[Paper Review] On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

Alexandre Girouard, Iosif Polterovich|arXiv (Cornell University)|Aug 21, 2008

Advanced Mathematical Modeling in Engineering24 references4 citations

TL;DR

This paper proves that the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues are sharp for all $ n \geq 1 $, with equality approached in the limit by a sequence of simply-connected planar domains degenerating to $ n $ identical disjoint disks. For $ n=2 $, it establishes a strict inequality, showing $ \sigma_2(\Omega)M(\Omega) < 4\pi $, resolving a long-standing conjecture on sharpness of the bound.

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.

Motivation & Objective

• To determine whether the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues are sharp for all $ n \geq 1 $.
• To investigate whether the bound $ \sigma_n(\Omega)M(\Omega) \leq 2\pi n $ is strict for $ n \geq 2 $, particularly for $ n=2 $.
• To construct a family of simply-connected domains degenerating to $ n $ identical disks that achieve the equality in the limit for the $ n $-th Steklov eigenvalue.
• To provide a new proof of the sharpness of the $ n=2 $ case using the Riemann mapping theorem, differing from the original method of Hersch-Payne-Schiffer.
• To explore the geometric and spectral implications of the eigenvalue bounds under domain degeneration and symmetry conditions.

Proposed method

• Constructing a one-parameter family of simply-connected Lipschitz domains $ \Omega_\varepsilon \subset \mathbb{R}^2 $ with $ \rho \equiv 1 $ on $ \partial\Omega_\varepsilon $, which degenerate to $ n $ identical disjoint disks as $ \varepsilon \to 0^+ $.
• Using the variational characterization of Steklov eigenvalues: $ \sigma_n(\Omega) = \inf_{E_n} \sup_{0 \neq u \in E_n} \frac{\int_\Omega |\nabla u|^2 \, dz}{\int_{\partial\Omega} u^2 \, ds} $, where $ E_n $ is an $ n $-dimensional subspace orthogonal to constants on $ \partial\Omega $.
• Applying a rearrangement argument on the boundary measure $ d\mu $, using hyperbolic caps and the concept of maximizing directions in $ \mathbb{R}P^1 $, to analyze extremal configurations.
• Proving that if the boundary measure $ d\mu $ is simple, then there exists a hyperbolic cap $ a $ such that the rearranged measure $ d\nu_a $ is multiple, leading to a contradiction in the topological structure of the space of caps.
• Using the Riemann mapping theorem to analyze the $ n=2 $ case, avoiding the techniques of Hersch-Payne-Schiffer and instead relying on conformal symmetry and extremal function spaces.
• Analyzing the behavior of maximizing directions $ [m(a)] $ as caps $ a $ degenerate to the full disk or to a point, showing that the map $ h(l,p) = [m(a_{l,p})] $ extends continuously to the closed cylinder and induces a non-contractible loop on $ \mathbb{RP}^1 $, leading to contradiction if all measures are simple.

Experimental results

Research questions

• RQ1 Is the Hersch-Payne-Schiffer inequality $ \sigma_n(\Omega)M(\Omega) \leq 2\pi n $ sharp for all $ n \geq 1 $?
• RQ2 Does the inequality $ \sigma_n(\Omega)M(\Omega) \leq 2\pi n $ hold with strict inequality for $ n \geq 2 $, or is equality achievable?
• RQ3 Can the sharpness of the inequality be achieved in the limit by a degenerating family of simply-connected domains?
• RQ4 What is the role of symmetry and boundary measure structure (simple vs. multiple) in determining the sharpness of the eigenvalue bounds?
• RQ5 Is there a geometric or spectral characterization of domains that achieve the eigenvalue bound in the limit?

Key findings

• The Hersch-Payne-Schiffer inequality $ \sigma_n(\Omega)M(\Omega) \leq 2\pi n $ is sharp for all $ n \geq 1 $, with equality approached in the limit by a family of simply-connected domains degenerating to $ n $ identical disjoint disks.
• For $ n=2 $, the inequality is strict: $ \sigma_2(\Omega)M(\Omega) < 4\pi $, proving that equality is not achieved for any bounded simply-connected planar domain.
• The limit of the eigenvalue-product inequality $ \sigma_n(\Omega)\sigma_{n+1}(\Omega)M(\Omega)^2 \leq 4\pi^2 n^2 $ is also sharp, with equality approached in the same degeneration limit.
• The proof for $ n=2 $ relies on the Riemann mapping theorem and a novel topological argument involving the space of hyperbolic caps and the real projective line.
• If the boundary measure $ d\mu $ is simple, then there exists a hyperbolic cap $ a $ such that the rearranged measure $ d\nu_a $ is multiple, leading to a contradiction via a non-contractible loop on $ \mathbb{RP}^1 $, which proves the strictness of the bound.
• When $ d\mu $ is multiple (e.g., under rotational symmetry), the bound $ \sigma_2(\Omega)M(\Omega) \leq 2\pi $ holds, which is stronger than the $ 4\pi $ bound, but equality is only achieved on the disk with constant density.