# Harnack’s principle

If the functions ${u}_{1}(z)$, ${u}_{2}(z)$, … are harmonic (http://planetmath.org/HarmonicFunction) in the domain $G\subseteq \u2102$ and

$${u}_{1}(z)\le {u}_{2}(z)\le \mathrm{\cdots}$$ |

in every point of $G$, then ${lim}_{n\to \mathrm{\infty}}{u}_{n}(z)$ either is infinite^{} in every point of the domain or it is finite in every point of the domain, in both cases uniformly (http://planetmath.org/UniformConvergence) in each closed (http://planetmath.org/ClosedSet) subdomain of $G$. In the latter case, the function $u(z)={lim}_{n\to \mathrm{\infty}}{u}_{n}(z)$ is harmonic in the domain $G$ (cf. limit function of sequence).

Title | Harnack’s principle |
---|---|

Canonical name | HarnacksPrinciple |

Date of creation | 2013-03-22 14:57:35 |

Last modified on | 2013-03-22 14:57:35 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 14 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 30F15 |

Classification | msc 31A05 |