Approximation in higher-order Sobolev spaces and Hodge systems
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AbstractLet d >= 2 be an integer, 1 <= l <= d - 1 and phi be a differential l-form on R-d with (W) over dot(1,5) coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form psi on R-d with coefficients in L-infinity boolean AND(W) over dot(1,d) such that d phi = d psi. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space (W) over dot(2,d/2) or more generally in the fractional Sobolev spaces (W) over dot(s,p) with sp = d. We give a positive answer to this question, provided that d - kappa <= l <= d - 1, where kappa is the largest positive integer such that kappa < min(p, d). The proof relies on an approximation result (interesting in its own right) for functions in (W) over dot(s,p) by functions in (W) over dot(s,p) boolean AND L-infinity, even though (W) over dot(s,p) does not embed into L-infinity in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.
Acceptance Date06/08/2018
All Author(s) ListPierre Bousquet, Emmanuel Russ, Yi Wang, Po-Lam Yung
Journal nameJournal of Functional Analysis
Volume Number276
Issue Number5
Pages1430 - 1478
LanguagesEnglish-United States

Last updated on 2020-26-03 at 02:29