domain in C~n (n ≥ 2). For p∈D, let G(z,p) be the Green function for D with pole at p associated to the standard Laplacian Δ=4n∑i=1(6)~2/(6)_(zi)(6)_(zi) on C~n≈R~(2n). Then G(z, p) is the unique function of z ∈ D satisfying the conditions that G(z,p) is harmonic on D{p}, G(z,p) → 0 as z→ (6)D, and G(z, p) - ∣z - p∣~(-2n+2) exists and is called the Robin constant for D at p. The function Λ: p→Λ(p) is called the Robin function for D.