# 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

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

Title | biharmonic equation |
---|---|

Canonical name | BiharmonicEquation |

Date of creation | 2013-03-22 16:03:19 |

Last modified on | 2013-03-22 16:03:19 |

Owner | perucho (2192) |

Last modified by | perucho (2192) |

Numerical id | 9 |

Author | perucho (2192) |

Entry type | Definition |

Classification | msc 31B05 |