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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 space ： ウィキペディア英語版 Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense. ==Motivation== There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class ''C''1 — see Differentiability class). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space ''C''1 (or ''C''2, etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations. Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the uniform norm. A typical example is measuring the energy of a temperature or velocity distribution by an ''L''2-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions. The integration by parts formula yields that for every ''u'' ∈ ''C''''k''(Ω), where ''k'' is a natural number and for all infinitely differentiable functions with compact support ''φ'' ∈ ''C''c∞(Ω), :$\backslash int\_\backslash Omega\; uD^\backslash alpha\backslash varphi\backslash ;dx=(-1)^\backslash int\_\backslash Omega\; \backslash varphi\; D^\backslash alpha\; u\backslash ;dx,$ where ''α'' a multi-index of order |''α''| = ''k'' and Ω is an open subset in ℝ''n''. Here, the notation :$D^f\; =\; \backslash frac\}\; \backslash dots\; \backslash partial\; x\_^,$ is used. The left-hand side of this equation still makes sense if we only assume ''u'' to be locally integrable. If there exists a locally integrable function ''v'', such that :$\backslash int\_\backslash Omega\; uD^\backslash alpha\backslash varphi\backslash ;dx=(-1)^\backslash int\_\backslash Omega\; \backslash varphi\; v\; \backslash ;dx,\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash varphi\backslash in\; C\_c^\backslash infty(\backslash Omega),$ we call ''v'' the weak ''α''-th partial derivative of ''u''. If there exists a weak ''α''-th partial derivative of ''u'', then it is uniquely defined almost everywhere. On the other hand, if ''u'' ∈ ''C''k(Ω), then the classical and the weak derivative coincide. Thus, if ''v'' is a weak ''α''-th partial derivative of ''u'', we may denote it by ''D''α''u'' := ''v''. For example, the function :$u(x)=\backslash begin$ 1+x & \text-110 & \textx=0\\ 1-x & \text00 & \text \end is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function :$v(x)=\backslash begin$ 1 & \text-1-1 & \text00 & \text \end satisfies the definition for being the weak derivative of $u(x)$, which then qualifies as being in the Sobolev space $W^$ (for any allowed ''p'', see definition below). The Sobolev spaces ''Wk,p''(Ω) combine the concepts of weak differentiability and Lebesgue norms. 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 space」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.