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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a connected [[open set|02BH3MCBQB]], and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\\Omega$.\n\n1. If there is a point $a \\in \\Omega$ with $u(a) = \\sup_{\\Omega} u$, then $u$ is constant on $\\Omega$.\n2. If $\\Omega$ is bounded and $\\limsup_{x \\to \\zeta,\\, x \\in \\Omega} u(x) \\le 0$ for every $\\zeta$ in the [[boundary|05D5EGMFYV]] $\\partial\\Omega$, then $u \\le 0$ on $\\Omega$.\n\n## Remarks\n\nProof idea for part 1: let $M = \\sup_\\Omega u$ and $A = \\set{x \\in \\Omega : u(x) = M}$.\nThe set $A = \\set{u \\ge M}$ is closed in $\\Omega$ because $u$ is [[upper semicontinuous|RN4G8SGP6P]].\nLet $b \\in A$.\nThe sub-mean inequality on small spheres about $b$ and $u \\le M$ force $u = M$ almost everywhere on a small ball about $b$.\nSince $A$ is closed, $u = M$ on that whole ball, so $A$ is open.\nConnectedness gives $A = \\Omega$ when $A$ is nonempty.\nFor part 2, extend $u$ to $\\overline{\\Omega}$ by its [[limit superior|0T3RZQXDYD]] at boundary points.\nThe extension is upper semicontinuous on a compact set, so it attains its maximum, and part 1 rules out a positive maximum inside.\n\nBoundedness of $\\Omega$ is needed in part 2.\nOn the half-space $\\set{x \\in \\R^n : x_1 \u003e 0}$, the function $u(x) = x_1$ is harmonic, hence subharmonic, and tends to $0$ at every boundary point, but it is positive everywhere.\nConnectedness is needed in part 1.\nOn two disjoint discs, the function equal to $0$ on one disc and $1$ on the other is subharmonic and attains its supremum, but it is not constant.\nApplied to $u$ and $-u$ for a harmonic $u$, it gives the maximum and minimum principles for harmonic functions.\n\n## Explanations\n\n### L1 human\n\nA subharmonic function cannot have a peak inside its domain unless it is flat.\nSo on a bounded region its largest values are found at the edge.\nIf it is at most $0$ near the edge, it is at most $0$ everywhere.\n\n### L2 human\n\nThis is the tool behind uniqueness for the [[Dirichlet problem|8D56GQ4AG8]] and behind the Perron method, where subharmonic functions are compared with harmonic ones.\nIt also gives the comparison principle: if $u$ is subharmonic, $v$ is superharmonic, and $\\limsup (u - v) \\le 0$ at the boundary of a bounded domain, then $u \\le v$ inside.\nUnbounded domains need extra growth conditions, which is the subject of Phragmén–Lindelöf type results.\n",
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