# Neumann problem

Suppose $\mathrm{\Omega}$ is a region of ${\mathbb{R}}^{n}$ and $\partial \mathrm{\Omega}$ is the boundary of $\mathrm{\Omega}$.
Further suppose $f$ is a function $f:\partial \mathrm{\Omega}\to \u2102$, and suppose $\frac{\partial}{\partial n}$ corresponds to taking a derivative in a direction normal to the boundary $\partial \mathrm{\Omega}$ at any point. Then the
*Neumann problem* is to find a function $\varphi :\mathrm{\Omega}\cup \partial \mathrm{\Omega}\to \u2102$
such that

$\frac{\partial \varphi}{\partial n}$ | $=$ | $f,\text{on}\partial \mathrm{\Omega},$ | ||

${\nabla}^{2}\varphi $ | $=$ | $0,\text{in}\mathrm{\Omega}.$ |

Here ${\nabla}^{2}$ represents the Laplacian operator and the second condition is that $\varphi $ be a harmonic function on $\mathrm{\Omega}$. The condition for the existence of a solution $\varphi $ of the Neumann problem is that integral of the normal derivative of the function $\varphi $, calculated over the entire boundary $\partial \mathrm{\Omega}$, vanish. This follows from the identic equation

${\int}_{\partial \mathrm{\Omega}}}{\displaystyle \frac{\partial \varphi}{\partial n}}\mathit{d}\sigma ={\displaystyle {\int}_{\mathrm{\Omega}}}\nabla \cdot (\nabla \varphi )\mathit{d}\tau ={\displaystyle {\int}_{\mathrm{\Omega}}}{\nabla}^{2}\varphi d\tau $ |

and from the fact that ${\nabla}^{2}\varphi =0$.

Title | Neumann problem |
---|---|

Canonical name | NeumannProblem |

Date of creation | 2013-03-22 15:19:59 |

Last modified on | 2013-03-22 15:19:59 |

Owner | dczammit (9747) |

Last modified by | dczammit (9747) |

Numerical id | 10 |

Author | dczammit (9747) |

Entry type | Definition |

Classification | msc 31B15 |

Classification | msc 31B05 |

Classification | msc 31A05 |

Related topic | HarmonicFunction |