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  "body": "\n## Summary\n\nThe biharmonic equation is the fourth-order equation $\\Delta^2 u = 0$, where $\\Delta^2 = \\Delta \\Delta$ is the [[Laplacian|0E0B6RA8K0]] applied twice.\nIts solutions are called biharmonic functions.\nIt models the small bending of a thin elastic plate and, in two dimensions, the stream function of slow viscous flow.\nIt is the basic example of a higher-order elliptic equation, for which tools such as the maximum principle no longer hold.\n\n## Explanations\n\n### L1 human\n\nA [[harmonic function|0DEB0JMN0W]] has zero Laplacian.\nA biharmonic function only needs its Laplacian to be harmonic.\nSo every harmonic function is biharmonic, but there are many more biharmonic functions.\nFor example, $u(x, y) = x^2 + y^2$ has Laplacian $4$, which is constant and so harmonic; hence $u$ is biharmonic, although it is not harmonic.\n",
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