TY - JOUR
T1 - From Steklov to Neumann and Beyond, via Robin
T2 - The SzegÅ' Way
AU - Freitas, Pedro
AU - Laugesen, Richard S.
N1 - Funding Information:
Received by the editors November 13, 2018. Published online on Cambridge Core March 7, 2019. his 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. AMS subject classification: 35P15, 30C70. Keywords: Robin, Neumann, Steklov, vibrating membrane, absorbing boundary condition, conformal mapping.
Publisher Copyright:
© 2020 Cambridge University Press. All rights reserved.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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, and lies between and. Corollaries include SzegÅ"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.
AB - 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, and lies between and. Corollaries include SzegÅ"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.
KW - AMS subject classification: 35P15 30C70
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U2 - 10.4153/S0008414X19000154
DO - 10.4153/S0008414X19000154
M3 - Article
AN - SCOPUS:85077219336
VL - 72
SP - 1024
EP - 1043
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
SN - 0008-414X
IS - 4
ER -