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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $u \\colon \\R^n \\to \\R$ be [[harmonic|0DEB0JMN0W]] on all of $\\R^n$.\nIf $u$ is bounded above or bounded below on $\\R^n$, then $u$ is constant.\n\n## Remarks\n\nProof idea for a bounded function, $\\abs{u} \\le M$: by the [[gradient estimate|N76R7JWMNJ]] on the ball $B(x, r)$, $\\abs{\\nabla u(x)} \\le n M / r$ for every $r \u003e 0$.\nLetting $r \\to \\infty$ gives $\\nabla u = 0$ everywhere, so $u$ is constant.\nIf $u$ is only bounded below, say $u \\ge m$, apply [[Harnack's inequality|YNH832D50W]] to the nonnegative harmonic function $u - m$ on larger and larger balls.\nIt shows that $u - m$ is bounded, and the first case applies.\nThe theorem fails on proper subsets: on the half-space $\\set{x_n \u003e 0}$, $u = x_n$ is bounded below and not constant.\nSome one-sided condition is needed: $u(x, y) = x$ is harmonic on $\\R^2$ and not constant.\nA harmonic function on $\\R^n$ with $\\abs{u(x)} \\le C(1 + \\abs{x})^k$ is a harmonic polynomial of degree at most $k$.\n\n## Explanations\n\n### L1 human\n\nA harmonic function defined on the whole space cannot stay below a ceiling, or above a floor, unless it is flat.\nEvery nonconstant harmonic function on the whole space takes arbitrarily large and arbitrarily small values.\n\n### L2 human\n\nFor $n = 2$ this is the real counterpart of Liouville's theorem for entire functions, and the two imply each other through harmonic conjugates.\nThe theorem gives uniqueness of solutions of Poisson's equation on $\\R^n$ that tend to $0$ at infinity.\nIt is the prototype of Liouville theorems for elliptic equations and for harmonic maps.\nOn $\\R^n$ minus a point, or on other spaces, the analogous question leads to the distinction between parabolic and hyperbolic spaces.\n",
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  "name": "Liouville's theorem for harmonic functions",
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