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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Harnack's inequality ： ウィキペディア英語版 Harnack's inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by , for the Ricci flow. Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. Harnack's inequality can also be used to show the interior regularity of weak solutions of partial differential equations. ==The statement== Harnack's inequality applies to a non-negative function ''f'' defined on a closed ball in R''n'' with radius ''R'' and centre ''x''0. It states that, if ''f'' is continuous on the closed ball and harmonic on its interior, then for any point ''x'' with |''x'' - ''x''0| = ''r'' < ''R'' :$\backslash displaystylef(x\_0)\backslash le\; f(x)\; \backslash le$ In the plane R2 (''n'' = 2) the inequality can be written: :$f(x\_0)\backslash le\; f(x)\backslash le\; f(x\_0).$ For general domains $\backslash Omega$ in $\backslash mathbf^n$ the inequality can be stated as follows: If $\backslash omega$ is a bounded domain with $\backslash bar\; \backslash subset\; \backslash Omega$, then there is a constant $C$ such that for every twice differentiable, harmonic and nonnegative function $u(x)$. The constant $C$ is independent of $u$; it depends only on the domains $\backslash Omega$ and $\backslash omega$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Harnack's inequality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.