Trace Operator
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High Quality Content by WIKIPEDIA articles! 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. Let be a bounded open set in the Euclidean space mathbbmathbb R^n with C1 boundary partial Omega. If u is a function that is C1 (or even just continuous) on the...
High Quality Content by WIKIPEDIA articles! 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. Let be a bounded open set in the Euclidean space mathbbmathbb R^n with C1 boundary partial Omega. If u is a function that is C1 (or even just continuous) on the closure bar Omega of , 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.
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