## The first Robin eigenvalue with negative boundary parameter.(English)Zbl 1317.35151

Let $$\Omega$$ be a domain in $$\mathbb{R}^d$$ of finite volume $$|\Omega|$$ and let $$\Delta:=-\partial_{x^1}^2-\dots-\partial_{xd}^2$$ be the Laplacian on $$\Omega$$. Let $$\nu$$ be the outward unit normal. For $$\alpha\in\mathbb{R}$$, consider the Robin boundary problem $$\Delta u=\lambda u$$ and $$\partial_\nu u+\alpha u|_{\partial\Omega}=0$$. Let $$\lambda_1^\alpha$$ be the first eigenvalue. For $$\alpha$$ positive, the ball of volume $$|\Omega|$$ minimizes $$\lambda_1^\alpha$$. Bareket’s conjecture [M. Bareket, SIAM J. Math. Anal. 8, 280–287 (1977; Zbl 0359.35060)] deals with the case $$\alpha\leq 0$$. \smallbreak Conjecture [Bareket]. Let $$\alpha\leq0$$ and let $$\Omega$$ be a bounded smooth domain. One has $$\lambda_1^\alpha(\Omega)\leq\lambda_1^\alpha(B)$$, where $$B$$ is a ball with $$|B|=|\Omega|$$. \medbreak The authors show this is false in general and provide a positive result if $$d=2$$. \smallbreak Theorem. The conjecture fails whenever $$\Omega$$ is a spherical shell of the same volume as the ball $$B$$ and for sufficiently large negative alpha. Moreover, for bounded planar domains of class $$C^2$$ and fixed area $$A$$, there exists a negative number $$\alpha_*(A)$$ so the conjecture holds if $$\alpha\in[\alpha_*(A),0]$$. \medbreakThe first section provides an introduction to the problem. The second section establishes the asymptotic necessary. The final section contains the proof of the results cited.

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs 35P05 General topics in linear spectral theory for PDEs 35P20 Asymptotic distributions of eigenvalues in context of PDEs

Zbl 0359.35060
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### References:

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