Abstract
We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 44, No. 2, pp. 33–47, 2010
Original Russian Text Copyright © by A. Girouard and I. Polterovich
The second author was supported by NSERC, FQRNT, and Canada Research Chairs Program.
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Girouard, A., Polterovich, I. On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues. Funct Anal Its Appl 44, 106–117 (2010). https://doi.org/10.1007/s10688-010-0014-1
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DOI: https://doi.org/10.1007/s10688-010-0014-1