Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ the closure of $\mathcal C^\infty_c(\Omega)$ in this space.

Let $E\subset \Omega$ be a Caccioppoli set (i.e. a set of finite perimeter in $\mathbb{R}^N$), $U\subset\Omega$ be open and $w\in W_0^{1,p}(\Omega)$. Suppose that
  $$ \int_E \operatorname{div}(\eta w) = 0 \qquad\forall\;\eta\in\mathcal C^1_c(U;\mathbb R^N).$$
  Question: Does there exist a sequence $w_n\in\mathcal C^1_c(\Omega)$ with $\mathcal H^{N-1}(U\cap\partial^*E \cap [w_n\neq 0])=0$ that converges to $w$ in $W^{1,p}(\Omega)$? (Here $[w_n\neq 0]=\{x\in\Omega:w_n(x)\ne 0\}$ and $\mathcal H^{N-1}$ is the $(N-1)$-dimensional Hausdorff-measure).


The assertion seems natural to me because for $w\in\mathcal C_c^1(\Omega)$ the equality implies (as $E$ is Caccioppoli)
$$ \int_{\partial^*E} w\eta \nu_E = 0 \qquad\forall\;\eta\in\mathcal C^1_c(U;\mathbb R^N),$$
where $\nu_E$ is the inner normal of $E$ which exists $\mathcal{H}^{N-1}$-a.e. on the reduced boundary $\partial^*E$. This then implies that $w=0$ $\mathcal H^{N-1}$-a.e. on $\partial^*E$. 
The same holds by the divergence theorem if $E$ has Lipschitz boundary and $w$ is Sobolev because then the trace of $w$ on $\partial E$ is well defined, and hence there exists a sequence of $\mathcal C_c^1(\Omega)$ functions vanishing on $\partial E\cap U$ that approximate $w$ in $W^{1,p}(\Omega)$.
I couldn't find much on other cases, but maybe the following reference helps for the case when $w\in W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$:
Chen, Gui-Qiang; Torres, Monica, Divergence-measure fields, sets of finite perimeter, and conservation laws, Arch. Ration. Mech. Anal. 175, No. 2, 245-267 (2005). ZBL1073.35156.
Are the above arguments for $w\in\mathcal C^1_c(\Omega)$ or if $E$ has Lipschitz boundary correct? If so, what about the general case where $w$ is only in $W^{1,p}_0(\Omega)$ and $E$ is merely Caccioppoli?
 A: Firstly, the condition in your question is sensible as it implies that $w$ vanishes $\mathcal{H}^{N-1}$-a.e. on $U\cap\partial^*E$. Moreover, your arguments for the two cases where $w\in C^1_\mathrm{c}(\Omega)$ or $E$ has Lipschitz boundary are correct. I expect that also the case $p>N$ where $w$ has a continuous representative works out. However, in general it is too much to ask that the approximating sequence is both smooth and vanishing on $U\cap\partial^*E$ up to a $\mathcal{H}^{N-1}$-nullset.

No, you cannot expect such a sequence to exist if $p\le N$ and $w\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$ and $E$ has finite perimeter in $\mathbb{R}^N$ (i.e. $E$ is Caccioppoli). 


As the construction of the example is somewhat technical, I shall first sketch the argument. An open set of finite perimeter $E$ can have massive amounts (say positive Lebesgue measure) of topological boundary $\partial E$ while the essential or reduced boundary $\partial^*E$ is small and very regular. In fact, one can arrange that there exists a nontrivial Sobolev function $w\in W^{1,p}_0(\Omega)$ supported in $\partial E\setminus E$ that vanishes $\mathcal{H}^{N-1}$-a.e. on $\partial^*E$ as $\partial^*E$ is regular.
Then $w$ satisfies the assumption in the question as it vanishes on $E$ (along with its gradient).
 Any continuous function that vanishes $\mathcal{H}^{N-1}$-a.e. on $\partial^*E$ actually vanishes everywhere on $\partial E$. So one cannot use such functions to approximate $w$ in $W^{1,p}(\Omega)$. 
Example: It suffices to consider the case where $\Omega$ is the unit ball in $\mathbb{R}^N$ and $U=\Omega$. For simplicity let $p=2$ and $N\ge 2$, but the argument works without change for $p\in(1,N]$. We make use of the same construction as in my answer https://mathoverflow.net/a/295459.
Let $E:=\bigcup_k B_k$ be a countable union of open balls with pairwise disjoint closures contained in $\Omega$, such that $E$ is dense in $\overline{\Omega}$ and $$\operatorname{cap}(E)\le\sum_k\operatorname{cap}(B_k)<\operatorname{cap}(\Omega).$$ Here for $A\subset\mathbb{R}^N$ we denote by $\operatorname{cap}(A)$ the Sobolev capacity $$
\operatorname{cap}(A)=\inf\{ \|u\|^p_{W^{1,p}} : u\in W^{1,p}(\mathbb{R}^N)\text{ and }u\ge1\text{ a.e. on neighbourhood of }A\}.
$$
Then $\sum_k\mathcal{H}^{N-1}(\partial B_k)<\infty$ because of the above estimate on the capacity. (Alternatively, just choose the balls small enough.) It is readily checked that $E$ has finite perimeter (i.e. is Caccioppoli) and $\partial^*E = \partial^*E\cap U = \bigcup_k\partial B_k$, possibly up to a $\mathcal{H}^{N-1}$-nullset.
As in the linked answer (the function is called $u$ there), there exists a nontrivial bounded $w\in W^{1,p}(\mathbb{R}^N)$ supported in $K := \overline{\Omega}\setminus E$. As $\Omega$ has a regular boundary, it follows by standard trace theory that $w\in W^{1,p}_0(\Omega)$. Observe that
$$\int_E\operatorname{div}(\eta w) = \int_\Omega \mathbf{1}_E(w\operatorname{div}(\eta) + \nabla w\cdot \eta)=0$$
for all $\eta\in C^1_\mathrm{c}(\Omega;\mathbb{R}^N)$ as both $w$ and $\nabla w$ vanish on $E$ (recall that $w$ is supported in $K$).
Suppose now that $w_n\in C^1_\mathrm{c}(\Omega)$ with $w_n=0$ $\mathcal{H}^{N-1}$-a.e. on $\partial^*E$. Then due to continuity and the structure of $\partial^*E$ one has $w_n=0$ everywhere on $\partial^*E\cup K$. As $w$ is nontrivial on $K$, the sequence $(w_n)$ cannot possibly converge to $w$ in $W^{1,p}(\Omega)$.

The above example shows that for general finite perimeter sets $E$ it is too much to ask that the approximating sequence is continuous and vanishes $\mathcal{H}^{N-1}$-a.e. on $\partial^*E$. I expect, however, that everything works out if one replaces the latter condition by something like $\mathcal{H}^{N-1}(U\cap\partial^*E\cap [w_n\ne 0])<\frac{1}{n}$ or $\|w_n\|_{L^1(U\cap\partial^*E,\mathcal{H}^{N-1})}<\frac{1}{n}$.
The assumptions in the question imply that $w$ (actually its approximately continuous representative) vanishes $\mathcal{H}^{N-1}$-a.e. on $U\cap\partial^* E$. I shall not go into details, but that claim follows from 
$$
\int_U \mathbf{1}_E w\operatorname{div}(\eta) = - \int_U \mathbf{1}_E\nabla w\cdot \eta,
$$
taking the supremum over $\eta\in C^1_\mathrm{c}(U;\mathbb{R}^N)$ with $|\eta(x)|\le 1$ for all $x\in U$ on both sides, and using that $\mathbf{1}_E w\in\mathrm{SBV}(\mathbb{R}^{N})$ with the jump behaviour that one would expect.
