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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 Mountain pass theorem ： ウィキペディア英語版 Mountain pass theorem The mountain pass theorem is an existence theorem from the calculus of variations. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points. == Theorem statement == The assumptions of the theorem are: * ''I'' is a functional from a Hilbert space ''H'' to the reals, * $I\backslash in\; C^1(H,\backslash mathbb)$ and $I\text{'}$ is Lipschitz continuous on bounded subsets of ''H'', * ''I'' satisfies the Palais-Smale compactness condition, * $I()=0$, * there exist positive constants ''r'' and ''a'' such that $I()\backslash geq\; a$ if $\backslash Vert\; u\backslash Vert\; =r$, and * there exists $v\backslash in\; H$ with $\backslash Vert\; v\backslash Vert\; >r$ such that $I()\backslash leq\; 0$. If we define: :$\backslash Gamma=\backslash (0)=0,\backslash mathbf(1)=v\backslash \}$ and: :$c=\backslash inf\_\backslash max\_\; I(),$ then the conclusion of the theorem is that ''c'' is a critical value of ''I''. 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「Mountain pass theorem」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.