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  "body": "\n## Summary\n\nA function on the boundary of a bounded open set is resolutive when the Perron method gives a well-defined harmonic function from it.\nThis function is then the generalised solution of the [[Dirichlet problem|8D56GQ4AG8]] with that boundary function as data.\nThe Perron method squeezes the solution between an upper envelope built from [[superharmonic functions|X4V2S56YQQ]] and a lower envelope built from [[subharmonic functions|Z1MHMGFNFW]].\nResolutivity says that the two envelopes meet and give a [[harmonic function|0DEB0JMN0W]].\nIt is the right notion of \"solvable data\" for the generalised Dirichlet problem of the [[Perron method|AEZ7PPCNQS]], and it holds for every continuous boundary function.\n\n## Explanations\n\n### L1 human\n\nImagine trying to fit a soap film to a wire frame whose height is given by a function on the frame.\nFrom above, press down with every \"superharmonic lid\" that stays above the frame; from below, push up with every \"subharmonic floor\" that stays below it.\nIf the lowest lid and the highest floor coincide, the data is resolutive, and the common surface is the answer.\nOn the unit disc, every continuous boundary function is resolutive, and the answer is its Poisson integral.\nA boundary function that is $+\\infty$ everywhere is not resolutive, because both envelopes are then $+\\infty$.\n",
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