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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded connected [[open set|02BH3MCBQB]] that has a [[Green's function|M36SD56MWQ]] $G$ for the Laplacian with zero boundary values.\nThen\n$$G(x, y) = G(y, x) \\quad \\text{for all } x, y \\in \\Omega \\text{ with } x \\ne y.$$\n\n## Remarks\n\nProof idea for a smooth boundary: fix $x \\ne y$ and apply Green's second identity to $v(z) = G(z, x)$ and $w(z) = G(z, y)$ on $\\Omega$ with small balls around $x$ and $y$ removed.\nBoth functions are harmonic there and vanish on $\\partial\\Omega$, so only the small spheres contribute.\nAs the radii tend to $0$, the sphere around $x$ gives $G(x, y)$ and the sphere around $y$ gives $G(y, x)$, up to the same constant, and the two must be equal.\nFor a general domain one approximates it from inside by smooth domains, whose Green's functions increase to $G$.\nExample: on the unit disc, $G(z, w) = \\log\\abs{\\frac{1 - \\bar w z}{z - w}}$, and this is symmetric because $\\abs{1 - \\bar w z} = \\abs{1 - \\bar z w}$.\n\n## Explanations\n\n### L1 human\n\nPut a unit charge at $y$ in a grounded box and measure the potential at $x$.\nThen put the charge at $x$ and measure at $y$.\nThe two numbers are the same.\nThis reciprocity is the symmetry of the Green's function.\n\n### L2 human\n\nSymmetry means that the Green operator $f \\mapsto \\int_\\Omega G(\\cdot, y) f(y) \\, dy$ is self-adjoint on $L^2(\\Omega)$, which matches the self-adjointness of the Dirichlet Laplacian.\nIt lets one treat $G(x, \\cdot)$ as a function of the second variable, for example to see that $G(x, y)$ is harmonic in $y$ away from $x$.\nFor operators that are not self-adjoint, the Green's function of the adjoint operator appears instead, and symmetry fails.\n",
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