Biharmonic equation

Remarks

In coordinates, $\Delta^2 u = \sum_{i,j=1}^n \partial_i^2 \partial_j^2 u$; in the plane, $\Delta^2 u = u_{xxxx} + 2 u_{xxyy} + u_{yyyy}$. Example: on $\R^n$, $u(x) = \abs{x}^2$ is biharmonic, since $\Delta \abs{x}^2 = 2n$ is constant and so $\Delta^2 u = 0$. Non-example: $u(x) = \abs{x}^4$ is not biharmonic, since $\Delta \abs{x}^4 = 4(n + 2) \abs{x}^2$ and then $\Delta^2 u = 8n(n + 2) \ne 0$. Every harmonic function is biharmonic. The inhomogeneous equation $\Delta^2 u = f$ is the plate equation; boundary conditions usually prescribe two quantities, such as $u$ and its normal derivative.