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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 Sobolev conjugate ： ウィキペディア英語版 Sobolev conjugate The Sobolev conjugate of ''p'' for :$p^$ *=\frac>p This is an important parameter in the Sobolev inequalities. ==Motivation== A question arises whether ''u'' from the Sobolev space $W^(\backslash mathbb^n)$ belongs to $L^q(\backslash mathbb^n)$ for some ''q''>''p''. More specifically, when does $\backslash |Du\backslash |\_$ control $\backslash |u\backslash |\_$? It is easy to check that the following inequality :$\backslash |u\backslash |\_\backslash leq\; C(p,q)\backslash |Du\backslash |\_$ ( *) can not be true for arbitrary ''q''. Consider $u(x)\backslash in\; C^\backslash infty\_c(\backslash mathbb^n)$, infinitely differentiable function with compact support. Introduce $u\_\backslash lambda(x):=u(\backslash lambda\; x)$. We have that :$\backslash |u\_\backslash lambda\backslash |\_^q=\backslash int\_|u(\backslash lambda\; x)|^qdx=\backslash frac\backslash int\_|u(y)|^qdy=\backslash lambda^\backslash |u\backslash |\_^q$ :$\backslash |Du\_\backslash lambda\backslash |\_^p=\backslash int\_|\backslash lambda\; Du(\backslash lambda\; x)|^pdx=\backslash frac\backslash int\_|Du(y)|^pdy=\backslash lambda^\backslash |Du\backslash |\_^p$ The inequality ( *) for $u\_\backslash lambda$ results in the following inequality for $u$ :$\backslash |u\backslash |\_\backslash leq\; \backslash lambda^C(p,q)\backslash |Du\backslash |\_$ If $1-n/p+n/q\backslash not\; =\; 0$, then by letting $\backslash lambda$ going to zero or infinity we obtain a contradiction. Thus the inequality ( *) could only be true for :$q=\backslash frac$, which is the Sobolev conjugate. 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「Sobolev conjugate」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.