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.
@article{119420, author = {Bernhard Kawohl and V. Fridman}, title = {Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {44}, year = {2003}, pages = {659-667}, zbl = {1105.35029}, mrnumber = {2062882}, language = {en}, url = {http://dml.mathdoc.fr/item/119420} }
Kawohl, Bernhard; Fridman, V. Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 659-667. http://gdmltest.u-ga.fr/item/119420/
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