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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set, let $m \\ge 1$ and $1 \u003c p \u003c \\infty$.\nLet $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$ be an [[integral functional|JRQZ803Q5Y]] for maps $u \\colon \\Omega \\to \\R^m$.\nA map $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\\Omega; \\R^m)$ is a *minimiser* of $J$ if $J(u) \\le J(u + \\varphi)$ for all $\\varphi \\in W^{1,p}_0(\\Omega; \\R^m)$.\nLet $k \\ge 1$ and $0 \u003c \\alpha \u003c 1$.\n\n1. The minimiser $u$ is *regular* (of class $C^{k,\\alpha}$) if it agrees almost everywhere with a map in $C^{k,\\alpha}(\\Omega; \\R^m)$.\n2. It is *partially regular* if there is a relatively closed set $\\Sigma \\subseteq \\Omega$ of Lebesgue measure zero such that $u$ is regular on $\\Omega \\setminus \\Sigma$.\n\nThe *regularity problem* for $J$ asks under which conditions on $L$ every minimiser is regular or partially regular.\n\n## Remarks\n\nHere $C^{k,\\alpha}$ is the space of $C^k$ maps whose $k$-th derivatives are locally [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$.\n\nExample: for $m = 1$ and $L(\\xi) = \\frac{1}{2} \\abs{\\xi}^2$, minimisers are weak solutions of $\\Delta u = 0$.\nBy Weyl's lemma they are harmonic, hence smooth, so they are regular for every $k$ and $\\alpha$.\n\nNon-example: for $n = 1$, $m = 1$ and $L(x, u, \\xi) = (\\xi^2 - 1)^2$ on $\\Omega = (0, 2)$, the function $u(x) = \\min(x, 2 - x)$ is a minimiser among maps with its boundary values, since $J(u) = 0$.\nIt is Lipschitz but not $C^1$, so it is not regular in the sense of clause 1.\nHere $L$ is not convex in $\\xi$.\n\nFor $m = 1$ and smooth, uniformly convex $L(\\xi)$ with bounded second derivatives, every minimiser in $W^{1,2}$ is smooth, by the De Giorgi–Nash–Moser theorem and Schauder estimates.\nFor $m \\ge 2$ this fails in general, and partial regularity is the typical result for smooth, uniformly convex or strongly quasiconvex integrands with suitable growth.\n",
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