Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.     

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.     

一类边界带特征参数的不连续高阶微分算子的自共轭性

考虑了一类具有特殊转移条件且边界条件中均带有特征参数的高阶边值问题,建立了一个与其相关的新空间H与新算子A,讨论了算子A在H中的自共轭性.     

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.     

一类两个边界带特征参数的不连续四阶微分算子的自共轭性

本文研究了一类具有特殊转移条件且两个边界条件中带有特征参数的四阶微分算子的自共轭性问题.建立了一个与其相关的新的空间H,将上述问题的研究转化为对此空间中一个线性算子A的研究.     

Non-classical Eigenvalue Asymptotic for Elliptic Operators of Second Order in Unbounded Trumpet-shaped Domains with Neumann Boundary Conditions

The paper deals with spectral properties of elliptic operators of second order in irregular unbounded domains with cusps. The eigenvalue asymptotic of the operator with Neumann boundary conditions is proved. The eigenvalue asymptotic in these domains is different from that with Dirichlet boundary conditions.     

边界条件中带有谱参数的不连续Sturm-Liouville算子的特征值问题

本文研究了具有转移条件且边界条件含特征参数的Sturm-Liouville算子L的特征值问题.首先,使用微分算子谱分析经典的方法,得到λ是该边值问题的特征值的充要条件,证明了该边值问题最多有可数个实的特征值、没有有限值的聚点.其次,通过渐近估计证得,所研究的Sturm-Liouville算子L有可数个离散的特征值且下方有界.     

On Boundary Conditions for Wedge Operators on Radial Sets

We present a theorem about calculation of fixed point index for k-ψ-contractive operators with 0 ≤ k < 1 defined on a radial set of a wedge of an infinite-dimensional Banach space. Then, results on the existence of eigenvectors and nonzero fixed points are obtained.     

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.     

On a Class of Population Dynamics Problem with Mixed Boundary Conditions

In this paper, we consider a linear population dynamics model with age dependence and spatial structure in which birth process is described by a nonlocal term. Carleman estimate for the backward nonlocal adjoint problem providing the observability results is fundamental to the analysis and by duality it provides the controllability result. More precisely, we observe that one can expect to compute the initial distribution using observations on a small part of the boundary.     

Heat Kernel Estimates for Operators with Boundary Conditions

We prove Gaussian upper bounds for kernels associated with non – symmetric, non – autonomous second order parabolic operators of divergence form subject to various boundary conditions. The growth of the kernel in time is determined by theboundary conditions and the geometric properties of the domain. The theory gives a unified treatment for Dirichlet, Neumann and Robin boundary conditions, and the existence of a Gaussian type bound is essentially reduced to verifying some properties of the Hilbert space in the weak formulation of the problem.     

Eigenvalue Problem of Doubly Stochastic Hamiltonian Systems with Boundary Conditions

In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii~erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.     

Minimization of the Principal Eigenvalue Under Neumann Boundary Conditions

We investigate the minimization of the principal eigenvalue of the Laplacian under Neumann boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, this minimization problem is motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive. The question may have practical importance in the context of reserve design.     

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [1–8], we prove theorems on the existence and uniqueness of the classical solutions of these problems.     

边界条件含特征参数的Sturm-Liouville问题的特征值比率

给出修正Prfer变换的几何意义以及相关性质,并利用上述性质讨论Sturm-Liouville问题-(py′)′+gy=λωy,x∈[0,1],参数边界条件为y(0)=0与((py′)(1))/(y(1))=aλ+b,得到了当a>0,q≥0的特征值比率上界,这把Ashbaugh等人的结果推广到边界条件含参数的情形.     

带有多点边条件的高阶微分算子的自共轭性

研究带有多点边条件的Sturm-Liouville问题,在新的Hilbert空间中,定义与多点边条件中连接特性相关的最大算子和最小算子,建立了带有多点边条件的高阶微分算子自共轭性的解析判别准则.     

Multipoint Boundary Conditions for Differential Operators

We establish differential properties of generalized solutions of multipoint boundary-value problems for ordinary differential equations.     

Faber-Krahn Inequality for Anisotropic Eigenvalue Problems with Robin Boundary Conditions

In this paper we study the main properties of the first eigenvalue λ ₁(Ω) and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain Ω ? ? _n , assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality for λ ₁(Ω).