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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\\xi$ be a point of its [[boundary|05D5EGMFYV]] $\\partial\\Omega$.\nA *barrier* at $\\xi$ is a function $w \\colon \\overline{\\Omega} \\to \\R$ such that:\n\n1. $w$ is [[continuous|0X3WGX95FE]] on $\\overline{\\Omega}$;\n2. $w$ is [[superharmonic|X4V2S56YQQ]] on $\\Omega$;\n3. $w(\\xi) = 0$ and $w(x) \u003e 0$ for every $x \\in \\overline{\\Omega} \\setminus \\set{\\xi}$.\n\n## Remarks\n\nExample: suppose there is a closed ball $\\overline{B}(y, R)$ with $\\overline{B}(y, R) \\cap \\overline{\\Omega} = \\set{\\xi}$, and let $n \\ge 3$.\nThen $w(x) = R^{2-n} - \\abs{x - y}^{2-n}$ is a barrier at $\\xi$.\nIt is continuous on $\\overline{\\Omega}$ and harmonic on $\\Omega$, since $y \\notin \\overline{\\Omega}$; it vanishes at $\\xi$ because $\\abs{\\xi - y} = R$; and $\\abs{x - y} \u003e R$ for the other points $x \\in \\overline{\\Omega}$, which gives $w(x) \u003e 0$.\nFor $n = 2$ the function $w(x) = \\log(\\abs{x - y}/R)$ plays the same role.\nNon-example: $w(x) = \\abs{x - \\xi}^2$ satisfies clauses 1 and 3, but $\\Delta w = 2n \u003e 0$, so it is strictly subharmonic and fails clause 2.\nA *local barrier* is a function with these properties on $\\overline{\\Omega} \\cap N$ for some neighbourhood $N$ of $\\xi$.\nA local barrier yields a barrier, so the two notions give the same regular points.\n",
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