Biharmonic equation

Summary

The biharmonic equation is the fourth-order equation $\Delta^2 u = 0$, where $\Delta^2 = \Delta \Delta$ is the Laplacian applied twice. Its solutions are called biharmonic functions. It models the small bending of a thin elastic plate and, in two dimensions, the stream function of slow viscous flow. It is the basic example of a higher-order elliptic equation, for which tools such as the maximum principle no longer hold.