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  "body": "\n## Definition\n\nLet $n \\ge 3$ and let $K \\subseteq \\R^n$ be a [[compact set|0R6EWFKRCF]].\nLet $C_c^\\infty(\\R^n)$ be the set of smooth real functions on $\\R^n$ with compact support.\nThe *Newtonian capacity* of $K$ is\n$$\\operatorname{cap}(K) = \\inf \\set{\\int_{\\R^n} \\abs{\\nabla u}^2 \\, dx : u \\in C_c^\\infty(\\R^n), \\ u \\ge 1 \\text{ on } K}.$$\n\n## Remarks\n\nThe infimum equals $\\int_{\\R^n \\setminus K} \\abs{\\nabla u}^2 \\, dx$ for the equilibrium potential $u$ of $K$.\nThis function is harmonic off $K$, equal to $1$ on $K$ when $K$ is regular, and tends to $0$ at infinity.\nSome authors multiply by a constant such as $1 / ((n-2) \\sigma_{n-1})$, where $\\sigma_{n-1}$ is the area of the unit sphere.\nExample: for the closed ball $\\bar B_r$ of radius $r$, the equilibrium potential is $u(x) = (r / \\abs{x})^{n-2}$ for $\\abs{x} \\ge r$.\nThen $\\abs{\\nabla u} = (n-2) r^{n-2} \\abs{x}^{1-n}$, and integration in polar coordinates gives $\\operatorname{cap}(\\bar B_r) = (n-2) \\sigma_{n-1} r^{n-2}$.\nFor $n = 3$ this is $4 \\pi r$.\nNon-example of an admissible function: $u = 1/2$ near $K$ does not satisfy $u \\ge 1$ on $K$, so it does not enter the infimum.\nCapacity is monotone, so a point has capacity at most $(n-2) \\sigma_{n-1} r^{n-2}$ for every $r \u003e 0$, hence capacity $0$.\n",
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