Title: Trace Result for Sobolev Extension Domains
Author(s): Djamel Ait-Akli, Abdelkader Merakeb
Pages: 503-511
Cite as:
Djamel Ait-Akli, Abdelkader Merakeb, Trace Result for Sobolev Extension Domains, Int. J. Anal. Appl., 19 (4) (2021), 503-511.

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.

Full Text: PDF

 

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. Google Scholar

  2. V.G. Maz’ya, Extension of functions from Sobolev spaces, J. Math. Sci. 22 (1983), 1851–1855. Google Scholar

  3. J.L. Lewis, Approximation of Sobolev functions in Jordan domains, Ark. Mat. 25 (1987), 255-264. Google Scholar

  4. G. Auchmuty, Sharp boundary trace inequalities, Proc. R. Soc. Edinburgh Sect. A: Math. 144 (2014), 1-12. Google Scholar

  5. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Americam Mathematical Society, Providence, 1998. Google Scholar

  6. D. Mitrea, I. Mitrea, On the Regularity of Green Functions in Lipschitz Domains, Commun. Part. Differ. Equ. 36 (2010), 304–327. Google Scholar


COPYRIGHT INFORMATION

Copyright © 2021 IJAA, unless otherwise stated.