Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N>1$ consists in minimizing the quotients $\mathrm{\mid }\mathrm{\partial }E\mathrm{\mid }\mathrm{/}\mathrm{\mid }E\mathrm{\mid }$ among all smooth subdomains $E\subset \mathrm{\Omega }$ and the Cheeger constant $h(\mathrm{\Omega })$ is the minimum of these quotients. Let ${\varphi }_p\in C^{1,\alpha }(\bar{\mathrm{\Omega }})$ be the $p$-torsion function, that is, the solution of torsional creep problem $-{\mathrm{\Delta }}_p{\varphi }_p=1$ in $\mathrm{\Omega }$, ${\varphi }_p=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where ${\mathrm{\Delta }}_{p}u:=\mathrm{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that ${\lim }_{p\to 1^{+}}(\mathrm{\parallel }{\varphi }_p{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}{)}^{1-p}=h(\mathrm{\Omega })={\lim }_{p\to 1^{+}}(\mathrm{\parallel }{\varphi }_p{\mathrm{\parallel }}_{L^1(\mathrm{\Omega })}{)}^{1-p}$. Moreover, we deduce the relation $\lim_{p\to1^{+}}|\phi_{p}|{L^{1}(\Omega)}\geq C{N}\lim_{p\to1^{+}}|\phi_{p}|{L^{\infty}(\Omega)}$where$C{N}$ is a constant depending only of $N$ and $h(\mathrm{\Omega })$, explicitely given in the paper. An eigenfunction $u\in BV(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$ of the Dirichlet 1-Laplacian is obtained as the strong $L^1$ limit, as $p\to 1^{+}$, of a subsequence of the family $\{{\varphi }_p\mathrm{/}\mathrm{\parallel }{\varphi }_p{\mathrm{\parallel }}_{L^1(\mathrm{\Omega })}{\}}_{p>1}$. Almost all $t$-level sets $E_t$ of $u$ are Cheeger sets and our estimates of $u$ on the Cheeger set $\mathrm{\mid }{E}_0\mathrm{\mid }$ yield $|B_{1}|h(B_{1})^{N}\leq |E_{0}|h(\Omega)^{N},$where$B_{1}$istheunitballin$$\mathbb{R}$^{N}$\mathrm{.}For$\Omega$convexweobtain$u=|E_{0}|^{-1}\chi_{E_{0}}$.