## 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
Full Text:

### References:

  (Abramowitz, M. S.; Stegun, I. A., Handbook of Mathematical Functions, (1965), Dover New York) · Zbl 0171.38503  Antunes, P. R.S.; Freitas, P.; Kennedy, J. B., Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian, ESAIM Control Optim. Calc. Var., 19, 438-459, (2013) · Zbl 1264.35151  Ashbaugh, M. S.; Benguria, R. D., On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78, 1-228, (1995) · Zbl 0833.35035  Bareket, M., On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal., 8, 280-287, (1977) · Zbl 0359.35060  Bossel, M.-H., Membranes élastiquement liées: extension du théoréme de Rayleigh-Faber-krahn et de l’inégalité de Cheeger, C. R. Acad. Sci. Paris Sér. I Math., 302, 47-50, (1986) · Zbl 0606.73018  Brock, F.; Daners, D., Conjecture concerning a Faber-krahn inequality for Robin problems, Oberwolfach Rep., 4, 1022-1023, (2007), Open Problem in Mini-Workshop: Shape Analysis for Eigenvalues (organized by: D. Bucur, G. Buttazzo, A. Henrot)  Daners, D., A Faber-krahn inequality for Robin problems in any space dimension, Math. Ann., 335, 767-785, (2006) · Zbl 1220.35103  Daners, D., Principal eigenvalues for generalised indefinite Robin problems, Potential Anal., 38, 1047-1069, (2013) · Zbl 1264.35152  Daners, D.; Kennedy, J. B., On the asymptotic behaviour of the eigenvalues of a Robin problem, Differential Integral Equations, 23, 601-799, (2010)  Exner, P.; Minakov, A.; Parnovski, L., Asymptotic eigenvalue estimates for a Robin problem with a large parameter, Port. Math., 71, 141-156, (2014) · Zbl 1295.35346  Faber, G., Beweis dass unter Allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt, Sitzungsber. Bayer. Akad. Wiss., 169-172, (1923) · JFM 49.0342.03  Ferone, V.; Nitsch, C.; Trombetti, C., On a conjectured reversed Faber-krahn inequality for a Steklov-type Laplacian eigenvalue, Commun. Pure Appl. Anal., 14, 63-81, (2015) · Zbl 1338.46046  Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag Berlin · Zbl 0148.12601  Krahn, E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann., 94, 97-100, (1924) · JFM 51.0356.05  Lacey, A. A.; Ockendon, J. R.; Sabina, J., Multidimensional reaction diffusion equations with nonlinear boundary conditions, SIAM J. Appl. Math., 58, 1622-1647, (1998) · Zbl 0932.35120  Levitin, M.; Parnovski, L., On the principal eigenvalue of a Robin problem with a large parameter, Math. Nachr., 281, 272-281, (2008) · Zbl 1136.35060  Nadirashvili, N. S., Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Ration. Mech. Anal., 129, 1-10, (1995) · Zbl 0826.73035  Pankrashkin, K., On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains, Nanosystems: Physics, Chemistry, Mathematics, 4, 474-483, (2013) · Zbl 1386.35302  Payne, L. E.; Weinberger, H. F., Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 2, 210-216, (1961) · Zbl 0098.39201  Rayleigh, J. W.S., The theory of sound, (1945), Dover New York, reprinted:  Savo, A., Lower bounds for the nodal length of eigenfunctions of the Laplacian, Ann. Global Anal. Geom., 16, 133-151, (2001) · Zbl 1010.58025  Szegö, G., Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal., 3, 343-356, (1954) · Zbl 0055.08802  Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal., 5, 633-636, (1956) · Zbl 0071.09902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.