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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 Nehari manifold ： ウィキペディア英語版 Nehari manifold In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differential manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation :$-\backslash triangle\; u\; =\; |u|^u,\backslash textu\backslash mid\_\; =\; 0.$ Here Δ is the Laplacian on a bounded domain Ω in R''n''. There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional :$J(v)\; =\; \backslash frac12\backslash int\_-\backslash frac1\backslash int\_$ on the Sobolev space . The Nehari manifold is defined to be the set of such that :$\backslash |\backslash nabla\; v\backslash |^2\_\; =\; \backslash |v\backslash |^\_\; >\; 0.$ Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear. More generally, given a suitable functional ''J'', the associated Nehari manifold is defined as the set of functions ''u'' in an appropriate function space for which :$\backslash langle\; J\text{'}(u),\; u\backslash rangle\; =\; 0.\; \backslash ,$ Here ''J''′ is the functional derivative of ''J''. == References == * A. Bahri and P. L. Lions (1988), Morse Index of Some Min-Max Critical Points. I. Applications to Multiplicity Results. Communications on Pure and Applied Mathematics. (XLI) 1027–1037. * * * 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「Nehari manifold」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.