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Biharmonic Functions 📂Functions

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,

$$ \begin{align*} \Delta^{2}g &= \sum\limits_{j=1}^{3} \partial_{j}\partial_{j} \left( \dfrac{\partial^{2} f}{\partial x^{2}} + \dfrac{\partial^{2} f}{\partial y^{2}} + \dfrac{\partial^{2} f}{\partial z^{2}}\right) \\ &= \dfrac{\partial^{4} f}{\partial x^{4}} + \dfrac{\partial^{4} f}{\partial y^{4}} + \dfrac{\partial^{4} f}{\partial z^{4}} + 2\dfrac{\partial^{4} f}{\partial x^{2} \partial y^{2}} + 2\dfrac{\partial^{4} f}{\partial y^{2} \partial z^{2}} + 2\dfrac{\partial^{4} f}{\partial z^{2} \partial x^{2}} \\ \end{align*} $$

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