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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $m \\ge 1$ be an integer, and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\nLet $\\Delta$ be the [[Laplacian|0E0B6RA8K0]], and let $\\Delta^m$ be $\\Delta$ applied $m$ times.\nA function $u \\colon \\Omega \\to \\R$ is *polyharmonic of order $m$* if it is of class $C^{2m}$ and\n$$\\Delta^m u = 0 \\quad \\text{on } \\Omega.$$\n\n## Remarks\n\nFor $m = 1$ this is the definition of a [[harmonic function|0DEB0JMN0W]], and for $m = 2$ that of a solution of the [[biharmonic equation|AS8A9F5SKE]].\nA function that is polyharmonic of order $m$ is polyharmonic of every order $m' \\ge m$.\nExample: $u(x) = \\abs{x}^4$ on $\\R^n$ is polyharmonic of order $3$: $\\Delta \\abs{x}^4 = 4(n+2)\\abs{x}^2$, $\\Delta^2 \\abs{x}^4 = 8n(n+2)$ and $\\Delta^3 \\abs{x}^4 = 0$.\nNon-example: the same $u$ is not polyharmonic of order $2$, since $\\Delta^2 u = 8n(n+2) \\ne 0$.\nNon-example: $u(x) = e^{x_1}$ is polyharmonic of no order, since $\\Delta^m u = e^{x_1} \\ne 0$ for all $m$.\n",
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