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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]] whose [[boundary|05D5EGMFYV]] $\\partial\\Omega$ is a $C^1$ hypersurface, with surface measure $\\sigma$.\nLet $\\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]],\n$$\\Phi(x) = -\\frac{1}{2\\pi} \\log\\abs{x} \\ (n = 2), \\qquad \\Phi(x) = \\frac{1}{(n-2)\\,\\omega_n}\\, \\abs{x}^{2-n} \\ (n \\ge 3),$$\nwhere $\\omega_n$ is the area of the unit sphere in $\\R^n$, so that $-\\Delta \\Phi = \\delta_0$.\nLet $\\varphi \\colon \\partial\\Omega \\to \\R$ be continuous.\nThe *single layer potential* with density $\\varphi$ is the function\n$$S\\varphi(x) = \\int_{\\partial\\Omega} \\Phi(x - y)\\, \\varphi(y)\\, d\\sigma(y), \\qquad x \\in \\R^n.$$\n\n## Remarks\n\nThe integral is a [[surface integral|065HE5V98E]].\nFor $x \\notin \\partial\\Omega$ the integrand is smooth in $x$, and $S\\varphi$ is [[harmonic|0DEB0JMN0W]] on $\\R^n \\setminus \\partial\\Omega$.\nFor $x \\in \\partial\\Omega$ the singularity of $\\Phi(x - y)$ is integrable on the $(n-1)$-dimensional surface, so $S\\varphi(x)$ is still defined.\nIt is the [[potential of the measure|JHKSF3XFY2]] $\\varphi\\,\\sigma$ carried by $\\partial\\Omega$.\n\nExample: let $\\Omega$ be the unit ball in $\\R^3$ and $\\varphi \\equiv 1$.\nThen $\\Phi(x) = 1/(4\\pi\\abs{x})$ and the mean of $1/\\abs{x - y}$ over the unit sphere in $y$ is $1/\\max(\\abs{x}, 1)$.\nSince the sphere has area $4\\pi$, $S\\varphi(x) = 1$ for $\\abs{x} \\le 1$ and $S\\varphi(x) = 1/\\abs{x}$ for $\\abs{x} \\ge 1$.\nIt is continuous across the sphere, while its radial derivative jumps from $0$ to $-1$.\n\nNon-example: the integral $\\int_\\Omega \\Phi(x - y) f(y)\\, dy$ over the open set itself is a volume potential, the [[Newtonian potential|DCMFQ01QGS]] of $f$, and not a single layer potential.\nIt is a potential of a charge spread through a solid, not over a surface.\n",
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