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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 \Omega be a bounded open set in the Euclidean space \mathbb R^n with ''C''1 boundary \partial \Omega. If u is a function that is C^1 (or even just continuous) on the closure \bar \Omega of \Omega, its function restriction is well-defined and continuous on \partial \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 \partial \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 \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 \Omega. Then, the restriction u_ of u to \partial \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)
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