Analytic approximation of the intrinsic distance to the boundary of an open subset

Question

Let $\Omega$ be an open subset of $\mathbb{R}^n$, do we have for any $x\in \Omega$ : $$\sup_\{\phi\in D(\Omega), \|d\phi\| \le 1\} |\phi(x)|=d_\Omega (x, \partial\Omega),$$

where the norm is the supremum norm on one-forms, $D(\Omega)$ is the space of compactly supported smooth functions on $\Omega$ and $d_\Omega$ is the intrinsic distance in $\Omega$ ? (Obviously, we have $\le$ in general.)

If not, are there necessary and/or sufficient (regularity / boundary) conditions on $\Omega$ for the result ?

Answer 1

Yes. Fix $x\in\Omega$ and let $r=d_\Omega(x,\partial\Omega)$, then $\Omega$ contains the Euclidean $r$-ball centered at $x$. So it suffices to construct $\phi$ supported in this ball with $\|d\phi\|\le 1$ and $\phi(x)\approx r$. A suitable smoothening of the 1-Lipschitz function $y\mapsto\max\{r-|x-y|,0\}$ does the job.