Approximation in higher-order Sobolev spaces and Hodge systems

Abstract

Let d≥2 be an integer, 1≤l≤d−1 and φ be a differential l-form on Rd with W˙1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on Rd with coefficients in L∞∩W˙1,d such that dφ=dψ. 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˙2,d/2 or more generally in the fractional Sobolev spaces W˙s,p with sp=d. We give a positive answer to this question, provided that d−κ≤l≤d−1, where κ is the largest positive integer such that κ<min⁡(p,d). The proof relies on an approximation result (interesting in its own right) for functions in W˙s,p by functions in W˙s,p∩L∞, even though W˙s,p does not embed into L∞ 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.