On the Lowest Eigenvalue of Laplace Operators with Mixed Boundary Conditions

In this paper we consider a Robin-type Laplace operator on bounded domains. We study the dependence of its lowest eigenvalue on the boundary conditions and its asymptotic behavior in shrinking and expanding domains. For convex domains we establish two-sided estimates on the lowest eigenvalues in terms of the inradius and of the boundary conditions.     

Minimization of inhomogeneous biharmonic eigenvalue problems

Biharmonic eigenvalue problems arise in the study of the mechanical vibration of plates. In this paper, we study the minimization of the first eigenvalue of a simplified model with clamped boundary conditions and Navier boundary conditions with respect to the coefficient functions which are of bang-bang type (the coefficient functions take only two different constant values). A rearrangement algorithm is proposed to find the optimal coefficient function based on the variational formula of the first eigenvalue. On various domains, such as square, circular and annular domains, the region where the optimal coefficient function takes the larger value may have different topologies. An asymptotic analysis is provided when two different constant values are close to each other. In addition, a symmetry breaking behavior is also observed numerically on annular domains.     

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.     

On ω-limit Sets of Triangular Maps on the Unit Cube

We consider Sturm–Liouville eigenvalue problems of second order with arbitary seperated boundary conditions and perform a suitabl discretization of them. The obtained discrete Sturm–Liouville eigenvalue problems are examined and the asymptotic behavior of their eigenvalue as the norm of the partition tends to zero is investigated.     

Principal Eigenvalues for Generalised Indefinite Robin Problems

We consider the principal eigenvalue of generalised Robin boundary value problems on non-smooth domains, where the zero order coefficient of the boundary operator is negative or changes sign. We provide conditions so that the related eigenvalue problem has a principal eigenvalue. We work with the framework involving measure data on the boundary due to Arendt and Warma (Potential Anal 19:341–363, 2003). Examples of simple domains with cusps are used to illustrate all possible phenomena.     

Layer potential techniques for the narrow escape problem

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.     

The emergence of eigenvalues of a PT-symmetric operator in a thin strip

The Schrödinger operator in a thin infinite strip with PT -symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained.     

Eigenvalue Distribution of Elliptic Operators of Second Order with Neumann Boundary Conditions in a Snowflake Domain

Spectral properties of strongly elliptic operators of second order on bounded snowflake domains without W¹₂–extension property are investigated. We prove that the operators have a pure point spectrum and the asymptotic eigenvalue distributions for the counting function N(λ) are of Weyl type. It is shown that the remainder estimate of N(λ) for Dirichlet and Neumann boundary conditions depends on the inner and outer Minkowski dimension of the boundary ∂Ω, respectively.     

Asymptotic Expansion of Eigenvalues of the Laplace Operator in Domains with Singularly Perturbed Boundary

In this paper, we consider eigenvalue problems for the Laplace operator in three-dimensional domains with singularly perturbed boundary. Perturbations are generated by a complementary Dirichlet boundary condition on a small nonclosed surface inside the domain. The convergence and the asymptotic behavior of simple eigenvalues of the problem are considered.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 299–307.Original Russian Text Copyright © 2005 by M. I. Cherdantsev.     

Zero dissipation limit of the 1D linearized Navier-Stokes equations for a compressible fluid

In this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one-dimensional compressible Navier-Stokes equations. We consider the characteristic boundary conditions, that is we assume that an eigenvalue of the associated inviscid Euler system vanishes uniformly on the boundary. The aim of this paper is to understand the evolution of the boundary layer, to construct the asymptotic ansatz which is uniformly valid up to the boundary, and to obtain rigorously the uniform convergence to the solution of the Euler equations without the weakness assumption on the boundary layer.     

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Faber-Krahn inequality for robin problems involving <Emphasis Type="Italic">p</Emphasis>-Laplacian

The eigenvalue problem for the p-Laplace operator with Robin boundary conditions is considered in this paper.A Faber-Krahn type inequality is proved.More precisely,it is shown that amongst all the domains of fixed volume,the ball has the smallest first eigenvalue.     

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Transfer Matrix for the Two-Dimensional Ising Model

The eigenvalue problem for the transfer matrix of the two-dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two-spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.     

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

1IntroductionIt is well-known that the Sturmian Theory is an important aid in solving many problemsin mathematical physics.Therefore this theory is one of the most actual and extensivelydeveloped field in spectral analysis of boundary-value problems of St…     

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     

The spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential

We study the effect of an eigenvalue appearing from the boundary of the essential spectrum of the Schrödinger operator perturbed by a rapidly oscillating compactly supported potential. We prove sufficient conditions for the existence and absence of such an eigenvalue and obtain the first few terms of its asymptotic expansion for the case where this eigenvalue exists.     

非自伴Dirac算子的迹公式

本文研究了非自伴Dirac算子的一般两点边值问题的渐近迹,首先运用平移算子得到了其Cauchy问题解的渐近式,并由此及边界条件,构造了整函数ω(λ),利用它将边界条件分为八种基本类型,最后采用留数的方法,得到了四种主要类型的特征值的渐近迹公式。     

On the spectral properties of odd-order differential operators with integrable potential

We consider a boundary value problem with irregular boundary conditions for a differential operator of arbitrary odd order. The potential in this operator is assumed to be an integrable function. We suggest a method for studying the spectral properties of differential operators with integrable coefficients. We analyze the asymptotic behavior of solutions of the differential equation in question for large values of the spectral parameter. The eigenvalue asymptotics for the considered differential operator is obtained.     

On the spectral properties of a second-order differential operator with a matrix potential

By applying the method of similar operators to a second-order differential operator with a matrix potential and semiperiodic boundary conditions, we obtain asymptotic estimates for the weighted mean eigenvalue and spectral projections and prove the equiconvergence of spectral expansions.