biharmonic equation

Definition. 1.

A real-valued function $V\mathrm{:}{\mathrm{R}}^n\mathrm{\to }\mathrm{R}$ of class (http://planetmath.org/http://planetmath.org/encyclopedia/Cn.html) $C^{\mathrm{4}}$, and satisfying the equation

${\nabla }^{4}V=0,$(1)

also defines a biharmonic function, and (1) is called the biharmonic equation. Biharmonic operator is defined as

$${\nabla }^4:=\sum _{k=1}^n\frac{{\partial }^4}{\partial x_k{}^4}+2\sum _{k=1}^{n-1}\sum _{l=k+1}^n\frac{{\partial }^4}{\partial x_k{}^2\partial x_l{}^2}\cdot$$