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

Neumann operator
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References:

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