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Keywords:
isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator
isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator
Summary:
First we recall a Faber-Krahn type inequality and an estimate for $\lambda_p(\Omega)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue $\lambda_p(\Omega)$ converges to the Cheeger constant $h(\Omega)$ as $p\to 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega$ which minimizes the ratio $|\partial D|/|D|$ among all simply connected $D\subset\subset\Omega$. As a byproduct we prove that for convex $\Omega$ the Cheeger set $\omega$ is also convex.
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