Lusin Lipschitz approximation in BV and Sobolev space

Question

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that

Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $u\in BV(\mathbb R^N)$ and $\lambda>0$ there exists a Lipschitz function $v_\lambda \in \mathrm{Lip}(\mathbb R^N)$ with Lipschitz constant $Lip(v_\lambda, \mathbb{R}^N) \le \kappa \lambda$ and $$ |x\in \mathbb R^N : u(x)\ne v_\lambda(x)| \le \frac{\kappa}{\lambda} |Du|(\mathbb R^N), $$ where $\kappa$ depends on $N$ and $m$.

1. Can the result be made more precise using the Radon-Nikodym decomposition $Du = D^{a.c.}u + D^{j}u+D^{c}u$? In particular, what is the role of the singular part $D^{sing}u = D^j u + D^c u$?

2. What (improved) Lusin approximation result by Lipschitz function can be stated if $u \in [W^{1,p}(\mathbb{R}^N)]^m$ (with $1 \le p < \infty$)?

Answer 1

With regards to Part 2, the paper Improved $C^{k,\lambda}$ Approximation of Higher Order Sobolev Functions in Norm and Capacity by Bojarski, Hajlasz, and Strzelecki should be helpful. In particular, a Lusin approximation by smooth functions (originally proven by Michael and Ziemer in 1985) is stated as Theorem 1.1, and improved results are given as Theorems 1.2, 1.3, and 1.4.