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  "body": "\n## Summary\n\nA polyharmonic function of order $m$ is a function whose $m$-fold iterated [[Laplacian|0E0B6RA8K0]] vanishes: $\\Delta^m u = 0$.\nFor $m = 1$ these are the [[harmonic functions|0DEB0JMN0W]], and for $m = 2$ the solutions of the [[biharmonic equation|AS8A9F5SKE]].\nPolyharmonic functions are the model solutions of higher-order elliptic equations.\nThey appear in elasticity and in the study of polyharmonic splines and interpolation.\n\n## Explanations\n\n### L1 human\n\nTake the Laplacian of a function, then the Laplacian of the result, and so on, $m$ times.\nIf you end up with zero, the function is polyharmonic of order $m$.\nFor example, $\\abs{x}^2$ on $\\R^n$ is not harmonic, but its Laplacian is the constant $2n$, so it is polyharmonic of order $2$.\nEvery polynomial of degree less than $2m$ is polyharmonic of order $m$, because each Laplacian lowers the degree by $2$.\n",
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