Geodesic
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 659-667.

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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.
Classification : 35J20, 35J70, 49Q20, 49R05, 49R50, 52A38
Keywords: isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator 
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     title = {Isoperimetric estimates for the first eigenvalue of the $p${-Laplace} operator and the {Cheeger} constant},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {659--667},
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     zbl = {1105.35029},
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Kawohl, B.; Fridman, V. Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 659-667. http://geodesic.mathdoc.fr/item/CMUC_2003__44_4_a9/