Biharmonic Functions
Definition1
Let’s call $\Delta = \nabla^{2}$ the Laplacian. $\Delta ^{2}$ is called the biharmonic operator or bilaplacian. The following equation is called the biharmonic equation.
$$ \Delta^{2} f = 0 $$
The solutions to the biharmonic equation are called biharmonic functions.
Explanation
Let’s say $\partial_{i} = \dfrac{\partial}{\partial x_{i}}$. In the Cartesian coordinate system, since $\Delta = \sum\limits_{i} \partial_{i}\partial_{i}$,
$$ \Delta^{2} f = \sum\limits_{j} \sum\limits_{i} \partial_{j}\partial_{j} \partial_{i}\partial_{i} f $$
Especially in 3 dimensions,
$$ \begin{align*} \Delta^{2}g &= \sum\limits_{j=1}^{3} \partial_{j}\partial_{j} \left( \dfrac{\partial^{2} f}{\partial x^{2}} + \dfrac{\partial^{2} f}{\partial y^{2}} + \dfrac{\partial^{2} f}{\partial z^{2}}\right) \\ &= \dfrac{\partial^{4} f}{\partial x^{4}} + \dfrac{\partial^{4} f}{\partial y^{4}} + \dfrac{\partial^{4} f}{\partial z^{4}} + 2\dfrac{\partial^{4} f}{\partial x^{2} \partial y^{2}} + 2\dfrac{\partial^{4} f}{\partial y^{2} \partial z^{2}} + 2\dfrac{\partial^{4} f}{\partial z^{2} \partial x^{2}} \\ \end{align*} $$