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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\nFor $a \\in \\R^n$ and $r \u003e 0$, let $B(a, r)$ be the open ball and $S(a, r)$ the sphere of centre $a$ and radius $r$.\nLet $\\sigma_{a,r}$ be the surface measure on $S(a, r)$ normalised to total mass $1$.\nA function $u \\colon \\Omega \\to (-\\infty, +\\infty]$ is *superharmonic* if:\n\n1. $u$ is [[lower semicontinuous|RN4G8SGP6P]].\n2. $u$ is not identically $+\\infty$ on any connected component of $\\Omega$.\n3. For every $a \\in \\Omega$ and $r \u003e 0$ with $\\overline{B(a, r)} \\subseteq \\Omega$,\n$$u(a) \\ge \\int_{S(a, r)} u \\, d\\sigma_{a,r}.$$\n\n## Remarks\n\nA function $u$ is superharmonic if and only if $-u$ is [[subharmonic|Z1MHMGFNFW]] and not identically $-\\infty$ on any component.\nIt suffices to require clause 3 for all small enough $r$ at each point.\nA function $u$ of class $C^2$ is superharmonic if and only if its [[Laplacian|0E0B6RA8K0]] satisfies $\\Delta u \\le 0$.\nA function is [[harmonic|0DEB0JMN0W]] if and only if it is both superharmonic and subharmonic.\n\nExample: for $n \\ge 3$ let $u(x) = \\abs{x}^{2-n}$ on $\\R^n$, with $u(0) = +\\infty$.\nIt is lower semicontinuous and finite off $0$.\nIts mean over $S(a, r)$ equals $\\max(\\abs{a}, r)^{2-n}$, which is at most $\\abs{a}^{2-n} = u(a)$, and at $a = 0$ the value $+\\infty$ dominates.\nSo $u$ is superharmonic.\nIn the plane, $u(x) = -\\log\\abs{x}$ is superharmonic in the same way.\n\nNon-example: $u(x) = \\abs{x}^2$ is continuous on $\\R^n$, but its mean over $S(a, r)$ is $\\abs{a}^2 + r^2 \u003e u(a)$, so clause 3 fails.\n",
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