Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant
Kawohl, Bernhard ; Fridman, V.
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 659-667 / Harvested from Czech Digital Mathematics Library

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.

Publié le : 2003-01-01
Classification:  35J20,  35J70,  49Q20,  49R05,  49R50,  52A38
@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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