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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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