In this work, a new approach to obtain a solenoidal Lipschitz truncation is presented. More precisely, the goal of the truncation is to modify a function $u \in W^{ 1,p}(\mathbb{ R}^3,\mathbb{ R}^3)$ that satisfies the additional constraint $\mathrm{ div}~ u=0$, such that its modification $\tilde{ u}$ is in $W^{ 1,\infty}(\mathbb{ R}^3,\mathbb{ R}^3)$ and still is divergence-free. We give an alternative approach to Lipschitz truncation compared to previous works by Breit, Diening & Fuchs (2012) and Breit, Diening & Schwarzacher (2013). The ansatz pursued here allows a rather strict bound on the $W^{ 1,p}$ distance of $u$ and $\tilde{ u}$.
Categories: math.AP