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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 Hopf lemma ： ウィキペディア英語版 Hopf lemma In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained. ==Statement for harmonic functions== Let Ω be a bounded domain in R''n'' with smooth boundary. Let ''f'' be a real-valued function continuous on the closure of Ω and harmonic on Ω. If ''x'' is a boundary point such that ''f''(''x'') > ''f''(''y'') for all ''y'' in Ω sufficiently close to ''x'', then the (one-sided) directional derivative of ''f'' in the direction of the outward pointing normal to the boundary at ''x'' is strictly positive. 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「Hopf lemma」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.