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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\nThe set $\\Omega$ is a *Lipschitz domain* if for every point $p$ of its [[boundary|05D5EGMFYV]] $\\partial\\Omega$ there are:\n\n1. a radius $r \u003e 0$;\n2. a rigid motion of $\\R^n$ giving coordinates $x = (x', x_n)$ with $x' \\in \\R^{n-1}$ and $x_n \\in \\R$;\n3. a [[Lipschitz|0AW1E6YTJY]] function $\\varphi \\colon \\R^{n-1} \\to \\R$,\n\nsuch that, in these coordinates,\n$$\\Omega \\cap B(p, r) = \\set{x \\in B(p, r) : x_n \u003e \\varphi(x')},$$\nwhere $B(p, r)$ is the [[open ball|0DC3MPR7C9]] of radius $r$ about $p$.\n\n## Remarks\n\nIt follows that $\\partial\\Omega \\cap B(p, r) = \\set{x \\in B(p, r) : x_n = \\varphi(x')}$.\nMany authors also require $\\Omega$ to be bounded, and sometimes connected.\nIf $\\Omega$ is bounded, finitely many such charts cover $\\partial\\Omega$, so one can use a common radius and a common Lipschitz constant.\nThese numbers are called the Lipschitz character of $\\Omega$.\nExample: the square $(0, 1)^2$.\nNear an edge point the square is the region above a horizontal line.\nNear the corner $(0, 0)$, in coordinates rotated by $45$ degrees, it is the region $\\set{x_2 \u003e \\abs{x_1}}$ near the corner, and $\\abs{x_1}$ is Lipschitz with constant $1$.\nNon-example: the slit disc $\\set{\\abs{z} \u003c 1} \\setminus [0, 1)$ in $\\R^2$.\nNear the point $1/2$ of the slit, the set lies on both sides of the slit, so it cannot be the region on one side of a graph.\nEvery bounded open set with $C^1$ boundary is a Lipschitz domain.\n",
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