B. Kawohl, V. Fridman
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

Comment.Math.Univ.Carolinae 44,4 (2003) 659-667.

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 $p\to 1$. 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.

Keywords: isoperimetric estimates, eigenvalue, Cheeger constant, $p$-Laplace operator, $1$-Laplace operator
AMS Subject Classification: 35J20, 35J70, 49R05, 49Q20, 52A38