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  "body": "\n## Summary\n\nThe Perron method solves the [[Dirichlet problem|8D56GQ4AG8]] for the Laplacian on an arbitrary bounded open set.\nIt takes the upper envelope of all [[subharmonic functions|Z1MHMGFNFW]] that lie below the boundary data at the boundary.\nThis envelope is always harmonic; the only question is whether it takes the right boundary values.\nThat question is local and leads to barriers and regular boundary points.\n\n## Explanations\n\n### L1 human\n\nYou want a harmonic function with given values on the boundary.\nLook at all subharmonic functions that stay below these values near the boundary, and at each point take the largest value any of them reaches.\nSubharmonic functions sag below harmonic ones, so this supremum is the best harmonic candidate.\nOn the unit disc with boundary values $\\cos\\theta$, the supremum is the harmonic function $x$.\n",
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