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An inequality for Steklov eigenvalues for planar domains. (English) Zbl 0868.35078

The author studies the spectrum of the Neumann operator \(N_{\partial\Omega}:u\mapsto \partial_\nu\widetilde u\biggl|_{\partial\Omega}\), where \(\widetilde u\) is the solution of the problem \(\Delta\widetilde u=0\) in \(\Omega\), \(\widetilde u\biggl|_{\partial\Omega}=u\). Let \(0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots\) be the spectrum of \(N_{\partial\Omega}\), \(\Omega\subset \mathbb{R}^2\), and \(0=\lambda_0'<\lambda_1'\leq \lambda_2'\leq\cdots\) be the spectrum of \(N_{S^1}\) (the Neumann operator on the unit ball in \(\mathbb{R}^2\)). The main result of the paper states that \(\sum_j(\lambda^2_j-(\lambda_j')^2)\geq 0\) with equality if and only if \(\Omega\) is a rigid copy of the unit ball.

MSC:

35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:

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