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  "body": "\n## Summary\n\nLet $K$ be a compact set inside an open set $\\Omega \\subseteq \\R^n$, and let $1 \u003c p \u003c \\infty$.\nThe variational capacity, or $p$-capacity, of $K$ is the least $p$-Dirichlet energy $\\int_\\Omega \\abs{\\nabla u}^p \\, dx$ of a suitable function $u$.\nHere $u$ is smooth, at least $1$ on $K$, and zero near the boundary of $\\Omega$.\nFor $p = 2$ it is the classical [[condenser capacity|NTEFMRR3HR]].\nIt is the natural measure of size in nonlinear potential theory and for the $p$-Laplace equation.\nSets of $p$-capacity zero are the sets that [[Sobolev functions|M4CEZ5K9HP]] with $p$-integrable gradient cannot see.\n\n## Explanations\n\n### L1 human\n\nThink of $\\Omega$ as a large room and $K$ as an object inside it.\nA test function must climb from $0$ near the walls to at least $1$ on $K$, and the capacity is the cheapest cost of this climb, measured by $\\int \\abs{\\nabla u}^p$.\nA large object close to the walls forces steep slopes and has a large capacity.\nA single point in $\\R^n$ has $p$-capacity zero when $p \\le n$: the climb can be made very cheap by spreading it over many scales.\nFor $p \u003e n$ even a point has positive $p$-capacity, which matches the fact that such Sobolev functions are continuous.\n",
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  "msc": [
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