# Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

• Volume: 44, Issue: 4, page 659-667
• ISSN: 0010-2628

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## Abstract

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First we recall a Faber-Krahn type inequality and an estimate for ${\lambda }_{p}\left(\Omega \right)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue ${\lambda }_{p}\left(\Omega \right)$ converges to the Cheeger constant $h\left(\Omega \right)$ 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.

## How to cite

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Kawohl, Bernhard, and Fridman, V.. "Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 659-667. <http://eudml.org/doc/249207>.

@article{Kawohl2003,
abstract = {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 $p\rightarrow 1$. 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.},
author = {Kawohl, Bernhard, Fridman, V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator; Faber-Krahn type inequality; eigenvalue for -Laplacian; Cheeger set; 1-Laplace operator},
language = {eng},
number = {4},
pages = {659-667},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant},
url = {http://eudml.org/doc/249207},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Kawohl, Bernhard
AU - Fridman, V.
TI - Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 659
EP - 667
AB - 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 $p\rightarrow 1$. 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.
LA - eng
KW - isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator; Faber-Krahn type inequality; eigenvalue for -Laplacian; Cheeger set; 1-Laplace operator
UR - http://eudml.org/doc/249207
ER -

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