A proof of the Faber-Krahn inequality for the first eigenvalue of the \(p\)-Laplacian. (English) Zbl 0966.35091

The aim of the paper is to prove the following theorem.
Theorem 1. Let \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), be a bounded domain and \(\Omega^*\) be the ball with the same volume as \(\Omega\) and centered at the origin. For \(1< p<\infty\) let \(\lambda_1= \lambda_1(p,n,\Omega)\) and \(u= u(p,n,\Omega)\) be the first eigenvalue and the first eigenfunction of the \(p\)-Laplacian on \(\Omega\), that is \[ L_pu+ \lambda_1|u|^{p- 2}u= 0, \] with \(u\in W^{1,p}_0(\Omega)\) and \(L_pu\equiv \text{div}(|Du|^{p-2} Du)\) the \(p\)-Laplacian. If \(\lambda^*_1= \lambda_1(p,n,\Omega^*)\) is the first eigenvalue of the ball \(\Omega^*\), then \(\lambda_1\geq \lambda^*_1\), where equality holds iff \(\Omega\) is a ball.
The proof is based on Talenti’s inequality and properties of the first eigenfunction.


35P15 Estimates of eigenvalues in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI


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