Geodesic
From Steklov to Neumann and Beyond, via Robin: The Szegő Way
Canadian journal of mathematics, Tome 72 (2020) no. 4, pp. 1024-1043

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008414X19000154
Mots-clés : Robin, Neumann, Steklov, vibrating membrane, absorbing boundary condition, conformal mapping 
Freitas, Pedro; Laugesen, Richard S. From Steklov to Neumann and Beyond, via Robin: The Szegő Way. Canadian journal of mathematics, Tome 72 (2020) no. 4, pp. 1024-1043. doi: 10.4153/S0008414X19000154
@article{10_4153_S0008414X19000154,
     author = {Freitas, Pedro and Laugesen, Richard S.},
     title = {From {Steklov} to {Neumann} and {Beyond,} via {Robin:} {The} {Szeg\H{o}} {Way}},
     journal = {Canadian journal of mathematics},
     pages = {1024--1043},
     year = {2020},
     volume = {72},
     number = {4},
     doi = {10.4153/S0008414X19000154},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000154/}
}
TY  - JOUR
AU  - Freitas, Pedro
AU  - Laugesen, Richard S.
TI  - From Steklov to Neumann and Beyond, via Robin: The Szegő Way
JO  - Canadian journal of mathematics
PY  - 2020
SP  - 1024
EP  - 1043
VL  - 72
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000154/
DO  - 10.4153/S0008414X19000154
ID  - 10_4153_S0008414X19000154
ER  - 
%0 Journal Article
%A Freitas, Pedro
%A Laugesen, Richard S.
%T From Steklov to Neumann and Beyond, via Robin: The Szegő Way
%J Canadian journal of mathematics
%D 2020
%P 1024-1043
%V 72
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000154/
%R 10.4153/S0008414X19000154
%F 10_4153_S0008414X19000154

[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, 1622–1651. https://doi.org/10.1137/0524091 Google Scholar

[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), 47–50. Google Scholar

[3] Brock, F., An isoperimetric inequality for eigenvalues of the Stekloff problem. ZAMM Z. Angew. Math. Mech. 81(2001), no. 1, 69–71. doi: 3.0.CO;2-#>Google Scholar

[4] Bucur, D., Freitas, P., and Kennedy, J., The Robin problem. In: Shape optimization and spectral theory. De Gruyter Open, Warsaw/Berlin, 2017. Google Scholar

[5] Bucur, D., Ferone, V., Nitsch, C., and Trombetti, C., Weinstock inequality in higher dimensions. J. Differential Geom.. to appear. Google Scholar

[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, 4488–4503. https://doi.org/10.1016/j.jde.2017.12.014 Google Scholar

[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, 927–961. https://doi.org/10.1007/s00205-010-0298-6 Google Scholar

[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, 757–824. https://doi.org/10.1007/s00205-015-0872-z Google Scholar

[9] Daners, D., A Faber–Krahn inequality for Robin problems in any space dimension. Math. Ann. 335(2006), 767–785. https://doi.org/10.1007/s00208-006-0753-8 Google Scholar

[10] Duren, P. L., Theory of H p spaces. Corrected and expanded ed. Dover, Mineola, New York, 2000. Google Scholar

[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), 63–82. https://doi.org/10.3934/cpaa.2015.14.63 Google Scholar

[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. Google Scholar

[13] Freitas, P. and Krejčiřík, D., The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280(2015), 322–339. https://doi.org/10.1016/j.aim.2015.04.023 Google Scholar

[14] Freitas, P. and Laugesen, R. S., From Neumann to Steklov and beyond, via Robin: the Weinberger way. Google Scholar

[15] Girouard, A. and Polterovich, I., Shape optimization for low Neumann and Steklov eigenvalues. Math. Methods Appl. Sci. 33(2010), 501–516. https://doi.org/10.1002/mma.1222 Google Scholar

[16] Girouard, A. and Polterovich, I., Spectral geometry of the Steklov problem. In: Shape optimization and spectral theory. De Gruyter Open, Warsaw/Berlin, 2017. Google Scholar

[17] Henrot, A., Shape optimization and spectral theory. De Gruyter Open, Warsaw, 2017. Google Scholar

[18] Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270(1970), A1645–A1648. Google Scholar

[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), 911–930. https://doi.org/10.7153/mia-2018-21-62 Google Scholar

[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, 627–633. https://doi.org/10.1090/S0002-9939-08-09704-9 Google Scholar

[21] Laugesen, R. S., The Robin Laplacian - spectral conjectures, rectangular theorems. Preprint, . Google Scholar

[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, 731–750. https://doi.org/10.1007/s00023-011-0142-z Google Scholar

[23] Laugesen, R. S. and Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mapping. J. Funct. Anal. 155(1998), no. 1, 64–108. https://doi.org/10.1006/jfan.1997.3222 Google Scholar

[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, 1795–1823. https://doi.org/10.1016/j.jfa.2010.12.018 Google Scholar

[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 Google Scholar

[26] Laugesen, R. S. and Siudeja, B. A., Sharp spectral bounds on starlike domains. J. Spectr. Theory 4(2014), no. 2, 309–347. https://doi.org/10.4171/JST/71 Google Scholar

[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. . Google Scholar

[28] Pólya, G. and Szegő, G., Isoperimetric inequalities in mathematical physics. Princeton University Press, Princeton, NJ, 1951. Google Scholar

[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 Google Scholar

[30] Szegő, G., Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3(1954), 343–356. https://doi.org/10.1512/iumj.1954.3.53017 Google Scholar

[31] Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem. J. Rational Mech. Anal. 5(1956), 633–636. https://doi.org/10.1512/iumj.1956.5.55021 Google Scholar

[32] Weinstock, R., Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal. 3(1954), 745–753. https://doi.org/10.1512/iumj.1954.3.53036 Google Scholar

Cité par Sources :