Stekloff Eigenvalue Research Articles

Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

<p style='text-indent:20px;'>A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medium. Second, we leverage regularity results for Maxwell's equations in order to establish stability results for the eigenvalues with respect to the material coefficients, and we show that this generalized class of Stekloff eigenvalues converges to the standard class as the smoothing parameter approaches zero.

Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scattering

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also on adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.

Analysis of a Trace Class Stekloff Eigenvalue Problem Arising in Inverse Scattering

The Stekloff eigenvalue problem has been a recent problem of interest due to its potential use in the nondestructive testing of materials, but it suffers from two shortcomings that are common among this class of eigenvalue methods: it is not known in general if Stekloff eigenvalues exist in the case of complex coefficients, and there exists no mechanism by which to increase their sensitivity to changes in the material properties of the medium under consideration. We present a variation of the Stekloff eigenvalue problem which overcomes both of these issues by the introduction of a certain operator in the boundary condition, and we provide numerical examples to examine the practical consequences of this variation. We discuss this idea in the broader context of what might be called tailored eigenvalue methods, in which the eigenvalue problem is designed to ensure favorable theoretical or practical results.

An inverse medium problem using Stekloff eigenvalues and a Bayesian approach††Dedicated to the memory of Armin Lechleiter, a dear friend of the third author.

This paper studies the reconstruction of Stekloff eigenvalues and the index of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum problem of Stekloff eigenvalues is investigated by the reciprocity gap method. Then a Bayesian approach is proposed to estimate the index of refraction using a few reconstructed eigenvalues. Since the multiplicities of the eigenvalues, which can be complex, are unknown, we employ the recently developed spectral indicator method to compute Stekloff eigenvalues. Numerical experiments validate the effectiveness of the proposed method.

Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem

In this paper, for a new Stekloff eigenvalue problem which is non-selfadjoint and not H1-elliptic, we establish and analyze two kinds of two-grid discretization scheme and a local finite element scheme. We present the error estimates of approximations of two-grid discretizations. We also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain. Numerical experiments are reported finally to show the efficiency of our schemes.

Modified transmission eigenvalues in inverse scattering theory

We consider the scattering of an acoustic plane wave by an inhomogeneous medium of compact support. Our aim is to introduce a new class of target signatures that can be used to detect changes in the material properties of the scattering object from a knowledge of the far field pattern of the scattered field. To this end we introduce a modified far field operator depending on a parameter η and show that this operator is injective with dense range provided η is not an eigenvalue of a new problem called the modified transmission eigenvalue problem. It is explained why this class of target signatures is preferable in some ways to previously studied target signatures that are based on scattering resonances, transmission eigenvalues, or Stekloff eigenvalues.

Electromagnetic Stekloff Eigenvalues in Inverse Scattering

In [F. Cakoni, D. Colton, S. Meng, and P. Monk, SIAM J. Appl. Math., 76 (2016), pp. 1737--1763] it was suggested to use Stekloff eigenvalues for the Helmholtz equation to detect changes in a scatterer using remote measurements of the scattered wave. This paper investigates the use of Stekloff eigenvalues for Maxwell's equations for the same purpose. Because the Stekloff eigenvalue problem for Maxwell's equations is not a standard eigenvalue problem for a compact operator, we propose a modified Stekloff problem that restores compactness. In order to measure the modified Stekloff eigenvalues of a domain from far field measurements we perturb the usual far field equation of the linear sampling method by using the far field pattern of an auxiliary impedance problem related to the modified Stekloff problem. We are then able to show (1) the existence of modified Stekloff eigenvalues and (2) the well-posedness of the corresponding auxiliary exterior impedance problem and (3) to provide theorems that support our claim to be able to detect modified Stekloff eigenvalues from far field measurements. Preliminary numerical results show that for some simple domains it is possible to measure a few modified Stekloff eigenvalues. (As for the Helmholtz equation, not all eigenvalues can be measured.) In addition the modified Stekloff eigenvalues are changed by perturbations of the scatterer. An open problem is to obtain a proof of the existence of modified Stekloff eigenvalues for absorbing media.

Stekloff Eigenvalues in Inverse Scattering

We consider a problem in nondestructive testing in which small changes in the (possibly complex valued) refractive index $n(x)$ of an inhomogeneous medium of compact support are to be determined from changes in measured far field data due to incident plane waves. The problem is studied by considering a modified far field operator ${\cal F}$ whose kernel is the difference of the measured far field pattern due to the scattering object and the far field pattern of an auxiliary scattering problem with the Stekloff boundary condition imposed on the boundary of a domain $B$, where $B$ is either the support of the scattering object or a ball containing the scattering object in its interior. It is shown that ${\cal F}$ can be used to determine the Stekloff eigenvalues corresponding to $B$, where, if $B\not=D$, the refractive index is set equal to one in $B\setminus\overline{D}$. A formula is obtained relating changes in $n(x)$ to changes in the Stekloff eigenvalues and numerical examples are given showing the effectiveness of determining changes to the refractive index in this way.

An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue σ2,p, providing various equivalent characterizations for it. We also prove an upper bound for σ2,p in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock–Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appear to be interesting in themselves.

Spectral optimization for the Stekloff–Laplacian: The stability issue

We consider the problem of minimizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock–Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (1999) [2]: we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.

Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem.

The Robin Function and Its Eigenvalues

Abstract The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.

Sums of reciprocal Stekloff eigenvalues

AbstractLet 0 = λ1 &lt; λ2 ≤ λ3 ≤ … be the Stekloff eigenvalues of a plane domain. The paper is concerned with formulas for ∑∞2λ(–2)j in simply and doubly connected domains. In the simply connected case it is proven that the disk minimized this sum. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Mixed Stekloff Eigenvalue Problem and New Extremal Properties of the Grötzsch Ring

We study the mixed Stekloff eigenvalue problem in doubly connected domains. Using circular symmetrization and a distortion theorem for conformal mappings of an annulus, we find lower bounds for the first eigenvalue. These bounds are sharp for the Grötzsch ring. We also solve an extremal problem for some polygonal doubly connected domains and prove some results concerning the existence of a closed nodal line. Bibliography: 19 titles.

Zur Konstruktion der Eigenfunktionen Stekloffscher Eigenwertaufgaben

The aim of this paper is a new method for the construction of the eigenfunctions and eigenvalues of the mixed Stekloff eigenvalue problem. Using conformal mappings, transplantation techniques and the reflection principle we construct an analytic function whose real part is the eigenfunction under consideration. A consequence of this is also an equation for all eigenvalues which is useful for the calculation of the eigenvalues. Numerical results are also given. The method presented here is outlined in detail for two exampels.

The Second Stekloff Eigenvalue and Energy Dissipation Inequalities for Functionals with Surface Energy

A functional with both bulk and interfacial surface energy is considered. It corresponds to the energy dissipated inside a two-phase electrical conductor in the presence of an electrical contact resistance at the two-phase interface. The effect of embedding a highly conducting particle into a matrix of lesser conductivity is investigated. We find the criterion that determines when the increase in surface energy matches or exceeds the reduction in bulk energy associated with the particle. This criterion is general and applies to any particle with Lipschitz continuous boundary. It is given in terms of the of the second Stekloff eigenvalue of the particle. This result provides the means for selecting energy-minimizing configurations.

The Geometry of the First Non-zero Stekloff Eigenvalue

Let (Mn,g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem \\begin{align}\\Delta\\varphi & = & 0\\qquad & on\\quad M,\\\\ \\frac{\\partial\\varphi}{\\partial \\eta} & = & \\ u_1\\varphi\\qquad & on\\quad\\partial M.\\end{align}\\eqno (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν1in terms of the geometry of the manifold (Mn,g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 for non-negative Ricci curvature manifolds we show thatν1>k0/2, wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue.

The second Stekloff eigenvalue of an inclusion and new size effects for composites with interface thermal barriers

Two-phase particle reinforced composites are considered. We treat the case when there is an interfacial thermal barrier between phases. We obtain the geometric criterion that determines when the effects of the interfacial thermal barrier overcome the benefits of a highly conducting reinforcement. This criterion is general and applies to any reinforcement with Lipschitz continuous boundary. It is given in terms of the second Stekloff eigenvalue of the reinforcement. This result is used to predict size effects for a general class of inclusion shapes.

The inclusion of Stekloff eigenvalues for nonsmooth bounded domains

The method used in [3] for including Stekloff eigenvalues is applied to domains with nonsmooth boundary. Considering special trial functions whose construction is adapted to the local singular behaviour of the eigenfunctions near corners leads to sharp eigenvalue intervals for rhombi and the L-shaped polygonal domain.

Elliptic Equations in Divergence Form, Geometric Critical Points of Solutions, and Stekloff Eigenfunctions

The Stekloff eigenvalue problem (1.1) has a countable number of eigenvalues $\{ p_n \} n = 1,2, \ldots $, each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of $p_n $, and the number of critical points and of nodal domains of the eigenfunctions corresponding to $p_n $. In view of a possible application to inverse conductivity problems, the result for the general case of elliptic equations with discontinuous coefficients in divergence form is proven by replacing the classical concept of critical point with the more suitable notion of geometric critical point.