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  "body": "\n## Summary\n\nThe Perron–Wiener–Brelot solution is the generalized solution of the [[Dirichlet problem|8D56GQ4AG8]] given by the [[Perron method|AEZ7PPCNQS]] with both upper and lower classes.\nThe upper solution is the infimum of [[superharmonic functions|X4V2S56YQQ]] lying above the boundary data.\nThe lower solution is the supremum of [[subharmonic functions|Z1MHMGFNFW]] lying below it.\nWhen the two agree and are harmonic, the boundary function is called resolutive, and the common value is the PWB solution.\nWiener showed that every continuous boundary function is resolutive, and Brelot characterized resolutive functions as those integrable for harmonic measure.\n\n## Explanations\n\n### L1 human\n\nSqueeze the unknown solution between functions that are too large at the boundary and functions that are too small.\nIf the best upper bound and the best lower bound meet, the meeting point is the solution.\nFor continuous boundary data on a bounded open set they always meet, even when the solution does not take the boundary values at every point.\nFor the punctured disc with data $0$ on the circle and $1$ at the centre, the PWB solution is $0$.\n",
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