Solutions of the Cheeger problem via torsion functions

Abstract

The Cheeger problem for a bounded domain $\Omega \subset {\mathbb{R}}^N$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subset \Omega$ and the Cheeger constant $h(\Omega )$ is the minimum of these quotients. Let ${\varphi }_p\in C^{1,\alpha }(\bar{\Omega })$ be the $p$-torsion function, that is, the solution of torsional creep problem $-{\Delta }_p{\varphi }_p=1$ in $\Omega$, ${\varphi }_p=0$ on $\partial \Omega$, where ${\Delta }_{p}u:=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that ${\lim }_{p\to 1^{+}}(\Vert {\varphi }_p{\Vert }_{L^{\infty }(\Omega )}{)}^{1-p}=h(\Omega )={\lim }_{p\to 1^{+}}(\Vert {\varphi }_p{\Vert }_{L^1(\Omega )}{)}^{1-p}$. Moreover, we deduce the relation ${\lim }_{p\to 1^{+}}\Vert {\varphi }_p{\Vert }_{L^1(\Omega )}\ge C_N{\lim }_{p\to 1^{+}}\Vert {\varphi }_p{\Vert }_{L^{\infty }(\Omega )}$ where $C_N$ is a constant depending only of $N$ and $h(\Omega )$, explicitely given in the paper. An eigenfunction $u\in BV(\Omega )\cap L^{\infty }(\Omega )$ of the Dirichlet 1-Laplacian is obtained as the strong $L^1$ limit, as $p\to 1^{+}$, of a subsequence of the family $\{{\varphi }_p/\Vert {\varphi }_p{\Vert }_{L^1(\Omega )}{\}}_{p>1}$. Almost all $t$-level sets $E_t$ of $u$ are Cheeger sets and our estimates of $u$ on the Cheeger set $|E_0|$ yield $|B_1|h(B_1{)}^N\le |E_0|h(\Omega {)}^N,$ where $B_1$ is the unit ball in ${\mathbb{R}}^N$. For $\Omega$ convex we obtain $u=|E_0{|}^{-1}{\chi }_{E_0}$.

Typos were corrected