The canonical Green kernel of a plane domain

Let $\mathrm{\Omega }\subset \mathbb{C}$ be a proper plane domain, that is, a nonempty connected open set with $\mathrm{\Omega }\mathrm{\ne }\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$), and let $a\in \mathrm{\Omega }$. Moduli are those of Real and imaginary parts, complex conjugation, and modulus, and harmonicity is that of Plane harmonic functions.

Logarithmic-pole candidates. A function $u:\mathrm{\Omega }\setminus \{a\}\to \mathbb{R}$ is a logarithmic-pole candidate at $a$ when:

1. $u(z)\ge 0$ for every $z\in \mathrm{\Omega }\setminus \{a\}$;
2. $u$ is harmonic on $\mathrm{\Omega }\setminus \{a\}$;
3. $u+\log \mathrm{\mid }z-a\mathrm{\mid }$ extends harmonically across $a$: there is a harmonic function $h$ on $\mathrm{\Omega }$ with $h(z)=u(z)+\log \mathrm{\mid }z-a\mathrm{\mid }(z\in \mathrm{\Omega }\setminus \{a\})\mathrm{.}$

The function $h$ in clause 3 is the harmonic corrector of $u$ at $a$; it is unique when it exists, because two harmonic functions on the connected open set $\mathrm{\Omega }$ that agree on the nonempty open set $\mathrm{\Omega }\setminus \{a\}$ agree on $\mathrm{\Omega }$.

Order. Candidates at $a$ are compared pointwise on $\mathrm{\Omega }\setminus \{a\}$.

Canonical Green function. Suppose the family of candidates at $a$ is nonempty. Its canonical Green function is its pointwise least member, when such a member exists: a candidate $g$ with $g\le u$ pointwise on $\mathrm{\Omega }\setminus \{a\}$ for every candidate $u$. A pointwise least member is unique, and it is written $z\mapsto g_{\mathrm{\Omega }}(z,a)$. If the family is empty, or if it is nonempty but has no pointwise least member, then $g_{\mathrm{\Omega }}(\cdot ,a)$ is not defined by this definition. The domain $\mathrm{\Omega }$ is called Greenian when $g_{\mathrm{\Omega }}(\cdot ,a)$ exists for every $a\in \mathrm{\Omega }$.

The coefficient of $\log \mathrm{\mid }z-a\mathrm{\mid }$ is fixed to equal exactly one, so for the canonical kernel the corrector $h_a:=g_{\mathrm{\Omega }}(\cdot ,a)+\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ is harmonic on all of $\mathrm{\Omega }$ and finite at $a$.

• Leastness is a genuine restriction. On the punctured disc $\mathrm{\Omega }=\mathbb{D}\setminus \{0\}$ with $a\mathrm{\ne }0$, if $g_{\mathrm{\Omega }}(\cdot ,a)$ exists then for every $t>0$ the function $z\mapsto g_{\mathrm{\Omega }}(z,a)+t\log (1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$ is another logarithmic-pole candidate at $a$. The added term is nonnegative on $\mathrm{\Omega }$ and harmonic there by Logarithmic modulus is harmonic off its centre, so the corrector still extends harmonically across $a\mathrm{\ne }0$. The new candidate is strictly larger on $\mathrm{\Omega }$. Thus the candidate clauses alone do not designate a unique function on this domain; the pointwise least-member clause does.

• Promised boundary behaviour. On a bounded domain the canonical kernel has zero limit at every regular boundary point, in the sense of Barriers and regular boundary points, and no pointwise limit is required or asserted at an irregular boundary point. Both statements are proved later on this page together with the existence theorem; they are not part of the definition and are not assumed here.

• Normalization relative to the PDE kernel. The published kernel $\mathrm{\Phi }(w)=-(2\pi {)}^{-1}\log \mathrm{\mid }w\mathrm{\mid }$ of Fundamental solution for the positive operator minus Laplacian satisfies $2\pi \mathrm{\Phi }(w)=-\log \mathrm{\mid }w\mathrm{\mid }$, so twice-$\pi$ times a Dirichlet Green function of that PDE page is a logarithmic-pole candidate with logarithmic coefficient one whenever the PDE correctors exist. The identification of the two normalizations, and the distributional identity $-{\mathrm{\Delta }}_z{g}_{\mathrm{\Omega }}(z,a)=2\pi {\delta }_a$, are not part of this definition: they are proved later on this page from that supplier.

• Properness. The setting is a proper domain, $\mathrm{\Omega }\mathrm{\ne }\mathbb{C}$; nothing below asserts the existence of a canonical kernel on the whole plane.