APPROXIMATION IN FRACTIONAL SOBOLEV SPACES AND HODGE SYSTEMS
Résumé
Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $\varphi$ be a differential $l$-form on $\R^d$ with $\dot{W}^{1,d}$ coefficients. It was proved by Bourgain and Brezis that there exists a differential $l$-form $\psi$ on $\R^d$ with coefficients in $L^{\infty}\cap \dot{W}^{1,d}$ such that $d\varphi=d\psi$. Bourgain and Brezis also asked whether this result can be extended to differential forms with coefficients in the fractional Sobolev space $\dot{W}^{s,p}$ with $sp=d$. We give a positive answer to this question, in the more general context of Triebel-Lizorkin spaces, provided that $d-\D\leq l\leq d-1$, where $\D$ is the largest positive integer such that $\D<\min(p,d)$. The proof relies on an approximation result for functions in $\dot{W}^{s,p}$ by functions in $\dot{W}^{s,p}\cap L^{\infty}$, even though $\dot{W}^{s,p}$ does not embed into $L^{\infty}$ in this critical case.
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