Trace Result for Sobolev Extension Domains

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Djamel Ait-Akli
Abdelkader Merakeb

Abstract

In this paper, we establish the existence and continuity of a trace operator for functions of the Sobolev space W1,p(Ω) with 1<p<∞ on the boundary of a domain Ω that has the Sobolev W1,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 W1,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.

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References

  1. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Ren. Sem. Mat. Univ. Padova. 27 (1957), 284-305.
  2. V.G. Maz'ya, Extension of functions from Sobolev spaces, J. Math. Sci. 22 (1983), 1851-1855.
  3. J.L. Lewis, Approximation of Sobolev functions in Jordan domains, Ark. Mat. 25 (1987), 255-264.
  4. G. Auchmuty, Sharp boundary trace inequalities, Proc. R. Soc. Edinburgh Sect. A: Math. 144 (2014), 1-12.
  5. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Americam Mathematical Society, Providence, 1998.
  6. D. Mitrea, I. Mitrea, On the Regularity of Green Functions in Lipschitz Domains, Commun. Part. Differ. Equ. 36 (2010), 304-327.