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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

Citations:

Zbl 0359.35060
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References:

[1] (Abramowitz, M. S.; Stegun, I. A., Handbook of Mathematical Functions, (1965), Dover New York) · Zbl 0171.38503
[2] 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
[3] 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
[4] 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
[5] 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
[6] 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)
[7] Daners, D., A Faber-krahn inequality for Robin problems in any space dimension, Math. Ann., 335, 767-785, (2006) · Zbl 1220.35103
[8] Daners, D., Principal eigenvalues for generalised indefinite Robin problems, Potential Anal., 38, 1047-1069, (2013) · Zbl 1264.35152
[9] Daners, D.; Kennedy, J. B., On the asymptotic behaviour of the eigenvalues of a Robin problem, Differential Integral Equations, 23, 601-799, (2010)
[10] 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
[11] 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
[12] 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
[13] Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag Berlin · Zbl 0148.12601
[14] Krahn, E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann., 94, 97-100, (1924) · JFM 51.0356.05
[15] 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
[16] 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
[17] 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
[18] 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
[19] 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
[20] Rayleigh, J. W.S., The theory of sound, (1945), Dover New York, reprinted:
[21] Savo, A., Lower bounds for the nodal length of eigenfunctions of the Laplacian, Ann. Global Anal. Geom., 16, 133-151, (2001) · Zbl 1010.58025
[22] Szegö, G., Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal., 3, 343-356, (1954) · Zbl 0055.08802
[23] Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal., 5, 633-636, (1956) · Zbl 0071.09902
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