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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be an open set.\nLet $u \\colon \\Omega \\to \\R$ be four times continuously differentiable.\nThe *biharmonic equation* on $\\Omega$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\Delta^2 u = 0,$$\nwhere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\\Delta^2 u = \\Delta(\\Delta u)$.\nA solution is called a *biharmonic function*.\n\n## Remarks\n\nIn coordinates, $\\Delta^2 u = \\sum_{i,j=1}^n \\partial_i^2 \\partial_j^2 u$; in the plane, $\\Delta^2 u = u_{xxxx} + 2 u_{xxyy} + u_{yyyy}$.\nExample: on $\\R^n$, $u(x) = \\abs{x}^2$ is biharmonic, since $\\Delta \\abs{x}^2 = 2n$ is constant and so $\\Delta^2 u = 0$.\nNon-example: $u(x) = \\abs{x}^4$ is not biharmonic, since $\\Delta \\abs{x}^4 = 4(n + 2) \\abs{x}^2$ and then $\\Delta^2 u = 8n(n + 2) \\ne 0$.\nEvery [[harmonic function|0DEB0JMN0W]] is biharmonic.\nThe inhomogeneous equation $\\Delta^2 u = f$ is the plate equation; boundary conditions usually prescribe two quantities, such as $u$ and its normal derivative.\n",
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