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  "body": "\n## Summary\n\nA superharmonic function is a lower semicontinuous function whose value at each point is at least its average over spheres around that point.\nA function $u$ is superharmonic exactly when $-u$ is [[subharmonic|Z1MHMGFNFW]], so the two theories are mirror images.\nThe potentials of positive charges, such as the Newtonian kernel $\\abs{x}^{2-n}$ in $\\R^n$ for $n \\ge 3$, are the basic examples.\nSuperharmonic functions form the upper class of the Perron method and are the main objects of classical [[potential theory|9PM8Q2E30Q]].\n\n## Explanations\n\n### L1 human\n\nA [[harmonic function|0DEB0JMN0W]] equals its average over every small sphere around a point.\nA superharmonic function sits at or above that average, like a tent pole pushing a membrane up from below.\nIt cannot have a strict local minimum inside its domain.\nThe gravitational potential $1/\\abs{x}$ of a point mass in space is superharmonic, with the value $+\\infty$ at the mass.\nAt the origin the average of $1/\\abs{x}$ over the sphere of radius $r$ is $1/r$, which is below the value $+\\infty$.\n",
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