Abstract. We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the n-th nonzero Steklov eigenvalue of a bounded simplyconnected planar domain is sharp for all n ≥ 1. The equality is attained in the limit by a sequence of simply-connected domains degenerating to the 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.
In REF , A. Girouard and I. Polterovich proved that the bound in (5) is sharp and attained by a sequence of simply-connected domains degenerating into a disjoint union of n identical balls.
ON THE HERSCH–PAYNE–SCHIFFER INEQUALITIES FOR STEKLOV EIGENVALUES
Funktsional. Anal. i Prilozhen. 44 (2010), no. 2, 33--47
Mathematics