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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.

MSC:

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
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References:

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