Canonical Green kernels are unique, symmetric and domain monotone

By [F1] the canonical Green function on a Greenian $\mathrm{\Omega }$ is defined as the pointwise least member of the family of logarithmic-pole candidates at $a$, and the definition records that a pointwise least member is unique; hence whenever it exists it is unique, as claimed in clause 1.

Domain monotonicity. Let ${\mathrm{\Omega }}_1\subseteq {\mathrm{\Omega }}_2$ be Greenian plane domains and let $a,z\in {\mathrm{\Omega }}_1$ be distinct. The restriction of $g_{{\mathrm{\Omega }}_2}(\cdot ,a)$ to ${\mathrm{\Omega }}_1\setminus \{a\}$ is a logarithmic-pole candidate at $a$ on ${\mathrm{\Omega }}_1$: it is nonnegative, it is harmonic on ${\mathrm{\Omega }}_1\setminus \{a\}\subseteq {\mathrm{\Omega }}_2\setminus \{a\}$, and the corrector $g_{{\mathrm{\Omega }}_2}(\cdot ,a)+\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ agrees with a harmonic function on ${\mathrm{\Omega }}_2$ by [F1] whose restriction to ${\mathrm{\Omega }}_1$ is harmonic, so the sum extends harmonically across $a$ inside ${\mathrm{\Omega }}_1$. Leastness of $g_{{\mathrm{\Omega }}_1}(\cdot ,a)$ gives $g_{{\mathrm{\Omega }}_1}(z,a)\le g_{{\mathrm{\Omega }}_2}(z,a)$, which is clause 3.

Symmetry setup. Assume Countable Choice and fix distinct $x,y\in \mathrm{\Omega }$. Apply [F3] with the finite set $A=\{x,y\}$: there are relatively compact connected open subsets $D_1\subseteq D_2\subseteq \cdots$ of $\mathrm{\Omega }$ with $\{x,y\}\subseteq D_1$, real-analytic regular one-sided boundaries, and every compact subset of $\mathrm{\Omega }$ contained in $D_j$ for all large $j$.

For each $j$, $D_j$ is a bounded complex domain, its boundary is a compact real-analytic curve in the sense of [F4], and $D_j$ is a bounded $C^1$ domain in the sense of [F7]. Indeed $D_j$ is nonempty, open, connected and relatively compact, hence bounded; for $\zeta \in \mathrm{\partial }{D}_j$ the regularity of [F3] provides $U$ and a real-analytic $g$ with $\mathrm{\partial }{D}_j\cap U=\{(x,y)\in U:y=g(x)\}$ and $D_j\cap U$ equal to one of the two components of $U\setminus \{(x,y)\in U:y=g(x)\}$. Parametrizing that graph, after translating the parameter, by $\gamma (t):=(t,g(t))$ gives a real-analytic curve with ${\gamma }^{\mathrm{\prime }}(t)=(1,g^{\mathrm{\prime }}(t))\mathrm{\ne }0$ for which $D_j\cap U$ is one of the two components of $U\setminus \gamma$, so [F4] applies; and since $g$ is $C^1$ by [F15], after relabelling the axes and if necessary reflecting one of them the boundary is locally a $C^1$ graph with $D_j$ locally the corresponding subgraph, which is the structure required by [F7].

Fix a pole $p\in \{x,y\}$ and $z\in \mathrm{\Omega }\setminus \{p\}$. Since the sequence exhausts $\mathrm{\Omega }$ and $z$ is an interior point, $z\in D_j$ for all large $j$, and $\{p\}\subseteq D_1$; step 1.2 applied to the Greenian domains $D_j\subseteq D_{j+1}$ shows $g_{D_j}(z,p)\le g_{D_{j+1}}(z,p)$, and applied to $D_j\subseteq \mathrm{\Omega }$ it shows $g_{D_j}(z,p)\le g_{\mathrm{\Omega }}(z,p)$, a finite bound by [F1]. Hence the limit $u_p(z):={\lim }_{j\to \mathrm{\infty }}{g}_{D_j}(z,p)$ exists and lies in $[0,g_{\mathrm{\Omega }}(z,p)]$.

For each $j$ and each pole $p\in \{x,y\}$ the canonical kernel $g_{D_j}(\cdot ,p)$ exists by [F2] because $D_j$ is a bounded complex domain, and [F4] applied to $D=D_j$ shows that the Perron corrector $h_p^{(j)}:=-\log \mathrm{\mid }\cdot -p\mathrm{\mid }-g_{D_j}(\cdot ,p)$ is harmonic on $D_j$, extends to $C^2(\overline{D_j})$, and that $g_{D_j}(\cdot ,p)=-\log \mathrm{\mid }\cdot -p\mathrm{\mid }-h_p^{(j)}$ extends to a $C^2$ function on $\overline{D_j}\setminus \{p\}$ with trace identically zero on $\mathrm{\partial }{D}_j$.

The limit $u_p$ is harmonic on $\mathrm{\Omega }\setminus \{p\}$. Let $W\subseteq \mathrm{\Omega }\setminus \{p\}$ be nonempty, open and relatively compact; by [F3] there is $j_0$ with $\bar{W}\subseteq D_{j_0}$, so $(g_{D_{j_0+n}}(\cdot ,p){\mathrm{\mid }}_W{)}_{n\ge 0}$ is an increasing sequence of harmonic functions on $W$ bounded above by $g_{\mathrm{\Omega }}(\cdot ,p)$, which is finite by [F1]. The first alternative of [F9] is therefore impossible and the second applies: the limit is harmonic on $W$ and the convergence is locally uniform there. As $W$ is arbitrary, $u_p$ is harmonic on $\mathrm{\Omega }\setminus \{p\}$.

For each $j$, $G_j:=g_{D_j}\mathrm{/}(2\pi )$ with correctors $H_p^{(j)}:=h_p^{(j)}\mathrm{/}(2\pi )$ is a Dirichlet Green function for $-\mathrm{\Delta }$ on $D_j$ in the sense of [F6] with designated $C^2(\overline{D_j})$ correctors. Correctors: for $z\in D_j\setminus \{p\}$, $\mathrm{\Phi }(z-p)-H_p^{(j)}(z)=-\frac{1}{2\pi }\log \mathrm{\mid }z-p\mathrm{\mid }-\frac{1}{2\pi }{h}_p^{(j)}(z)=\frac{1}{2\pi }(-\log \mathrm{\mid }z-p\mathrm{\mid }-h_p^{(j)}(z))=\frac{1}{2\pi }{g}_{D_j}(z,p)=G_j(z,p)$ by [F8] and step 3.1, and $H_p^{(j)}$ is harmonic with $H_p^{(j)}=\mathrm{\Phi }(\cdot -p)$ on $\mathrm{\partial }{D}_j$ because $g_{D_j}(\cdot ,p)$ has zero boundary trace; harmonicity off the pole and the zero trace of $G_j(\cdot ,p)$ are step 3.1. Dirac identity: by [F2] and step 2.1, $-\mathrm{\Delta }{T}_{g_{D_j}(\cdot ,p)}=2\pi {\delta }_p$ in ${\mathcal{D}}^{\mathrm{\prime }}(D_j)$, and linearity of the embedding and of distributional differentiation [F14] gives $-\mathrm{\Delta }{T}_{G_j(\cdot ,p)}=\frac{1}{2\pi }(-\mathrm{\Delta }{T}_{g_{D_j}(\cdot ,p)})={\delta }_p$.

The limit $u_p$ is a logarithmic-pole candidate at $p$ on $\mathrm{\Omega }$: it is nonnegative by step 2.2, harmonic by step 3.2, and ${\mathrm{\Phi }}_p:=u_p+\log \mathrm{\mid }\cdot -p\mathrm{\mid }$ is harmonic on $\mathrm{\Omega }\setminus \{p\}$ by [F12] and [F13]. Near $p$ the function ${\mathrm{\Phi }}_p$ is bounded: below, $u_p\ge g_{D_1}(\cdot ,p)$ by step 2.2, so ${\mathrm{\Phi }}_p\ge g_{D_1}(z,p)+\log \mathrm{\mid }z-p\mathrm{\mid }=-h_p^{(1)}(z)$ by step 3.1, and $h_p^{(1)}\in C^2(\overline{D_1})$ is bounded on a neighbourhood of $p$; above, $u_p\le g_{\mathrm{\Omega }}(\cdot ,p)$ by step 2.2, so ${\mathrm{\Phi }}_p\le g_{\mathrm{\Omega }}(z,p)+\log \mathrm{\mid }z-p\mathrm{\mid }$, and the right-hand side is harmonic on $\mathrm{\Omega }$, hence bounded on a neighbourhood of $p$ by [F13]. Thus ${\mathrm{\Phi }}_p$ is harmonic and bounded on a punctured disc about $p$, and [F10] extends it harmonically across $p$; so $u_p+\log \mathrm{\mid }\cdot -p\mathrm{\mid }$ extends harmonically to $\mathrm{\Omega }$ and $u_p$ is a candidate in the sense of [F1].

Symmetry on the exhaustion domains. [F5] applies to the bounded $C^1$ domain $D_j$ of step 2.1, to the Dirichlet Green function $G_j$ of step 4.1 and to its correctors $H_p^{(j)}\in C^2(\overline{D_j})$: hence $G_j(u,v)=G_j(v,u)$ for all distinct $u,v\in D_j$, and multiplying by $2\pi$, $g_{D_j}(u,v)=g_{D_j}(v,u)$.

Identification of the limit. Leastness of $g_{\mathrm{\Omega }}(\cdot ,p)$ among the candidates on the Greenian domain $\mathrm{\Omega }$ [F1] gives $g_{\mathrm{\Omega }}(\cdot ,p)\le u_p$, while step 2.2 gives $u_p(z)\le g_{\mathrm{\Omega }}(z,p)$ for every $z\in \mathrm{\Omega }\setminus \{p\}$; hence $u_p=g_{\mathrm{\Omega }}(\cdot ,p)$, that is ${\lim }_j{g}_{D_j}(z,p)=g_{\mathrm{\Omega }}(z,p)$ for every $z\in \mathrm{\Omega }\setminus \{p\}$.

Symmetry. Step 5.1 gives $g_{D_j}(x,y)=g_{D_j}(y,x)$ for every $j$. Taking $j\to \mathrm{\infty }$ and applying step 5.2 with $p=y$ on the left and with $p=x$ on the right yields $g_{\mathrm{\Omega }}(x,y)=g_{\mathrm{\Omega }}(y,x)$, which is clause 2 for the given pair; as $x,y$ were arbitrary distinct points of $\mathrm{\Omega }$, symmetry holds throughout.

Choice accounting and scope. Countable Choice is used exactly through the bounded-domain existence theorem [F2], applied to each $D_j$ in step 3.1, and through the PDE Green symmetry theorem [F5] in step 5.1; the exhaustion [F3], the monotone bound of step 2.2, the Harnack limit of step 3.2, the removable-singularity step 4.2 and the comparison steps 1.1-1.2 and 5.2 use no choice principle. Steps 1.1-1.2, 6.1 establish the three clauses: 1.1 the uniqueness, 1.2 the domain monotonicity for arbitrary Greenian pairs, and 6.1 the symmetry for the Greenian $\mathrm{\Omega }$ fixed in step 1.3.