An alternative approach to solenoidal Lipschitz truncation

In this work, we present an alternative approach to obtain a solenoidal Lipschitz truncation result in the spirit of D. Breit, L. Diening and M. Fuchs [Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Equ. 253 (2012), 1910–1942.]. More precisely, the goal of the truncation is to modify a function $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^N)$ that satisfies the additional constraint $\operatorname {div} u=0$ , such that its modification $\tilde {u}$ is Lipschitz continuous and divergence-free. This approach is different to the approaches outlined in the aforementioned work and D. Breit, L. Diening and S. Schwarzacher [Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013), 2671–2700, Section 4] and is able to obtain the rather strong bound on the difference between $u$ and $\tilde {u}$ from the former article. Finally, we outline how the approach pursued in this work may be generalized to closed differential forms.