On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

By A. Girouard, Iosif Polterovich. 2008

66 citations recorded by the source corpus.

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6c858fde1629eb8377cd2bd1752b7c4e35025b1

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Isoperimetric control of the Steklov spectrum (2011, background): “For higher eigenvalues in the planar situation, see [14, 15].”
Eigenvalue inequalities for mixed Steklov problems (2009, methodology): “…4.1 is quite short: it uses domain monotonicity for eigenvalues of mixed Steklov problems (see sections 3.1 and 3.2), the properties of mixed Steklov eigenvalues for cylindrical domains (see section 2.1), and the classical inequalities between the Dirichlet and Neumann eigenvalues [11, 8, 23].”
Shape optimization for low Neumann and Steklov eigenvalues (2008, methodology): “In fact, it was proved in [ 13 ] that for the domains " defined above, the Steklov spectrum is collapsing:”
The legacy of Vladimir Andreevich Steklov (2014, unknown): “It was shown in [16] that the inequality λn(D)L(∂D) à 2πn, n = 1,2,3 .”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “Indeed, it was proved in [47] that equality is attained in the limit by a family of domains degenerating to a disjoint union of k identical disks (see Figure measure j j rather than its perimeter.”
Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem (2011, methodology): “The relationships between the geometry of the domain and the spectrum of the corresponding Steklov operator have been investigated by several authors (see for example [CEG2], [FS] and [GP]).”