Construction of The Trace Operator
To rigorously define the notion of restriction to a function in a Sobolev space, let be a real number. Consider the linear operator
defined on the set of all functions on the closure of with values in the Lp space given by the formula
The domain of is a subset of the Sobolev space It can be proved that there exists a constant depending only on and such that
- for all in
Then, since the functions on are dense in, the operator admits a continuous extension
defined on the entire space is called the trace operator. The restriction (or trace) of a function in is then defined as
This argument can be made more concrete as follows. Given a function in consider a sequence of functions that are on with converging to in the norm of Then, by the above inequality, the sequence will be convergent in Define
It can be shown that this definition is independent of the sequence approximating
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