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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 Maximum principle ： ウィキペディア英語版 Maximum principle :''This article describes the maximum principle in the theory of partial differential equations. For the maximum principle in optimal control theory, see Pontryagin's minimum principle.'' In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain. Specifically, the ''strong'' maximum principle says that if a function achieves its maximum in the interior of the domain, the function is uniformly a constant. The ''weak'' maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary. In convex optimization, the maximum principle states that the maximum of a convex function on a compact convex set is attained on the boundary.〔Chapter 32 of Rockafellar (1970).〕 ==The classical example== Harmonic functions are the classical example to which the strong maximum principle applies. Formally, if ''f'' is a harmonic function, then ''f'' cannot exhibit a true local maximum within the domain of definition of ''f''. In other words, either ''f'' is a constant function, or, for any point $x\_0\backslash ,$ inside the domain of ''f'', there exist other points arbitrarily close to $x\_0\backslash ,$ at which ''f'' takes larger values.〔Berenstein and Gay.〕 Let ''f'' be an harmonic function defined on some connected open subset ''D'' of the Euclidean space R''n''. If $x\_0\backslash ,$ is a point in ''D'' such that :$f(x\_0)\backslash ge\; f(x)\; \backslash ,$ for all ''x'' in a neighborhood of $x\_0\backslash ,$, then the function ''f'' is constant on ''D''. By replacing "maximum" with "minimum" and "larger" with "smaller", one obtains the minimum principle for harmonic functions. The maximum principle also holds for the more general subharmonic functions, while superharmonic functions satisfy the minimum principle.〔Evans.〕 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「Maximum principle」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.