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.

Let ${\displaystyle \Omega }$ be a bounded open set in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with C¹ boundary ${\displaystyle \partial \Omega .}$ If ${\displaystyle u}$ is a function that is ${\displaystyle C^{1}}$ (or even just continuous) on the closure ${\displaystyle {\bar {\Omega }}}$ of ${\displaystyle \Omega ,}$ its function restriction is well-defined and continuous on ${\displaystyle \partial \Omega .}$ If however, ${\displaystyle u}$ 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 ${\displaystyle \partial \Omega }$ 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 ${\displaystyle \Omega .}$

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