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From Steklov to Neumann and Beyond, via Robin: The Szegő Way

  • Pedro Freitas (a1) (a2) and Richard S. Laugesen (a3)

Abstract

The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$ , and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$ . Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.

The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

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This research was supported by the Fundação para a Ciência e a Tecnologia (Portugal) through project PTDC/MAT-CAL/4334/2014 (Pedro Freitas), by a grant from the Simons Foundation (#429422 to Richard Laugesen), by travel support for Laugesen from the American Institute of Mathematics to the workshop on Steklov Eigenproblems (April–May 2018), and support from the University of Illinois Scholars’ Travel Fund.

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[1]Ashbaugh, M. S. and Benguria, R. D., More bounds on eigenvalue ratios for Dirichlet Laplacians in N dimensions. SIAM J. Math. Anal. 24(1993), no. 6, 16221651. https://doi.org/10.1137/0524091
[2]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(1986), 4750.
[3]Brock, F., An isoperimetric inequality for eigenvalues of the Stekloff problem. ZAMM Z. Angew. Math. Mech. 81(2001), no. 1, 6971. doi:10.1002/1521-4001(200101)81:1 <69::AID-ZAMM69 >3.0.CO;2-#
[4]Bucur, D., Freitas, P., and Kennedy, J., The Robin problem. In: Shape optimization and spectral theory. De Gruyter Open, Warsaw/Berlin, 2017.
[5]Bucur, D., Ferone, V., Nitsch, C., and Trombetti, C., Weinstock inequality in higher dimensions. J. Differential Geom.. to appear. arxiv:1710.04587
[6]Bucur, D., Ferone, V., Nitsch, C., and Trombetti, C., The quantitative Faber-Krahn inequality for the Robin Laplacian. J. Differential Equations 264(2018), no. 7, 44884503. https://doi.org/10.1016/j.jde.2017.12.014
[7]Bucur, D. and Giacomini, A., A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal. 198(2010), no. 3, 927961. https://doi.org/10.1007/s00205-010-0298-6
[8]Bucur, D. and Giacomini, A., Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach. Arch. Ration. Mech. Anal. 218(2015), no. 2, 757824. https://doi.org/10.1007/s00205-015-0872-z
[9]Daners, D., A Faber–Krahn inequality for Robin problems in any space dimension. Math. Ann. 335(2006), 767785. https://doi.org/10.1007/s00208-006-0753-8
[10]Duren, P. L., Theory of H p spaces. Corrected and expanded ed. Dover, Mineola, New York, 2000.
[11]Ferone, V., Nitsch, C., and Trombetti, C., On a conjectured reversed Faber–Krahn inequality for a Steklov-type Laplacian eigenvalue. Commun. Pure Appl. Anal. 14(2015), 6382. https://doi.org/10.3934/cpaa.2015.14.63
[12]Freitas, P. and Kennedy, J. B., Extremal domains and Pólya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles. Int. Math. Res. Not. IMRN. to appear. arxiv:1805.10075
[13]Freitas, P. and Krejčiřík, D., The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280(2015), 322339. https://doi.org/10.1016/j.aim.2015.04.023
[14]Freitas, P. and Laugesen, R. S., From Neumann to Steklov and beyond, via Robin: the Weinberger way. arxiv:1810.07461
[15]Girouard, A. and Polterovich, I., Shape optimization for low Neumann and Steklov eigenvalues. Math. Methods Appl. Sci. 33(2010), 501516. https://doi.org/10.1002/mma.1222
[16]Girouard, A. and Polterovich, I., Spectral geometry of the Steklov problem. In: Shape optimization and spectral theory. De Gruyter Open, Warsaw/Berlin, 2017.
[17]Henrot, A., Shape optimization and spectral theory. De Gruyter Open, Warsaw, 2017.
[18]Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270(1970), A1645A1648.
[19]Keady, G. and Wiwatanapataphee, B., Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds. Math. Inequal. Appl. 21(2018), 911930. https://doi.org/10.7153/mia-2018-21-62
[20]Kennedy, J., An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions. Proc. Amer. Math. Soc. 137(2009), no. 2, 627633. https://doi.org/10.1090/S0002-9939-08-09704-9
[21]Laugesen, R. S., The Robin Laplacian - spectral conjectures, rectangular theorems. Preprint, arxiv:1905.07658.
[22]Laugesen, R. S., Liang, J., and Roy, A., Sums of magnetic eigenvalues are maximal on rotationally symmetric domains. Ann. Henri Poincaré 13(2012), no. 4, 731750. https://doi.org/10.1007/s00023-011-0142-z
[23]Laugesen, R. S. and Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mapping. J. Funct. Anal. 155(1998), no. 1, 64108. https://doi.org/10.1006/jfan.1997.3222
[24]Laugesen, R. S. and Siudeja, B. A., Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane. J. Funct. Anal. 260(2011), no. 6, 17951823. https://doi.org/10.1016/j.jfa.2010.12.018
[25]Laugesen, R. S. and Siudeja, B. A., Sums of Laplace eigenvalues: rotations and tight frames in higher dimensions. J. Math. Phys. 52(2011), no. 9, 093703. https://doi.org/10.1063/1.3635379
[26]Laugesen, R. S. and Siudeja, B. A., Sharp spectral bounds on starlike domains. J. Spectr. Theory 4(2014), no. 2, 309347. https://doi.org/10.4171/JST/71
[27]NIST Digital Library of Mathematical Functions. Release 1.0.18 of 2018-03-27. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. http://dlmf.nist.gov/.
[28]Pólya, G. and Szegő, G., Isoperimetric inequalities in mathematical physics. Princeton University Press, Princeton, NJ, 1951.
[29]Pommerenke, C., Boundary behaviour of conformal maps. (Grundlehren der Mathematischen Wissenschaften, 299), Springer–Verlag, Berlin, 1992. https://doi.org/10.1007/978-3-662-02770-7
[30]Szegő, G., Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3(1954), 343356. https://doi.org/10.1512/iumj.1954.3.53017
[31]Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem. J. Rational Mech. Anal. 5(1956), 633636. https://doi.org/10.1512/iumj.1956.5.55021
[32]Weinstock, R., Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal. 3(1954), 745753. https://doi.org/10.1512/iumj.1954.3.53036
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From Steklov to Neumann and Beyond, via Robin: The Szegő Way

  • Pedro Freitas (a1) (a2) and Richard S. Laugesen (a3)

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