## Isoperimetric estimates for the first eigenvalue of the $$p$$-Laplace operator and the Cheeger constant.(English)Zbl 1105.35029

The eigenvalue $$\lambda =\lambda _p(\Omega )$$ corresponding to a positive weak solution $$u\in W^{1,p}_0(\Omega )$$ of the homogeneous Dirichlet boundary-value problem for the equation $$\text{div}(| \nabla u| ^{p-2}\nabla u) +\lambda | u| ^{p-2}u=0$$, characterized by the Rayleigh quotient $$\lambda _p(\Omega )= \min _{v\in W^{1,p}_0(\Omega )}\int _\Omega | \nabla v| ^p\,\text dx/ \int _\Omega | v| ^p\,\text dx$$, is addressed and related with Cheeger’s constant $$h(\Omega )$$ defined as the infimum of $$\text{meas}_{n-1}(\partial D)/\text{meas}_n(D)$$ over all smooth subdomains $$D\subset \Omega \subset {\mathbb R}^n$$ whose boundary $$\partial D$$ does not touch $$\partial \Omega$$. A subset $$D\subset \Omega$$ with $$\text{meas}_{n-1}(\partial D)/\text{meas}_n(D)=h(\Omega )$$ is called a Cheeger domain. It is proved that $$\lambda _p(\Omega )\to h(\Omega )$$ for $$p\to 1$$, and the corresponding eigenfunction $$u_p$$ (if normalized), converges to the characteristic function of the Cheeger set. It implies, in particular, that the Cheeger set is convex if $$\Omega$$ is convex.

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J70 Degenerate elliptic equations 49R50 Variational methods for eigenvalues of operators (MSC2000)
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