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Functional Analysis and Its Applications

, Volume 44, Issue 2, pp 106–117 | Cite as

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

  • A. Girouard
  • I. Polterovich
Article

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.

Key words

Steklov eigenvalue problem eigenvalue isoperimetric inequality 

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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Université de NeuchâtelNeuchâtelSwitzerland
  2. 2.Université de MontréalMontréalCanada

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