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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $1 \u003c p \u003c \\infty$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $K \\subseteq \\Omega$ be a [[compact set|0R6EWFKRCF]].\nLet $C_c^\\infty(\\Omega)$ be the set of smooth real functions with compact support in $\\Omega$.\nThe *variational capacity* of $K$ relative to $\\Omega$ is\n$$\\operatorname{cap}_p(K, \\Omega) = \\inf \\set{ \\int_\\Omega \\abs{\\nabla u}^p \\, dx : u \\in C_c^\\infty(\\Omega), \\ u \\ge 1 \\text{ on } K }.$$\n\n## Remarks\n\nFor an open set $U \\subseteq \\Omega$ one sets $\\operatorname{cap}_p(U, \\Omega) = \\sup \\set{ \\operatorname{cap}_p(K, \\Omega) : K \\subseteq U \\text{ compact} }$.\nFor an arbitrary set $E \\subseteq \\Omega$ one sets $\\operatorname{cap}_p(E, \\Omega) = \\inf \\set{ \\operatorname{cap}_p(U, \\Omega) : E \\subseteq U \\subseteq \\Omega, \\ U \\text{ open} }$.\nThe infimum does not change if the admissible functions are taken in the [[Sobolev space|M4CEZ5K9HP]] $W_0^{1,p}(\\Omega)$ and continuous, with $u \\ge 1$ on $K$.\n\nExample: let $0 \u003c r \u003c R$, $p \\ne n$, $\\alpha = (p - n)/(p - 1)$, and let $\\sigma_{n-1}$ be the area of the unit sphere in $\\R^n$.\nThe radial function $u$ with $u = 1$ for $\\abs{x} \\le r$, $u = (\\abs{x}^\\alpha - R^\\alpha)/(r^\\alpha - R^\\alpha)$ for $r \\le \\abs{x} \\le R$, and $u = 0$ outside, solves the $p$-Laplace equation on the shell.\nIts gradient has size $\\abs{\\alpha} \\abs{x}^{\\alpha - 1} / \\abs{r^\\alpha - R^\\alpha}$, and integration in polar coordinates gives\n$$\\operatorname{cap}_p(\\bar B_r, B_R) = \\sigma_{n-1} \\abs{\\alpha}^{p-1} \\abs{r^\\alpha - R^\\alpha}^{1-p}.$$\nFor $p = 2$ and $n = 3$ this is $4 \\pi r R / (R - r)$.\n\nNon-example of an admissible function: a function $u \\in C_c^\\infty(\\Omega)$ with $u = 1/2$ on $K$ fails the condition $u \\ge 1$ on $K$ and does not enter the infimum.\n\nA related notion is the Sobolev $p$-capacity, which uses $\\int_{\\R^n} (\\abs{u}^p + \\abs{\\nabla u}^p) \\, dx$ and no reference set $\\Omega$; for subsets of a bounded open set $\\Omega$ both capacities have the same null sets.\n",
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