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  "body": "\n## Definition\n\nLet $n \\ge 3$ and let $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R^n$ with finite total mass and compact [[support|FQTXV6BQZG]].\nThe *Newtonian potential* of $\\mu$ is the function $U^\\mu \\colon \\R^n \\to [0, +\\infty]$,\n$$U^\\mu(x) = \\int \\frac{d\\mu(y)}{\\abs{x - y}^{n-2}}.$$\nIt is the [[potential|JHKSF3XFY2]] of $\\mu$ with respect to the [[Newtonian kernel|6TJTWQ5HCR]].\n\n## Remarks\n\nExample: for $n = 3$ and $\\mu = m \\, \\delta_a$ a point mass, $U^\\mu(x) = m / \\abs{x - a}$.\nNon-example: in $\\R^3$ the function $u(x) = \\abs{x}^2$ is not the Newtonian potential of any such $\\mu$, since a Newtonian potential tends to $0$ at infinity.\nThe function $U^\\mu$ is lower semicontinuous, [[harmonic|0DEB0JMN0W]] on $\\R^n \\setminus \\operatorname{supp} \\mu$, and $U^\\mu(x) \\abs{x}^{n-2} \\to \\mu(\\R^n)$ as $\\abs{x} \\to \\infty$.\nIf $\\mu = f \\, dx$ with $f \\ge 0$ twice continuously differentiable with compact support, then $U^\\mu$ is twice continuously differentiable and $\\Delta U^\\mu = -(n-2)\\sigma_n f$, where $\\sigma_n$ is the area of the unit sphere.\nEquivalently, $U^\\mu = (n-2)\\sigma_n \\, \\Phi * \\mu$, where $\\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].\nSome texts call $\\Phi * f$ itself the Newtonian potential of $f$, and use the same name in the plane with the logarithmic kernel.\n",
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