Abstract
In this paper, we establish the existence and continuity of a trace operator for functions of the Sobolev space W 1,p (Ω) with 1<p<∞ on the boundary of a domain Ω that has the Sobolev W 1,p −extension property. First, we prove the existence and the continuity of such an operator when it is applied to the elements of the subspace of the up to boundary smooth functions by using a uniform estimate. The essential ingredients used in the proof of this estimate are Green’s representation of a function on a disk as well as Banach’s isomorphism theorem. Finally, we conclude the trace result using the density of smooth functions in W 1,p (Ω). The presented proof fully exploits the extensibility hypothesis of the domain Ω. The relevance of the result lies in the existence of extension domains which are not Lipschitz and under this point of view it constitutes a generalization of the usual trace theorem.
Highlights
The trace operator, when applied to the functions of the Sobolev space W 1,p(Ω) defined on a Lipschitz domain Ω, is a standard notion in the theory of Sobolev spaces
We conclude the trace result using the density of smooth functions in W 1,p(Ω)
The relevance of the result lies in the existence of extension domains which are not Lipschitz and under this point of view it constitutes a generalization of the usual trace theorem
Summary
The trace operator, when applied to the functions of the Sobolev space W 1,p(Ω) defined on a Lipschitz domain Ω, is a standard notion in the theory of Sobolev spaces. We establish the existence and continuity of a trace operator for functions of the Sobolev space W 1,p(Ω) with 1 < p < ∞ on the boundary of a domain Ω that has the Sobolev W 1,p−extension property. We prove the existence and the continuity of such an operator when it is applied to the elements of the subspace of the up to boundary smooth functions by using a uniform estimate.
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