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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $K \\subseteq \\Omega$ be a [[compact set|0R6EWFKRCF]].\nThe pair $(K, \\Omega)$ is called a *condenser*.\nLet $C_c^\\infty(\\Omega)$ be the set of smooth real functions with compact support in $\\Omega$.\nThe *capacity* of the condenser $(K, \\Omega)$ is\n$$\\operatorname{cap}(K, \\Omega) = \\inf \\set{ \\int_\\Omega \\abs{\\nabla u}^2 \\, dx : u \\in C_c^\\infty(\\Omega), \\ u \\ge 1 \\text{ on } K }.$$\n\n## Remarks\n\nWhen $\\Omega$ is bounded and its boundary and that of $K$ are regular, the infimum is attained in a larger class by the potential $u$ of the condenser.\nThis is the [[harmonic function|0DEB0JMN0W]] on $\\Omega \\setminus K$ with $u = 1$ on $\\partial K$ and $u = 0$ on $\\partial \\Omega$.\n\nExample: in $\\R^3$ let $0 \u003c r \u003c R$, $K = \\bar B_r$ and $\\Omega = B_R$.\nThe potential is $u(x) = (1/\\abs{x} - 1/R) / (1/r - 1/R)$ for $r \\le \\abs{x} \\le R$, and $\\abs{\\nabla u} = \\abs{x}^{-2} / (1/r - 1/R)$.\nIntegration over spheres gives $4 \\pi \\int_r^R s^{-2} \\, ds / (1/r - 1/R)^2 = 4 \\pi / (1/r - 1/R)$.\nSo $\\operatorname{cap}(\\bar B_r, B_R) = 4 \\pi r R / (R - r)$.\nIn the plane the same computation with $u(x) = \\log(R/\\abs{x}) / \\log(R/r)$ gives $\\operatorname{cap}(\\bar B_r, B_R) = 2 \\pi / \\log(R/r)$.\n\nNon-example: if $K$ is compact but not contained in $\\Omega$, the pair is not a condenser.\nNo $u \\in C_c^\\infty(\\Omega)$ can be at least $1$ at a point of $K$ outside $\\Omega$.\n\nSome authors normalize by a constant, such as $1/(4\\pi)$ in $\\R^3$ or $1/(2\\pi)$ in the plane, or define the capacity with $\\abs{\\nabla u}^n$, which makes it invariant under conformal maps.\n",
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