Steklov Eigenvalue Research Articles

Determination of the modified exterior Steklov eigenvalues via the reciprocity gap method

The modified exterior Steklov eigenvalues (MESEs) arise from the inverse scattering problem for inhomogeneous media with a cavity and may serve as potential target signatures in nondestructive testing. In this paper we are interested in the determination of the MESEs from the measured Cauchy data of the total field on some manifold inside the cavity due to interior point sources. To this end, the reciprocity gap (RG) method based on a linear integral equation is employed. We provide the related theory and show that the blow-up property of the approximate solution to the integral equation can be used to characterize the MESEs. Numerical examples are presented to demonstrate the viability of our method.

Critical Lengths of Steklov Eigenvalues of Hypersurfaces of Revolution in Euclidean Space

We study the Steklov problem on hypersurfaces of revolution with two boundary components in Euclidean space and focus on the phenomenon of critical length, at which a Steklov eigenvalue is maximized. In this article, we conjecture that, in any dimension, there is a finite number of infinite critical length. To investigate this, we develop an algorithm to efficiently perform numerical experiments, providing support to our conjecture. Furthermore, we prove the conjecture in dimension n = 3 and n = 4.

Isospectral graphs via inner symmetries

In this paper a new class of isospectral graphs is presented. These graphs are isospectral with respect to both the normalized Laplacian on the discrete graph and the standard differential Laplacian on the corresponding metric graph. The new class of graphs is obtained by gluing together subgraphs with the Steklov maps possessing special properties. It turns out that isospectrality is related to the degeneracy of the Steklov eigenvalues.

Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove theexistence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

An adaptive algorithm for Steklov eigenvalues of the Lamé operator in linear elasticity

An adaptive algorithm for Steklov eigenvalues of the Lamé operator in linear elasticity

Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi-Laplacian and gi:∂Ω×R×R→R are Carathéodory functions that satisfy general growth and structure conditions for i=1,2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of gi near zero related to the first eigenvalue of the pi-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g1,g2)=∇g with a smooth function (s1,s2)↦g(x,s1,s2). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the pi-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

Conforming element method and shifted-inverse two-grid dicretization for the fourth-order Steklov eigenvalue problem

Conforming element method and shifted-inverse two-grid dicretization for the fourth-order Steklov eigenvalue problem

Steklov eigenvalues of nearly hyperspherical domains

We consider Steklov eigenvalues of nearly hyperspherical domains in R d + 1 with d ≥ 3 . In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of Pochhammer’s and Wigner 3 j -symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (i) for an infinite subset of Steklov eigenvalues, the ball is not optimal and (ii) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

Monotonicity of Steklov eigenvalues on graphs and applications

Monotonicity of Steklov eigenvalues on graphs and applications

Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

R'esum'eWe investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution in Euclidean space with two boundary components, each isometric to $${\mathbb {S}}^{n-1}$$ S n - 1 . For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound $$B_n(L)$$ B n ( L ) (that depends only on the dimension $$n \ge 3$$ n ≥ 3 and the meridian length $$L>0$$ L > 0 ) which is reached by a degenerated metric $$g^*$$ g ∗ that we compute explicitly. We also give a sharp upper bound $$B_n$$ B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.

Multiple tubular excisions and large Steklov eigenvalues

Multiple tubular excisions and large Steklov eigenvalues

Second-Order Three-Scale Asymptotic Analysis and Algorithms for Steklov Eigenvalue Problems in Composite Domain with Hierarchical Cavities

Second-Order Three-Scale Asymptotic Analysis and Algorithms for Steklov Eigenvalue Problems in Composite Domain with Hierarchical Cavities

Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the kth nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order p when k≤p−1. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when k≥p. In particular, we show that as k→∞, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.

The upper bound of the harmonic mean of the Steklov eigenvalues in curved spaces

AbstractConsidering an ‐dimensional Riemannian manifold whose sectional curvature is bounded above by and Ricci curvature is bounded below by , we obtain an upper bound of the harmonic mean of the first nonzero Steklov eigenvalues for domains contained in . This can be viewed as certain isoperimetric inequality and generalizes the result on comparing the first nonzero Steklov eigenvalues for such domains (Li–Wang–Wu, 2020, arXiv:2003.03093).

Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface M and a fixed conformal class c one defines \Lambda_{k}(M,c) to be the supremum of the k -th nontrivial Laplacian eigenvalue over all metrics g\in c of unit volume. It has been observed by Nadirashvili that the metrics achieving \Lambda_{k}(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify \Lambda_{1}(M,c) and \Lambda_{2}(M,c) with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing \Lambda_{1}(M,c) , \Lambda_{2}(M,c) , and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

A Steklov version of the torsional rigidity

Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of [Formula: see text] into [Formula: see text]. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.

Some recent developments on the Steklov eigenvalue problem

AbstractThe Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.

Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary

Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary

Manifolds with Density and the First Steklov Eigenvalue

Manifolds with Density and the First Steklov Eigenvalue