## 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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