Dimensional lower bounds for contact surfaces of Cheeger sets

Abstract

We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $\Omega$. We show that this size is strongly related to the regularity of $\partial \Omega$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partial \Omega$. In particular we show that, if $\partial \Omega$ has $C^{1,\alpha }$ regularity then ${\mathcal{H}}^{d-2+\alpha }(\partial E\cap \partial \Omega )>0$. This shows that a sufficient condition to ensure that ${\mathcal{H}}^{d-1}(\partial E\cap \partial \Omega )>0$ is that $\partial \Omega$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $\partial E$ as well, we obtain that $\Omega$ convex, which yields $\partial E\in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.