Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Determination of Dirac operator with eigenvalue-dependent boundary and jump conditions

Dirac operator with eigenvalue-dependent boundary and jump conditions is studied. Uniqueness theorems of the inverse problems from either Weyl function or the spectral data (the sets of eigenvalues and norming constants except for one eigenvalue and corresponding norming constant; two sets of different eigenvalues except for two eigenvalues) are proved. Finally, we investigate two applications of these theorems and obtain analogues of a theorem of Hochstadt-Lieberman and a theorem of Mochizuki-Trooshin.     

The principal eigenvalue of a space–time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.      

Trapped Modes for an Elastic Plate with a Perturbation of Young's Modulus

We consider a linear elastic plate with stress-free boundary conditions and zero Poisson coefficient. We prove that under a local change of Young's modulus infinitely many eigenvalues arise in the essential spectrum which accumulate at a positive threshold. We give estimates on the accumulation rate and on the asymptotical behaviour of the eigenvalues.     

Minimal compact global attractor for a damped semilinear wave equation

Abstract

The asymptotic behavior of eigenvalues of an elliptic operator with a divergence form is discussed. The coefficients of the operator are discontinuous through a boundary of a subdomain and degenerate to zero on the subdomain when a parameter tends to zero. We will prove that the eigenvalues approach eigenvalues of the Laplacian on the subdomain or on the complement. We will obtain precise asymptotic behavior of their convergence.     

A class of second‐order differential operators with eigenparameter‐dependent boundary and transmission conditions

In this study, we investigate a Sturm–Liouville type problem with eigenparameter‐dependent boundary conditions and eigenparameter‐dependent transmission conditions. By establishing a new self‐adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. Also we investigate some properties of its spectrum. Copyright © 2013 John Wiley & Sons, Ltd.     

EIGENVALUES OF STURM-LIOUVILLE OPERATOR WITH GENERAL BOUNDARY CONDITIONS

We study the asymptotic behavior for the eigenvalues of Sturm-Liouville operators with smooth potential.The precise asymptotic expressions for the eigenvalues of the operators with general boundary conditions are given.     

A quantum waveguide with Aharonov–Bohm magnetic field

In a previous study, we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disk window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. In the present work, we study the effect of the presence of a magnetic field of Aharonov–Bohm type on this system. Precisely, we prove that in the presence of such field, there is some critical values of a₀>0, for which we have absence of the discrete spectrum for . We give a sufficient condition for the existence of discrete eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.     

Spectral theory of left definite difference operators

This paper is concerned with the spectral theory for the second-order left definite difference boundary value problems. Existence of eigenvalues of boundary value problems is proved, numbers of their eigenvalues are calculated and fundamental spectral results are obtained.     

Absence of Critical Mass Densities for a Vibrating Membrane

We study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.     

Eigenvalues of the Dirac Operator on Fibrations over S1

We consider the Dirac operator on fibrations overS ¹ which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.Moreover, we discuss the dependence of the spectrum of theholonomy and obtain bounds for the multiplicity of the eigenvalues.     

Characteristic frequencies of bodies with thin spikes III. Frequency splitting

The eigenvalue problem for the Laplace operator with the Neumann boundary conditions in a domain that has a thin spike of finite length is considered for the case in which the limit value is an eigenvalue both for the main body and the spike. The method of matched asymptotic expansions is used to construct total asymptotics of the eigenvalues of the perturbed problem and obtain closed formulas for the leading asymptotic terms. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 494–502, April, 1997. Translated by V. E. Nazaikinskii     

On the eigenvalues for a weighted p-Laplacian operator on metric graphs

We provide an explicit upper bound of the eigenvalues corresponding to a weighted p-Laplacian operator defined on a connected metric graph with finite total length.     

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.     

Analysis of Two Linearized Problems Modeling Viscous Two‐Layer Flows

Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip‐like domain Π = {x = (x₁, x₂) ∈ ℝ² : x₁ ∈ ℝ¹, (0 < x₂ < h_*) ∨ (h_* < x₂ < 1)}. The first problem arises in the linearization of a two‐layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two‐layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π. The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.     

The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems

For domains with concave corners, the solutions to elliptic boundary values have the typical r^α‐singularity. The so‐called singularity exponents α are the eigenvalues of an eigenvalue problem which is associated with the given boundary value problem. This paper is aimed at deriving the mentioned eigenvalue problems for two examples, the Laplace equation and the linear elasticity problem. We will show interesting properties of these eigenvalue problems. For the linear elasticity problem, we explain in addition why the classical symmetry and positivity assumptions of the material tensor have to be used with care. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Convergence of Solutions of Singularly Perturbed Boundary-Value Problems for the Laplace Operator

In this paper, we study the convergence of solutions and eigenvalues of singularly perturbed boundary-value problems for the Laplace operator in three-dimensional bounded domains with thin tubes cut out and variation of boundary conditions on narrow strips.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

一类具有转移条件的不连续微分算子的渐近状态

研究一类具有转移条件且边界条件依赖于特征参数的Sturm-Liouville算子,建立一个与其相关的新的空间框架,给出其特征的相关性质与特征函数的完备性,利用函数论方法得出其特征的渐近表示,并获得Green函数的表达式.     

双调和算子本征值的上界

设Ω是 Rn中的有界区域 ,其边界足够光滑 ,λk为双调和算子在自由边界条件下的第 k个本征值 ,利用变分原理及 Fourier变换 ,给出了本征值部分和 ∑kj=1λj的一个上界 ,该上界仅依赖于区域的体积 .