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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $u \\ge 0$ be a [[superharmonic function|X4V2S56YQQ]] on $\\Omega$.\nLet $E \\subseteq \\Omega$.\nThe *reduced function* of $u$ on $E$ is the function $R^E_u \\colon \\Omega \\to [0, +\\infty]$ given by\n$$R^E_u(x) = \\inf \\set{v(x) : v \\ge 0 \\text{ is superharmonic on } \\Omega \\text{ and } v \\ge u \\text{ on } E}.$$\nThe *regularised reduced function* $\\widehat{R}^E_u$ is its lower semicontinuous regularisation:\n$$\\widehat{R}^E_u(x) = \\min\\left(R^E_u(x), \\liminf_{y \\to x} R^E_u(y)\\right).$$\n\n## Remarks\n\nThe function $u$ itself is admissible, so $0 \\le \\widehat{R}^E_u \\le R^E_u \\le u$, and $R^E_u = u$ on $E$.\nBy the [[fundamental convergence theorem|RJK6ZKNF6R]], $\\widehat{R}^E_u$ is superharmonic and differs from $R^E_u$ only on a [[polar set|C95QJ7YH5T]] contained in $E$.\nThe function $\\widehat{R}^E_u$ is harmonic on $\\Omega \\setminus \\overline{E}$.\nExample: let $\\Omega = \\R^3$, $u = 1$ and $E = \\set{x : \\abs{x} \\le 1}$.\nThe function $v(x) = \\min(1, 1/\\abs{x})$ is positive, superharmonic and equal to $1$ on $E$.\nAny admissible $v$ satisfies $v \\ge 1/\\abs{x}$ on $\\abs{x} \u003e 1$ by the minimum principle, so $R^E_1 = \\widehat{R}^E_1 = \\min(1, 1/\\abs{x})$.\nNon-example of an admissible function: in the same example, the indicator function of the closed ball $E$ equals $u = 1$ on $E$.\nIt is not superharmonic, since it is not lower semicontinuous at the points of the unit sphere.\nOther notations are $R_u^E$, $\\operatorname{red}_E u$, and the French name réduite.\n",
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