On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

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.

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

References in corpus (1)

• Average growth of the spectral function on a Riemannian manifold

Cited by in corpus (1)

• The first Steklov eigenvalue, conformal geometry, and minimal surfaces