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# biharmonic equation

###### Definition. 1.

A real-valued function $V\colon\mathbb{R}^{n}\to\mathbb{R}$ of class $C^{4}$, and satisfying the equation

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

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

$\nabla^{4}:=\sum_{{k=1}}^{n}\frac{\partial^{4}}{\partial{x_{k}}^{4}}+2\sum_{{k% =1}}^{{n-1}}\sum_{{l=k+1}}^{n}\frac{\partial^{4}}{\partial{x_{k}}^{2}\partial{% x_{l}}^{2}}\cdot$ |

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## Mathematics Subject Classification

31B05*no label found*

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