On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

Details On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

2008-08-21

arxiv.org/abs/0808.2968v2

Funktsional. Anal. i Prilozhen. 44 (2010), no. 2, 33--47

Mathematics

Spectral Theory

15 pages, 3 figures. Significantly revised version, new title and abstract

Scientific paper

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.

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