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Trace Operator


Let Omega be a bounded open set in R^d whose boundary partialOmega is at least C^1 smooth and let

 T:C_c^1(Omega^_)->L^p(partialOmega)
(1)

be a linear operator defined by

 T(u)=u|partialOmega
(2)

on the collection of all real-valued compactly-supported C^1 functions with domain in the topological closure Omega^_ of Omega. In functional analysis, the trace operator is defined to be the extension

 T^~:W^(1,p)(Omega)->L^p(partialOmega)
(3)

of T to functions whose domain is the Sobolev space W^(1,p)(Omega).

Intuitively, the trace operator literally "traces" the boundary of a function u in W^(1,p)(Omega). This piece of data is of particular important when studying function spaces and partial differential equations due to the existence of various boundary-value parameters in these contexts.


See also

Stampacchia Theorem

This entry contributed by Christopher Stover

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References

Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011.

Cite this as:

Stover, Christopher. "Trace Operator." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TraceOperator.html

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