Biharmonic equation (${\nabla }^{4}f=0$)

The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

For angular Fourier modes of integer order $n\ge 2$, the biharmonic equation in polar coordinates has radial basis $r^n,r^{-n},r^{n+2},r^{2-n}$. Apply the Laplacian twice to obtain the displayed indicial equation. For $n=0$, repeated roots give the radial basis $1,\log r,r^2,r^2\log r$; for $n=1$, the basis is $r^{-1},r,r\log r,r^3$. These are Cauchy-Euler equations. Regularity at the origin and boundary conditions select the admissible powers. For $n=2$, a constant radial factor still multiplies $\cos 2\theta$ or $\sin 2\theta$, so it is bounded but fails continuity at the origin.

If $H,G$ are holomorphic functions, both real and imaginary parts of $\bar{z}H(z)+G(z)$ solve the biharmonic equation. In Wirtinger derivatives, $\Delta =4{\partial }_z{\partial }_{\bar{z}}$, so $\Delta (\bar{z}H+G)=4H^{\prime }$ is harmonic and its next Laplacian is zero. This construction gives local representations of planar solutions of the biharmonic equation; it is useful for the biharmonic stream function for planar Stokes flow.

For a smooth three-dimensional biharmonic function regular throughout a ball, its surface average has exactly this form. Taylor expansion and rotationally invariant spherical moments turn each even term into a multiple of an iterated Laplacian; terms from the fourth derivative onward vanish when ${\Delta }^{2}f=0$. This is the mean-value identity that converts the reciprocal surface average into Faxén's first law.