looking for a Poincare-type lemma for BV functions

Question

Given a smooth bounded open subset $\Omega$ of $\mathbb{R}^n$, does there exist $A >0$ such that if $f\in BV(\Omega)$ with zero trace on $\partial \Omega$, and $\int_\Omega |Df| = 1$, then $\|f\|_{L^1(\Omega)} \leq A$? This is not a homework problem.

Answer 1

Let $\epsilon > 0$ and consider the set $$ \Omega_\epsilon = \{x\in \Omega : \text{dist}(x,\partial \Omega) > \epsilon\}. $$ with characteristic function $\chi_{\Omega_\epsilon}$. By the coarea formula the function $f_\epsilon = \frac{1}{\mathcal{H}^{n-1}(\partial \Omega_\epsilon)} \cdot \chi_{\Omega_\epsilon}$ has zero trace and fulfils $\int_\Omega|Df_\epsilon| = 1$ and $|| f_\epsilon ||_{L^1(\Omega)} = \frac{\lambda(\Omega_\epsilon)}{\mathcal{H}^{n-1}(\partial \Omega_\epsilon)}$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}^n$ and $\mathcal{H}^{n-1}$ the $(n-1)$-dimensional Hausdorff measure. By letting $\epsilon \rightarrow 0$ you get the bound $A = \frac{\lambda(\Omega)}{\mathcal{H}^{n-1}(\partial \Omega)}$.