dg.differential geometry - Trace of Sobolev Functions to the reduces boundary of an Caccioppoli set

Let $W^{k,p}$ denote the sobolev space of all $k$-times weak-differentiable $L^p$ functions such that each derivative is $L^p$ and $W_0^{k,p}(Omega)$ be the closure of $mathcal C^infty_c(Omega)$ in this space.

Given $Omegasubsetsubsetmathbb R^N$, a Caccioppoli set $Esubset Omega$, $Usubseteq Omega$ open and $win W_0^{1,p}(Omega)$. Can we conclude in the case of
$$ int_E div(eta w) = 0 qquadforall;etainmathcal C^1_c(U;mathbb R^N) $$
that there exists a sequence $w_ninmathcal C^1_c(Omega)$ with $mathcal H^{N-1}(Ucappartial^*E cap [w_nneq 0])=0$ which converges to $w$ in $W^{1,p}(Omega)$? (Where $[w_nneq 0]=Omegasetminus w_n^{-1}(0)$ and $mathcal H^{N-1}$ is the $N-1$-dimensional Hausdorff-measure).

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The assertion seems naturally to me, because for $winmathcal C_c^1(Omega)$ the equality would per definition of a caccippoli set imply
$$ int_{partial^*E} weta nu_E = 0 qquadforall;etainmathcal C^1_c(U,mathbb R^N)$$
where $nu_E$ is the inner normal of $E$ existing almost-everywhere on the reduced boundary. This would imply that $w=0$ $mathcal H^{N-1}$-almost everywhere on the reduced boundary, wouldn't it? Just the same is by the divergence-theorem true if $partial E$ is lipschitz and $w$ sobolev, because in that case the trace of $w$ on $partial E$ is well defined, which means that there exists a sequence of $mathcal C_c^1(Omega)$ functions vanashing on $partial Ecap U$ approximating $w$ in $W^{1,p}(Omega)$.

But I didn't find informations about this special case, except in the case of $win W_0^{1,p}(Omega)cap L^infty(Omega)$ in which this seems also to be correct -- see Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws, by Gui-Qiang Chen, Monica Torres in Arch. Rational Mech. Anal. 175, 2005. They do actually not need $W^{1,p}$ but it works in that case...

Are at all the statements above for $winmathcal C^1_c$ or $partial E$ lipschitz correct? What is with the case of $w$ only in $W^{1,p}$ and $E$ only caccioppoli?