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  "body": "\n## Summary\n\nA barrier at a boundary point $\\xi$ of a bounded open set is a [[superharmonic function|X4V2S56YQQ]] that vanishes at $\\xi$.\nIt is positive at every other point of the closure.\nBarriers are the local test for whether the [[Perron method|AEZ7PPCNQS]] gives the right boundary value at $\\xi$.\nA boundary point is regular for the [[Dirichlet problem|8D56GQ4AG8]] if and only if it has a barrier.\nSimple geometric conditions, such as an exterior ball or an exterior cone at $\\xi$, produce explicit barriers.\n\n## Explanations\n\n### L1 human\n\nA barrier is a superharmonic function that dips down to zero exactly at the chosen boundary point.\nIt acts like a funnel that forces the Perron solution to approach the boundary value at that point.\nIn $\\R^3$, suppose a closed ball with centre $y$ and radius $R$ meets the closure of the set only at $\\xi$.\nThen the function $\\frac{1}{R} - \\frac{1}{\\abs{x - y}}$ is a barrier at $\\xi$.\n",
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