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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 minimum principle.) If there is $a \\in U$ with $u(a) = \\inf_U u$, then $u$ is constant on $U$.\n2. (Weak minimum principle.) If $U$ is bounded and $u$ extends continuously to the closure $\\overline{U}$, then $\\min_{\\overline{U}} u = \\min_{\\partial U} u$.\n\n## Remarks\n\nProof: $-u$ is harmonic, and the [[maximum principle|T62HHSRHP5]] for $-u$ is exactly this statement for $u$.\nConsequence: if $U$ is bounded, $u$ is continuous on $\\overline{U}$ and $u \\ge 0$ on $\\partial U$, then $u \\ge 0$ on $U$; if moreover $u$ is not constant, then $u \u003e 0$ on $U$.\nExample: on the unit disc, $u(x, y) = 1 + x$ is $\\ge 0$ on the boundary circle, with value $0$ only at $(-1, 0)$, and it is positive inside.\nBoundedness is needed in part 2: on the half-plane $\\set{x \u003e 0}$, $u = -x$ is $0$ on the boundary and negative inside.\nFor subharmonic functions only the maximum principle holds, and for superharmonic functions only the minimum principle.\n\n## Explanations\n\n### L1 human\n\nA harmonic function cannot have a dip inside its domain.\nIts smallest values on a bounded region are reached on the edge.\nFor a steady temperature distribution, the coldest point is on the boundary.\n\n### L2 human\n\nTogether with the maximum principle, it says that the values of a harmonic function inside a bounded domain lie between its smallest and largest boundary values.\nThe positivity form, nonnegative boundary values give nonnegative solutions, is the comparison principle used everywhere in potential theory, for example to show that Green's functions are positive.\n",
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