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An isoperimetric inequality for eigenvalues of the Stekloff problem. (English) Zbl 0971.35055

Summary: Let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^n\) and let \(0= \lambda_1\leq \lambda_2\leq\cdots\) denote the eigenvalues of the Stekloff problem: \(\Delta u=0\) in \(\Omega\) and \((\partial u)/(\partial v)=\lambda u\) on \(\partial\Omega\). We show that \(\sum^{n+1}_{i=2} \lambda^{-1}_i\geq n/\lambda^*_2\), where \(\lambda^*_2\) denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as \(\Omega\). The proof is based on a weighted isoperimetric inequality.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
26D15 Inequalities for sums, series and integrals
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:

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