Symmetry of the Dirichlet Green function

Fix distinct $x,y\in \mathrm{\Omega }$. Openness gives radii $r_x,r_y>0$ with ${\bar{B}}_2(x,r_x),{\bar{B}}_2(y,r_y)\subset \mathrm{\Omega }$. For $0<\epsilon <\min \{r_x\mathrm{/}4,r_y\mathrm{/}4,\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}5\}$ put $D_{\epsilon }=\mathrm{\Omega }\setminus ({\bar{B}}_2(x,\epsilon )\cup {\bar{B}}_2(y,\epsilon ))$. The balls are disjoint and lie strictly inside $\mathrm{\Omega }$. The point $x+2\epsilon e_1$ lies in $D_{\epsilon }$, so it is nonempty; it is bounded and open. Its boundary is the disjoint union of $\mathrm{\partial }\mathrm{\Omega }$ and the two spheres. The outer boundary remains locally a $C^1$ graph; each sphere is a regular level set and the positive separations prevent any boundary intersections. Thus $D_{\epsilon }$ is a bounded $C^1$ domain under [F5] even if it is disconnected. On either spherical boundary, with $\theta =(z-a)\mathrm{/}\epsilon$ for its centre $a$, the outward normal of $D_{\epsilon }$ is $-\theta$.

On $D_{\epsilon }$ set $u(z)=G_{\mathrm{\Omega }}(z,x)$ and $v(z)=G_{\mathrm{\Omega }}(z,y)$. By [F1] and [F14], both are harmonic there. Their $C^2(\bar{\mathrm{\Omega }})$ correctors and the smooth-kernel fact [F15] show $u,v\in C^2(\overline{D_{\epsilon }})$. Apply [F3]. The volume integral is zero; on $\mathrm{\partial }\mathrm{\Omega }$ both functions vanish, so the outer boundary integral is zero. Therefore $0=I_x(\epsilon )+I_y(\epsilon )$, where $I_a(\epsilon ):={\int }_{S_2(a,\epsilon )}(v{\mathrm{\partial }}_{\nu }u-u{\mathrm{\partial }}_{\nu }v)dS$ and each normal is outward from $D_{\epsilon }$.

By [F2] and [F9], for $s>0$ the radial derivative is ${\mathrm{\Phi }}_n^{\mathrm{\prime }}(s)=-1\mathrm{/}({\omega }_{n-1}{s}^{n-1})$ when $n\ge 3$. For $n=2$ the same formula follows from [F9] and [F11]. The direction of the hole normal is $-\theta$ by step 1.1, and [F7]–[F8] identify the normal derivative with differentiation in that direction. Thus on a hole sphere, ${\mathrm{\partial }}_{\nu }\mathrm{\Phi }(z-a)=-{\mathrm{\Phi }}_n^{\mathrm{\prime }}(\epsilon )=1\mathrm{/}({\omega }_{n-1}{\epsilon }^{n-1})$. By [F4] and [F13], its sphere area is ${\omega }_{n-1}{\epsilon }^{n-1}$; the singular normal derivative consequently has integral $1$. The compact-sphere hypothesis for [F13] follows from [F16]. The corrector and its first derivatives are bounded near each pole by their continuity.

On $S_2(x,\epsilon )$, [F1] and [F14] make $v$ and $\mathrm{\nabla }v$ continuous across $x$, so $v(z)\to v(x)$ uniformly there and ${\mathrm{\partial }}_{\nu }v$ is bounded. The singular profile and local boundedness of $H_x$ give $\mathrm{\mid }u(z)\mathrm{\mid }\le C{\epsilon }^{2-n}$ for $n\ge 3$, and $\mathrm{\mid }u(z)\mathrm{\mid }\le C(1+\mathrm{\mid }\log \epsilon \mathrm{\mid })$ for $n=2$. By [F12] and the sphere-area formula in [F4], $\mid {\int }_{S_2(x,\epsilon )}u{\mathrm{\partial }}_{\nu }vdS\mid \le \sup \mathrm{\mid }{\mathrm{\partial }}_{\nu }v\mathrm{\mid }\sup \mathrm{\mid }u\mathrm{\mid }{\omega }_{n-1}{\epsilon }^{n-1}$, which is $O(\epsilon )$ for $n\ge 3$ and $O(\epsilon (1+\mathrm{\mid }\log \epsilon \mathrm{\mid }))$ for $n=2$. Also [F7]–[F8] and step 2.2 give ${\mathrm{\partial }}_{\nu }u=1\mathrm{/}({\omega }_{n-1}{\epsilon }^{n-1})-{\mathrm{\partial }}_{\nu }{H}_x$. Since $v$ and $D{H}_x$ are bounded near $x$, [F12] and [F4] bound the corrector contribution by $O({\epsilon }^{n-1})$. The remaining term is the surface average of $v$: its difference from $v(x)$ is at most ${\sup }_{S_2(x,\epsilon )}\mathrm{\mid }v-v(x)\mathrm{\mid }$, which tends to zero by continuity and the area formula [F4]. Finally, for $0<\epsilon <1$, [F10] with $t=1\mathrm{/}\epsilon$ gives $\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid }=\log (t)\mathrm{/}t\to 0$. Therefore $I_x(\epsilon )\to v(x)=G_{\mathrm{\Omega }}(x,y)$.

At $y$, the same estimates as in step 3.1 with the pole roles reversed give ${\int }_{S_2(y,\epsilon )}v{\mathrm{\partial }}_{\nu }udS\to 0$ and ${\int }_{S_2(y,\epsilon )}u{\mathrm{\partial }}_{\nu }vdS\to u(y)$. Hence $I_y(\epsilon )\to -u(y)=-G_{\mathrm{\Omega }}(y,x)$.

Taking limits in the identity from step 2.1 using the two limits in steps 3.1 and 4.1 gives $G_{\mathrm{\Omega }}(x,y)-G_{\mathrm{\Omega }}(y,x)=0$. The estimates cover the logarithmic case $n=2$ and the power cases $n\ge 3$; the theorem excludes $n=1$ and the coincident poles because its Green function is defined only for distinct poles. Countable Choice is inherited through [A1] and the cited kernel and surface-measure conventions; radius choices are pointwise, and no full Axiom of Choice is used. There is no iff claim.