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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\\partial\\Omega$, and let $f \\colon \\partial\\Omega \\to \\R$ be a bounded function.\nThe *Perron family* of $f$ is the set $\\mathcal{S}_f$ of all [[subharmonic functions|Z1MHMGFNFW]] $v$ on $\\Omega$ such that\n$$\\limsup_{x \\to \\xi, \\, x \\in \\Omega} v(x) \\le f(\\xi) \\quad \\text{for every } \\xi \\in \\partial\\Omega.$$\nThe *Perron solution* of $f$ is the function $H_f \\colon \\Omega \\to \\R$ given by the [[supremum|0BTMQ04556]]\n$$H_f(x) = \\sup_{v \\in \\mathcal{S}_f} v(x).$$\nThe *Perron method* is the construction of $H_f$ as a candidate solution of the [[Dirichlet problem|8D56GQ4AG8]] with boundary data $f$.\n\n## Remarks\n\nThe family $\\mathcal{S}_f$ is not empty, since it contains the constant $\\inf f$, and each $v \\in \\mathcal{S}_f$ is at most $\\sup f$ by the maximum principle; so $H_f$ is finite.\nPerron's theorem says that $H_f$ is harmonic on $\\Omega$.\nIts proof replaces a function of the family by its Poisson integral on a small ball, which keeps it in the family and makes it larger.\nExample: on the unit disc with $f(\\xi) = \\xi_1$, the function $v(x) = x_1$ belongs to $\\mathcal{S}_f$, and the maximum principle gives $v \\le x_1$ for every $v \\in \\mathcal{S}_f$, so $H_f(x) = x_1$.\nNon-example: $v(x) = x_1 + \\frac12$ is subharmonic but not in $\\mathcal{S}_f$, since its boundary values exceed $f$ by $\\frac12$.\nA boundary point $\\xi$ at which $H_f(x) \\to f(\\xi)$ for every continuous $f$ is called a regular boundary point.\n",
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