I am currently studying the theory of functions of Bounded Variation on $\mathbb{R}^d$. Trying to solve an exercise, I got stuck wondering if given a function $u$ in $BV(\Omega)$, under some suitable geometrical conditions, the Jump set $\mathcal{J_u}$ of $u$ is the reduced boundary of a set of finite perimeter $F\subset\mathbb{R}^d$ (so that I may consider the characteristic function of $F$). This in general is clearly not the case, since if one considers in $\mathbb{R}^2$ the set $E=$" boundary of a rectangle with one side removed", there exists a $u\in BV(\Omega)$ such that $J_u=E$, but clearly $E$ is not the reduced boundary of any set of finite perimeter in $\mathbb{R}^2$.
To be more precise, I give a brief definition of the objects at hand and a precise statement to what I would like to know. I denote with $\mathcal{H}^{d-1}$ the $d-1$-dimensional Hausdorff measure on $\mathbb{R}^d$. A set $E\subset\mathbb{R}^d$ is said to be a $\mathcal{H}^{d-1}$-rectifiable set if $\mathcal{H}^{d-1}(E)<+\infty$, and there exists a countable sequence of Lipschitz maps $f_i$ from some hyperplane $\pi_i\cong\mathbb{R}^{d-1}$ into $\mathbb{R}^d$ such that $\mathcal{H}^{d-1}(E\setminus\cup f(\pi_i))=0$. For $k\in\{0,...,d\}$, I denote with $\Theta^{k}(E,x)$ the $k$-dimensional density of $E$ at $x\in\mathbb{R}^d$, that is, if it exists, the quantity
$$\Theta^k(E,x)=\lim_{\rho\to0^+}\frac{\mathcal{H}^{k}(E\cap B_\rho(x))}{\omega_k\rho^k},$$
where $B_\rho(x)$ is the ball of center $x$ and radius $\rho$, $\omega_k$ the $k$-dimensional Lebesgue measure of the unit ball. It is known (see for instance Ambrosio, Fusco, Pallara, "Functions of Bounded variations and Free discontinuity Problems", Theorem $2.66$), that $E$ is $\mathcal{H}^{d-1}$-rectifiable iff $\Theta^{d-1}(E,x)=1$ for $\mathcal{H}^{d-1}$-a.e. $x\in E$.
A set $F\subset\mathbb{R}^d$ is said to be of finite perimeter if
$$\sup\left\{\int_F\text{div}\varphi dx:\,\, \varphi\in C^\infty_c(\mathbb{R}^d,\mathbb{R^d}), \,\,|\varphi|\leq 1 \right\}<+\infty.$$
It is well known that if a set has finite perimeter, then there exists a Radon measure $\mu_F$, for which the Gauss-Green formula for integrations holds true. I denote by $\partial^*F$ the reduced boundary of $F$, that is the set on which $\mu_F$ concentrates. More precisely $\partial^*F$ is the set of points for which $|\mu_F|(B_\rho(x))>0$ for every $\rho>0$ and for which $$\nu(x):=\lim_{\rho\to 0^+}\frac{\mu_F(B_\rho(x)}{|\mu_F|(B_\rho(x))},$$ exists and $|\nu(x)|=1.$ I denote by $\partial_eF$ the essential boundary of $F$, i.e. the set
$$\partial_eF:=\mathbb{R}^d\setminus\{x:\Theta^d(F,x)=1 \,\,or \,\,\Theta^d(F,x)=0\}.$$
Again, I know that for a set of finite perimeter $F$, $\mathcal{H}^{d-1}(\partial_eF\setminus\partial^*F)=0$ and that $\partial^*F$ is $\mathcal{H}^{d-1}$-rectifiable (again, see Ambrosio,..., Chapter 2).
Let $E\subset \mathbb{R}^d$ be $\mathcal{H}^{d-1}$-rectifiable. I would like to know if there exists some general condition on $E$ that assures me that there exists a set of finite perimeter $F\subset\mathbb{R}^d$, with positive $d$-dimensional Lebesgue measure such that $\partial^*F=E$ or, equivalently $\partial_eF=E$ (the equalities are intended up to $\mathcal{H}^{d-1}$-negligeable sets). In the previous "counter-example" in $2$-dimensions, what was going wrong I think may be rephrased as: There exists a subset $E_0\subset E$, with $\mathcal{H}^0(E_0)\neq 0$ such that $\Theta^{1}(E,x)=\frac{1}{2}$ for $x\in E_0$ ($E_0$ in that case is made of the two vertices of the missing side). Since these types of counterexamples are the only ones I am able to think of also in higher dimension, do you know if the following might be true?
If $\Theta^{d-1}(E,x)=1$ for $\mathcal{H}^{d-2}$-a.e.$x\in E$, then there exists $F$ of finite perimiter such that $\partial^*F=E$ (or equivalently $\partial_eF=E$).
Do you have any references for these kinds of problems? Any known results with stronger/different hypotheses? I tried also to look on Maggi's book but I did not find anything about it.