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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 trace operator ： ウィキペディア英語版 trace operator In mathematics, the concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions. The trace operator makes it possible to extend the notion of restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. ==Informal discussion== Let $\backslash Omega$ be a bounded open set in the Euclidean space $\backslash mathbb\; R^n$ with ''C''1 boundary $\backslash partial\; \backslash Omega.$ If $u$ is a function that is $C^1$ (or even just continuous) on the closure $\backslash bar\; \backslash Omega$ of $\backslash Omega,$ its function restriction is well-defined and continuous on $\backslash partial\; \backslash Omega.$ If however, $u$ is the solution to some partial differential equation, it is in general a weak solution, so it belongs to some Sobolev space. Such functions are defined only up to a set of measure zero, and since the boundary $\backslash partial\; \backslash Omega$ does have measure zero, any function in a Sobolev space can be completely redefined on the boundary without changing the function as an element in that space. It follows that simple function restriction cannot be used to meaningfully define what it means for a general solution to a partial differential equation to behave in a prescribed way on the boundary of $\backslash Omega.$ The way out of this difficulty is the observation that while an element $u$ in a Sobolev space may be ill-defined as a function, $u$ can be nevertheless approximated by a sequence $(u\_n)$ of $C^1$ functions defined on the closure of $\backslash Omega.$ Then, the restriction $u\_$ of $u$ to $\backslash partial\; \backslash Omega$ is defined as the limit of the sequence of restrictions $(u\_)$. 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「trace operator」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.