Definition:Biharmonic Operator

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Definition

Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $\map U {x_1, x_2, \ldots, x_n}$ be a real-valued function on $\R^n$.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

Let the partial derivative of $V$ with respect to $x_k$ exist for all $x_k$.


The biharmonic operator on $U$ is defined as:

\(\ds \nabla^4 U\) \(:=\) \(\ds \map {\nabla^2} {\nabla^2 U}\)
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} } \paren {\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} } U\) Definition of Del Operator


In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:

\(\ds \nabla^4 U\) \(:=\) \(\ds \map {\nabla^2} {\nabla^2 U}\)
\(\ds \) \(=\) \(\ds \paren {\dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2} + \dfrac {\partial^2} {\partial z^2} } \paren {\dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2} + \dfrac {\partial^2} {\partial z^2} } U\) Definition of Del Operator
\(\ds \) \(=\) \(\ds \dfrac {\partial^4 U} {\partial x^4} + \dfrac {\partial^4 U} {\partial y^4} + \dfrac {\partial^4 U} {\partial z^4} + 2 \dfrac {\partial^4 U} {\partial x^2 \partial y^2} + 2 \dfrac {\partial^4 U} {\partial x^2 \partial z^2}\)


Also see

  • Results about the biharmonic operator can be found here.


Sources

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