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On the Hersch–Payne–Schiffer inequalities for Steklov eigenvalues

A. Girouard^a, I. V. Polterovich<sup>b

a</sup> Universite de Neuchatel
^b Université de Montréal

Abstract: We prove that the Hersch–Payne–Schiffer isoperimetric inequality for the $n$th nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all $n\ge 1$. 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.

Keywords: Steklov eigenvalue problem, eigenvalue, isoperimetric inequality.

UDC: 517.956.227

Received: 15.09.2008

DOI: 10.4213/faa2984

 English version: Functional Analysis and Its Applications, 2010, 44:2, 106–117

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