Green Functions

Abstract

Let D be a nonempty open subset of ℝ^N. If N > 2 or if N = 2 and D is bounded, the function G(ξ, ·) is lower bounded for each point ξ of D; so GM _D G(ξ, ·) exist (Section III.1.) If N = 2, if D is unbounded, and if G(ξ, ·) has a subharmonic minorant on D for some ξ in D, then the minorant GM _D G(ξ, ·) exists for every ξ in D. In fact G(ξ , ·)–G(ξ, ·) is bounded below outside each neighborhood of ξ, and G(ξ, ·) is bounded below on each compact neighborhood of ξ so that if GM _D G(ξ, ·) exists, G(ξ , ·) ≥ c + GM _D G(ξ, ·) GM _D G(ξ , ·) ≥ c + GM _D G(ξ, ·) for some constant c depending on ξ and ξ.