[proofplan] The proof uses blow-up analysis at points of the essential boundary $\partial^* E$ to show that $D\mathbb{1}_E$ has $(n-1)$-dimensional flat tangent measures, then invokes the Preiss--Mattila rectifiability criterion. At $\mathcal{H}^{n-1}$-a.e. point of $\partial^* E$, the rescaled sets $r^{-1}(E - x_0)$ converge in $L^1_{\mathrm{loc}}$ to a half-space, giving the measure-theoretic outward normal $\nu_E$ and the representation $D\mathbb{1}_E = -\nu_E\,\mathcal{H}^{n-1} \lfloor \partial^* E$. [/proofplan] [step:Define the essential boundary and classify points by density] For $x \in \mathbb{R}^n$, define the upper and lower densities: \begin{align*} \overline{D}(E, x) = \limsup_{r \to 0} \frac{\mathcal{L}^n(E \cap B(x,r))}{\mathcal{L}^n(B(x,r))}, \qquad \underline{D}(E, x) = \liminf_{r \to 0} \frac{\mathcal{L}^n(E \cap B(x,r))}{\mathcal{L}^n(B(x,r))}. \end{align*} The essential boundary is $\partial^* E = \{x : 0 < \underline{D}(E, x) \leq \overline{D}(E, x) < 1\}$. The Lebesgue density theorem gives $\mathcal{L}^n(\partial^* E) = 0$. [/step] [step:Show the blow-up at $\partial^* E$ converges to a half-space indicator] At $\mathcal{H}^{n-1}$-a.e. $x_0 \in \partial^* E$, the rescaled sets $r^{-1}(E - x_0)$ converge in $L^1_{\mathrm{loc}}$ to a half-space $\{y : y \cdot \nu(x_0) > 0\}$ for some unit vector $\nu(x_0)$. The rescaled measures $r^{1-n}D\mathbb{1}_E(x_0 + r\,\cdot)$ converge (in the sense of Radon measures) to $\nu(x_0)\,\mathcal{H}^{n-1} \lfloor \{y : y \cdot \nu(x_0) = 0\}$, a flat $(n-1)$-dimensional measure. This is established using the monotonicity formula for the perimeter ratio $r^{1-n}|D\mathbb{1}_E|(B(x_0, r))$ (which is monotone non-decreasing in $r$ at $\mathcal{H}^{n-1}$-a.e. point of $\partial^* E$) and the characterisation of flat tangent measures as half-space boundaries. [/step] [step:Conclude rectifiability of $\partial^* E$ via the Preiss--Mattila criterion] By the previous step, at $\mathcal{H}^{n-1}$-a.e. point of $\partial^* E$, the tangent measure of $|D\mathbb{1}_E|$ is a flat $(n-1)$-dimensional measure $c\,\mathcal{H}^{n-1} \lfloor \Pi$ for some hyperplane $\Pi$. The Preiss--Mattila theorem states that a Radon measure whose tangent measures are $(n-1)$-dimensional a.e. is supported on a countably $\mathcal{H}^{n-1}$-rectifiable set. This gives the rectifiability of $\partial^* E$. [/step] [step:Assemble the representation formula $D\mathbb{1}_E = -\nu_E\,\mathcal{H}^{n-1} \lfloor \partial^* E$] Combining the blow-up analysis and rectifiability: \begin{align*} D\mathbb{1}_E = -\nu_E\,\mathcal{H}^{n-1} \lfloor \partial^* E, \end{align*} where $\nu_E(x) = \nu(x)$ is the measure-theoretic outward unit normal from the blow-up step. The normalisation constant is $c = 1$, determined by $|D\mathbb{1}_E|(B(x,r)) / (\omega_{n-1}r^{n-1}) \to 1$ at $\mathcal{H}^{n-1}$-a.e. $x \in \partial^* E$. In particular: \begin{align*} \operatorname{Per}(E; U) = |D\mathbb{1}_E|(U) = \mathcal{H}^{n-1}(\partial^* E \cap U). \end{align*} [/step]