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 R d with W ˙ 1 , d coefficients. It was proved by Bourgain and Brezis ( [5, Theorem 5] ) that there exists a differential l -form ψ on R d with coefficients in L ∞ ∩ W ˙ 1 , d such that d φ = d ψ . 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 s p = 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.