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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be a domain, that is, a connected open set, and let $u \\colon U \\to \\R$ be [[harmonic|0DEB0JMN0W]].\n\n1. (Strong maximum principle.) If there is $a \\in U$ with $u(a) = \\sup_U u$, then $u$ is constant on $U$.\n2. (Weak maximum principle.) If $U$ is bounded and $u$ extends continuously to the closure $\\overline{U}$, then $\\max_{\\overline{U}} u = \\max_{\\partial U} u$.\n\n## Remarks\n\nProof idea for part 1: the set $A = \\set{x \\in U : u(x) = \\sup_U u}$ is closed in $U$ by continuity.\nIt is open: if $x \\in A$ and $\\overline{B}(x, r) \\subseteq U$, the [[mean value property|0HYPR64JM2]] over the ball makes $u(x)$ the average of values $\\le u(x)$, which forces $u = u(x)$ on the ball.\nAs $U$ is connected and $A$ is not empty, $A = U$.\nPart 2 follows: the maximum of $u$ on the compact set $\\overline{U}$ is attained, and if it were attained only inside, part 1 would make $u$ constant.\nExample: $u(x, y) = x^2 - y^2$ on the unit disc takes its maximum $1$ on the boundary, at $(\\pm 1, 0)$, and the value $0$ at the centre.\nBoundedness is needed in part 2: on the half-plane $\\set{x \u003e 0}$, $u = x$ is $0$ on the boundary and positive inside.\nConnectedness is needed in part 1.\nOn two disjoint discs, the function equal to $0$ on one and $1$ on the other is harmonic and attains its maximum without being constant.\nThis is the harmonic case of the general [[maximum principle|D2JF19HF91]] for elliptic operators.\n\n## Explanations\n\n### L1 human\n\nA harmonic function cannot have a peak inside its domain.\nIts largest values on a bounded region are reached on the edge.\nThink of the height of a soap film spanned on a wire frame: the highest point lies on the wire.\n\n### L2 human\n\nThe weak maximum principle gives uniqueness for the Dirichlet problem.\nTwo harmonic functions with the same boundary values differ by a harmonic function that is $0$ on the boundary, hence $0$ inside.\nIt also gives stability: if the boundary values of two solutions differ by at most $\\varepsilon$, then the solutions differ by at most $\\varepsilon$ everywhere.\nSubharmonic functions satisfy the same maximum principle, and Perron's method builds solutions of the Dirichlet problem from it.\n",
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