On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

TL;DR

This paper proves the sharpness of the Hersch-Payne-Schiffer inequality for Steklov eigenvalues in planar domains, providing new proofs and exploring related eigenvalue product results.

Contribution

It establishes the sharpness of the inequality for all n and offers a new proof for the case n=2, advancing understanding of Steklov eigenvalue bounds.

Findings

The inequality is sharp for all n with equality approached by degenerate domains.

A new proof for the n=2 case shows the inequality is strict.

Results on the product of consecutive Steklov eigenvalues are also presented.

Abstract

We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. We give a new proof of this inequality for n=2 and show that it is strict in this case. Related results are also obtained for the product of two consecutive Steklov eigenvalues.