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Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain $Ω\subset\mathbb{R}^{N}$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subsetΩ$ and the Cheeger constant $h(Ω)$ is the minimum of these quotients. Let $ϕ_{p}\in C^{1,α}(\barΩ)$ be the $p$-torsion function, that is, the solution of torsional creep problem $-Δ_{p}ϕ_{p}=1$ in $Ω$, $ϕ_{p}=0$ on $\partialΩ$, where $Δ_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that $\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{\infty}(Ω)})^{1-p}=h(Ω)=\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{1}(Ω)})^{1-p}$. Moreover, we deduce the relation $\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{1}(Ω)}\geq C_{N}\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{\infty}(Ω)}$ where $C_{N}$ is a constant depending only of $N$ and $h(Ω)$, explicitely given in the paper. An eigenfunction $u\in BV(Ω)\cap L^{\infty}(Ω)$ of the Dirichlet 1-Laplacian is obtained as the strong $L^{1}$ limit, as $p\rightarrow1^{+}$, of a subsequence of the family $\{ϕ_{p}/\|ϕ_{p}\|_{L^{1}(Ω)}\}_{p>1}$. Almost all $t$-level sets $E_{t}$ of $u$ are Cheeger sets and our estimates of $u$ on the Cheeger set $|E_{0}|$ yield $|B_{1}|h(B_{1})^{N}\leq |E_{0}|h(Ω)^{N},$ where $B_{1}$ is the unit ball in $\mathbb{R}^{N}$. For $Ω$ convex we obtain $u=|E_{0}|^{-1}χ_{E_{0}}$.