American Institute of Mathematical Sciences

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  2012, 19: 77-85. doi: 10.3934/era.2012.19.77

Upper bounds for Steklov eigenvalues on surfaces

Alexandre Girouard 1, and  Iosif Polterovich 2,

1. 

Laboratoire de Mathématiques (LAMA), Université de Savoie campus scientifique, 73376 Le Bourget-du-Lac, France
2. 

Département de Mathématiques et de Statistique, Université de Montréal, C. P. 6128, Succ. Centre-ville, Montréal, Québec, H3C 3J7, Canada

Received  March 2012 Published  August 2012

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.
Keywords: eigenvalue inequalities., Steklov problem, Riemannian surface.
Mathematics Subject Classification: Primary: 58J50; Secondary: 35P1.
Citation: Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77
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show all references

References:
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Comment. Math. Helv., 24 (1950), 100-134. doi: 10.1007/BF02567028.  Google Scholar

[2]

Pitman, Boston, Mass., 1980.  Google Scholar

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Z. Angew. Math. Mech., 81 (2001), 69-71. doi: 10.1002/1521-4001(200101)81:1<69::AID-ZAMM69>3.0.CO;2-#.  Google Scholar

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J. Anal. Math., 83 (2001), 243-258. doi: 10.1007/BF02790263.  Google Scholar

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Discrete Appl. Math., 9 (1984), 105-109.  Google Scholar

[6]

in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro), Soc. Brasil. Mat., Rio de Janeiro, 1980.  Google Scholar

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J. Funct. Anal., 261 (2011), 1384-1399. doi: 10.1016/j.jfa.2011.05.006.  Google Scholar

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in "Séminaire de Théorie Spectrale et Géométrie, Année 1983-1984," VII.1-VII.15, Univ. Grenoble I, Saint-Martin-d'Héres, 1984.  Google Scholar

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[11]

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[13]

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[14]

Journal of Functional Analysis, 261 (2011), 3419-3436. doi: 10.1016/j.jfa.2011.08.003.  Google Scholar

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J. Convex Anal., 15 (2008), 581-592.  Google Scholar

[16]

Arch. Rational Mech. Anal., 57 (1975), 99-114.  Google Scholar

[17]

preprint, arXiv:1103.2448, 2011. Google Scholar

[18]

J. Differential Geom., 37 (1993), 73-93.  Google Scholar

[19]

Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

[20]

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[21]

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SIAM J. Appl. Math., 31 (1976), 385-391. doi: 10.1137/0131032.  Google Scholar

[23]

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7 (1980), 55-63.  Google Scholar

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