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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 Direct method in the calculus of variations ： ウィキペディア英語版 Direct method in the calculus of variations In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional,〔Dacorogna, pp. 1–43.〕 introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. == The method == The calculus of variations deals with functionals $J:V\; \backslash to\; \backslash bar\}\; =\; \backslash mathbb\; \backslash cup\; \backslash$. The main interest of the subject is to find ''minimizers'' for such functionals, that is, functions $v\; \backslash in\; V$ such that:$J(v)\; \backslash leq\; J(u)\backslash forall\; u\; \backslash in\; V.$ The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand. The functional $J$ must be bounded from below to have a minimizer. This means :$\backslash inf\backslash \; >\; -\backslash infty.\backslash ,$ This condition is not enough to know that a minimizer exists, but it shows the existence of a ''minimizing sequence'', that is, a sequence $(u\_n)$ in $V$ such that $J(u\_n)\; \backslash to\; \backslash inf\backslash .$ The direct method may broken into the following steps # Take a minimizing sequence $(u\_n)$ for $J$. # Show that $(u\_n)$ admits some subsequence $(u\_)$, that converges to a $u\_0\backslash in\; V$ with respect to a topology $\backslash tau$ on $V$. # Show that $J$ is sequentially lower semi-continuous with respect to the topology $\backslash tau$. To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions. :The function $J$ is sequentially lower-semicontinuous if ::$\backslash liminf\_\; J(u\_n)\; \backslash geq\; J(u\_0)$ for any convergent sequence $u\_n\; \backslash to\; u\_0$ in $V$. The conclusions follows from :$\backslash inf\backslash \; =\; \backslash lim\_\; J(u\_n)\; =\; \backslash lim\_\; J(u\_)\; \backslash geq\; J(u\_0)\; \backslash geq\; \backslash inf\backslash$, in other words :$J(u\_0)\; =\; \backslash inf\backslash$. 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「Direct method in the calculus of variations」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.