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.


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
Full Text: DOI arXiv


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