散度形式二阶椭圆算子Neumann本征值的上界

设Ω是Rm（m≥2）中的一个有界区域,其边界足够光滑。本文旨在给出散度形式二阶椭圆算子-·（A·）的Neumann本征值的一个上界,该上界除了与维数m及系数张量A的迹trA有关外,仅依赖于区域Ω的体积．     

半简单本征值有限元外推

林群等的工作(见[1—3])奠定了本征值有限元外推的理论基础,证明了外推方法对简单本征值有效.本文要证明外推对半简单本征值也有效.由[1—3]的证明过程易知,只要证明存在λ_h,λ_(h/2)的本征函数 u_h,u_(h/2),它们都逼近λ的同一个本征函数 u 就可以了.但由于重本征值在离散化后一般被分离,给证明造成困难.本文提出了一个实施林群外推方法的新方案,巧妙地解决了这个问题.这方案花较少代价就能提高半简单本征值有限元近似解的精度阶.     

一般非均匀凸介质迁移算子本征值的分布

本文讨论一般非均匀凸介质所确定的迁移算子的本征值的分布问题，利用Ｈｉｌｂｅｒｔ空间的Ｈ算子理论，完整地解决了一般非均匀凸介质中迁移算子本征值的分布问题，若｛λｎ｝ｎ＝１＾∞是迁移算子本征值的一种计数，我们证明了Σ↓ｎ＝１↑∞ｅ＾６Ｒｅλｎτ〈＋∞，其中τ是粒子的最大逃逸时间，并对本征值的发散程度以及本征值的个数函数作了相应的讨论。     

关于一类整函数的级与下级

§1.引言本文研究零点聚集在有穷条半射线argz=θ:θ_2,…,θ_k附近的整函数f(z)的级λ(f)与下级μ(f)的关系。以往的研究(参见[1]、[2]、[3]、[4]、[5]、[6])表明:如果f(z)的全部零点仅分布在argz=θ_1,θ_2,…θ_k上,若μ(f)<∞,则λ(f)<∞。张广厚和伍鹏程曾研究上述结果的推广问题。Norbert steinmetz在[6]中提出:是否存在仅与μ(f)和k有关的λ(f)的明确上界?他证明了k=1,2时有λ(f)≤[μ(f)]+k。而对k≥3没有任何一般性结果。本文寻求新的途径,拓广以前的结果,并给出k=3时λ(f)的一个明确上界。     

一类高阶微分方程第二特征值的上界

本考虑形如（-1）^tD^t(p(x)D^ty)=λ(-D^2)^ry，x∈（a,b），D^ky（a）=D^ky(b)=0,k=0,1,2,…，t-1的第二特征值入λ2的上界问题，得到了定理1和定理2，其中定理1的估计系数与[a,b]无关，定理2的结果在一定条件下比定理1的好。     

一类扰动半群生成元的谱分布及其应用

本文研究了扰动半群生成元Ａ＝Ｂ＋Ｋ在带形区域Ｓ（Ｂ）〈Ｒｅλ〈ω（Ｂ）＋‖Ｋ‖内的谱分布，目的是寻找Ａ在带形区域中只分布孤立的有限重本征值以及可能的聚点只分布在直线Ｒｅλ＝Ｓ（Ｂ）上的条件。作为结论的直接应用，本文非常简洁地证明了中子迁移算子的相关结论。     

关于Schrodinger算子的任意相邻两特征值之差的上界估计

设Ω是Ｒ＾ｎ中有界光滑区域，用γ记Ω上Ｓｃｈｒｏｄｉｎｇｅｒ算子的具Ｄｉｒｉｃｈｅｔ边界条件的特征值问题的第ｉ个特征值。本文利用极大极小原理研究任意相邻两特征值之差λｋ＋１－λＫ的上界，推广了Ｐａｙｎｅ和Ｓ．Ｔ．Ｙａｕ等人的有关结果。     

任意阶Laplace算子的特征值估计

设D为n维Euclid空间Rn的一个有界区域,且0＜λ1≤λ2≤…≤λk≤…是l阶Laplace算子的Dirichlet问题{(-△)lu=λu, 在D中,u=(e)u/(e)n=…=(e)l-1u/(e)nl-1=0,在(e)D上的特征值.得到了该问题用其前k个特征值来估计第(k+1)个特征值λk+1的不等式k∑i=1(λk+1-λi)≤1/n(4l(n+2l-2)]1/2{k∑i=1(λk+1-λi)1/2λil-1/lk∑i=1(λk+1-λi)1/2λi1/l}1/2,此不等式不依赖于区域D.对l≥3,上述不等式比所有已知的结果都要好.陈庆民与杨洪苍考虑了l=2的情形.我们的结果是他们结果的自然推广.当l=1时,我们的不等式蕴含杨洪苍不等式的弱形式.文中还给出了陈和杨的一个断言的直接证明.     

非光滑边界条件下具时滞的Rotenberg方程主算子的谱分析

该文考虑细胞分裂的时滞性,引入了具时滞的Rotenberg迁移方程,在L₁空间中非光滑边界条件下(边界算子无界),证明迁移算子生成C₀-半群,并进一步分析迁移算子的谱,得到该迁移算子的谱在区域Γ=σ(A_H)∩{λ∈C|Reλ>γ}(γ>max{λ₀,-σ₀})中仅由有限个具有有限代数重数的离散本征值组成等结果(λ₀,-σ₀的具体意义见正文).     

关于几类图的L(2,1)标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) ,使得若d(x ,y) =1 ,则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y) ｜≥ 1 .图G的L( 2 ,1 ) 标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .Griggs和Yeh猜想对最大度为Δ的一般图G ,有λ(G) ≤Δ2 .本文给出了Kneser图 ,Mycieklski图 ,Descartes图 ,Halin图的λ值的上界 ,并证明了上述猜想对以上几类图成立     

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals

A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in Fazio (SIAM J. Numer. Anal. 33 (1996) 1473). We consider here a survey on recent developments related to the free boundary identification of the truncated boundary. The goals of this survey are: to recall the reasoning for a free boundary identification of the truncated boundary, to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature, and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line.     

Characterizations of boundary pluripolar hulls

We present some basic properties of the so-called boundary relative extremal function and discuss boundary pluripolar sets and boundary pluripolar hulls. We show that for B-regular domains the boundary pluripolar hull is always trivial on the boundary of the domain and present a “boundary version” of Zeriahi’s theorem on the completeness of pluripolar sets.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Stability of hyperbolic–parabolic mixed type equations with partial boundary condition

In this paper, we are concerned with the hyperbolic–parabolic mixed type equations with the non-homogeneous boundary condition. If it is degenerate on the boundary, the part of the boundary whose boundary value should be imposed, is determined by the entropy condition from the convection term. If there is no convection term in the equation, we show that the stability of solutions can be proved without any boundary condition. If the equation is completely degenerate, we show that the stability of solutions can be established just based on the partial boundary condition.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.