Steklov Eigenvalue Research Articles

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8pi k, which is the best upper bound for the k^{text {th}} area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

Approximation of the inverse scattering Steklov eigenvalues and the inverse spectral problem

In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method is studied where the basis functions are the Neumann eigenfunctions of the Laplacian. Error estimates for the eigenvalues and eigenfunctions are proven by appealing to Weyl’s law. We will test this method against separation of variables in order to validate the theoretical convergence. We also consider the inverse spectral problem of estimating/recovering the refractive index from the knowledge of the Steklov eigenvalues, since the eigenvalues are monotone with respect to a real-valued refractive index implying that they can be used for nondestructive testing. Some numerical examples are provided for the inverse spectral problem.

Weinstock inequality in higher dimensions

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n \geq 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.

An HDG method for the Steklov eigenvalue problem

Abstract We propose a hybridizable discontinuous Galerkin (HDG) method for approximating the Steklov eigenvalue problem. We prove optimal convergence rates for the eigenvalues and the eigenfunctions, and under some regularity assumptions we obtain a superconvergent rate for the eigenvalues. Moreover, after we eliminate the flux variable and the scalar variable, the reduced eigenvalue problem is linear and our result holds on any sufficiently regular mesh made of general polyhedral elements. Finally, we present numerical experiments to confirm our theoretical results.

The orthotropic p-Laplace eigenvalue problem of Steklov type as p → +∞

We study the Steklov eigenvalue problem for the ∞-orthotropic Laplace operator defined on convex sets of RN, with N≥2, considering the limit for p→+∞ of the Steklov problem for the p-orthotropic Laplacian. We find a limit problem that is satisfied in the viscosity sense and a geometric characterization of the first non trivial eigenvalue. Moreover, we prove Brock-Weinstock and Weinstock type inequalities among convex sets, stating that the ball in a suitable norm maximizes the first non trivial eigenvalue for the Steklov ∞-orthotropic Laplacian, once we fix the volume or the anisotropic perimeter.

Steklov eigenvalues for the Lamé operator in linear elasticity

In this paper we introduce the notion of Steklov eigenvalues for the Lamé operator in the theory of linear elasticity. In this eigenproblem the spectral parameter appears on a Robin boundary condition, linking the traction and the displacement. We investigate the spectrum of this problem and study the existence of eigenpairs on Lipschitz domains as well as show that any conforming Galerkin method is able to provide good approximations to this problem. A standard conforming finite element method is used to obtain numerical experiments on 2D and 3D domains to support our theoretical findings.

Degenerating sequences of conformal classes and the conformal Steklov spectrum

AbstractLet $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

New blow‐up conditions to p‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

In this paper, we study blow‐up phenomena of the following p‐Laplace type nonlinear parabolic equations under nonlinear mixed boundary conditions and on Γ2 × (0, t∗) such that , where f and h are real‐valued C1‐functions. To discuss blow‐up solutions, we introduce new conditions: For each x ∈ Ω, z ∈ ∂Ω, t > 0, u > 0, and v > 0, for some constants α, β1, β2, γ1, γ2, and δ satisfying where , , , and . Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p‐Laplace operator, respectively.

Rigidity of a trace estimate for Steklov eigenvalues

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Shi and Yu (2016) [13]). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a splitting theorem for flat and totally geodesic Riemannian submersions which may be of independent interests.

Maximization of the Steklov Eigenvalues With a Diameter Constraint

In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume, and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first nontrivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm.

From Neumann to Steklov and beyond, via Robin: The Weinberger way

The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative, and the problem admits no maximiser when $\alpha$ is positive.

Liouville-type theorems on manifolds with nonnegative curvature and strictly convex boundary

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

Electromagnetic Steklov eigenvalues: approximation analysis

We continue the work of Camano et al. [SIAM J. Math. Anal. 49 (2017) 4376–4401] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues do not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function A(⋅). We construct a “test function operator function” T(⋅) so that A(λ) is weakly T(λ)-coercive for all suitable λ, i.e. T(λ)*A(λ) is a compact perturbation of a coercive operator. The construction of T(⋅) relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property which concerns the tangential trace. The existence and construction of such projection operators remain open questions.

Impedance Eigenvalues in Linear Elasticity

This paper is devoted to studying impedance eigenvalues (that is, eigenvalues of a particular Dirichlet-to-Neumann map) for the time harmonic linear elastic wave problem, and their potential use as target-signatures for fluid-solid interaction problems. We first consider several possible families of eigenvalues of the elasticity problem, focusing on certain impedance eigenvalues that are an analogue of Steklov eigenvalues. We show that one of these families arises naturally in inverse scattering. We also analyse their approximation from far field measurements of the scattered pressure field in the fluid, and illustrate several alternative methods of approximation in the case of an isotropic elastic disk.

Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of mathbb {Z}^n. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

AN EFFICIENT NUMERICAL METHOD BASED ON LEGENDRE-GALERKIN APPROXIMATION FOR THE STEKLOV EIGENVALUE PROBLEM IN SPHERICAL DOMAIN

<abstract> We present in this paper an efficient numerical method based on Legendre-Galerkin approximation for the Steklov eigenvalue problem in spherical domain. Firstly, by means of spherical coordinate transformation and spherical harmonic expansion, the original problem is reduced to a sequence of equivalent one-dimensional eigenvalue problems that can be solved individually in parallel. Through the introduction of the appropriate weighted Sobolev spaces, the weak form and corresponding discrete scheme are established for each one-dimensional eigenvalue problem. Then from the approximate property of orthogonal polynomials in the weighted Sobolev spaces, we prove the error estimates of approximate eigenvalues for each one dimensional eigenvalue problem. Finally, some numerical examples are provided to illustrate the validity of our algorithms. </abstract>

Stability and instability issues of the Weinstock inequality

Given two planar, conformal, smooth open sets Ω \Omega and ω \omega , we prove the existence of a sequence of smooth sets ( Ω ϵ ) (\Omega _\epsilon ) which geometrically converges to Ω \Omega and such that the (perimeter normalized) Steklov eigenvalues of ( Ω ϵ ) (\Omega _\epsilon ) converge to the ones of ω \omega . As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.

An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space

In this article, we prove an isoperimetric inequality for the harmonic mean of the first $$(n-1)$$ nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Our approach to prove this result also gives a similar inequality for the first n nonzero Steklov eigenvalues on bounded domains in n-dimensional Euclidean space.

From Steklov to Neumann via homogenisation

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.