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

Part of: Geometric function theory Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  07 March 2019

Pedro Freitas  and
Richard S. Laugesen
Pedro Freitas
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, P-1049-001 Lisboa, Portugal Grupo de Física Mátematica, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, P-1749-016 Lisboa, Portugal Email: psfreitas@fc.ul.pt
Richard S. Laugesen
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Email: Laugesen@illinois.edu
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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.

Keywords

RobinNeumannSteklovvibrating membraneabsorbing boundary conditionconformal mapping

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds
Secondary: 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 1024 - 1043
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

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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