数值积分下四阶方程协调有限元解的L_∞估计

|u|_(m,Ω), ‖u‖_(m,Ω)(以下下标为Ω时略去),p=∞时采用通常的修正定义.H(?)(Ω)是C_0~∞(Ω)在模‖·‖(?)下的闭包,(·,·)表示L_2内积。另外,记‖u‖m, ,h=(sum from e ((‖u‖_m~p),p,e)p。 讨论下列四阶方程的有限元逼近问题:     

简单分歧数值解的L_p估计

§1.问题的提出 考虑单参数二阶椭圆拟线性微分方程:λ∈R,Ω?R~N(N=1,2)是多角形凸域(要求?Ω是Lipschitz连续的)或光滑域.[a_(ij)]∈C~1满足正定条件.f(x,y)∈C~2f(x,0)≡0,f_y(x,θ)≥0,但f_y(x,0)?0,?x∈Ω.记||·||_(j,p,Ω),p≥1,j=0,1,2为通常的W~(j,p)(Ω)范.H_0~1?W_0~(1,2),(·,·)为H_0~j中通     

简单分歧数值解的L_p估计

§1.问题的提出 考虑单参数二阶椭圆拟线性微分方程:λ∈R,Ω?R~N(N=1,2)是多角形凸域(要求?Ω是Lipschitz连续的)或光滑域.[a_(ij)]∈C~1满足正定条件.f(x,y)∈C~2f(x,0)≡0,f_y(x,θ)≥0,但f_y(x,0)?0,?x∈Ω.记||·||_(j,p,Ω),p≥1,j=0,1,2为通常的W~(j,p)(Ω)范.H_0~1?W_0~(1,2),(·,·)为H_0~j中通     

有限元导数的一致超收敛估计

§1 引言 设ΩR~2是边界为Γ的有界区域,Ω=Ω∪Γ。Sobolev空间W_p~m(Ω)的范数、半范数分别用‖·‖_(m,p,Ω),|·|_(m,p,Ω)或‖·‖_(m,p),|·|_(m,p)表示。在W_p~1(Ω)×W_p~1(Ω) (1≤p≤∞,1/p+1/q=1)上定义双线性泛函: 我们假定系数a_(if)定义在Ω上且满足     

磁流体力学方程组弱解的适定性

考虑由磁流体力学方程组控制的二维不可压缩流体的初边值问题,在边界光滑的有界区域中,当(u_0,B_0)∈((W~(m,p)(Ω))~2×W~(m,p)(Ω))时,利用Galerkin方法和先验估计,得到了相应的初边值问题存在唯一的弱解(u(·,t),B(·,t))∈((W~(m,p)(Ω))~2×W~(m,p)(Ω)),并证明了弱解对初值(u_0,B_0)具有连续依赖性.     

L~∞-Bounds of Solutions for Strongly Nonlinear Elliptic Problems with Two Lower Order Terms

In this work,we will prove the existence of bounded solutions in W_0~(1,p)(Ω)∩L~∞(Ω) for nonlinear elliptic equations-div(a(x,u,▽u))+g(x,u,▽u)+H(x,▽u) =f,where a,g and H are Caratheodory functions which satisfy some conditions,and the right hand side/belongs to W~(-1,q)(Ω).     

二阶退化双曲型方程带奇性斜导数的混合问题

本文研究二阶弱双曲型方程具有奇性斜导数的混合问题其中场v在Г=?Ω的子流形Г_0上与Г相切,而与Г_0横截,dim Г_0=dim Г-1,当x’(∈Г)沿v(x’)的切向通过奇点时,〈v(x’),n(x’)〉不变号(n(x’))表Г的单位外法向量),证明了若f∈W_(1/2)~(l+1,l)(Q),g∈W_(1/2)~(l+3/2,l+2)(?Q),则问题(Ⅰ)有唯一解u∈W_(1/2)~(l+2?l+2)(Q)。当〈v(x’),n(x’)〉由正到负时,在Г_0上补充条件u|_(Г_0)=u_0(x’,t)∈W_(1/2)~(l+3/2,l+2)(?_2Q),?_2Q=Г_0×R_+~1以后,问题(Ⅰ)存在唯一解u∈W_(1/2)~(l+2,l+2)(Q)。     

关于含临界指数的p-平均曲率算子

This paper is concerned with the existence of positive solutions of the following Dirichlet problem for p-mean curvature operator with critical exponent: -div((1 |▽u|~2)(p-2)/2▽u)=λu~(p*-1) μu~(q-1),u＞0,x∈Ω, u=0,x∈■Ω, where u∈W_0~(1,p)(Ω),Ωis a bounded domain in R~N(N＞p＞1)with smooth boundary ■Ω,2＜=p＜=q＜=P~*,P~*=(Np)/(N-p),λ,P＞0.It reaches the conclusions that this problem has at least one positive solution in the different cases.It is discussed the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with critical exponent by using Nehari-type duality property firstly.As p=2,q=p,the result is correspond to that of Laplace operator.     

拟凸条件下变分问题解的正则性

本文研究变分问题(1)■(u:Ω)=∫_Ωf(x,u,Du)dx的极小函数的正则性,其中Ω■R~n是有界开域,u:Ω→R-~N,Du:Ω→R(nN),f: ×R~N×R(nN)→R。定义称函数f满足严格拟凸条件,是指存在常数v>0,使得对任意的(x_0,u_0,p_0)∈Ω×R~N×R(nN)和φ∈C~∞_0(Ω,R~N),都有■(2)其中|Ω|是Ω的Lebesgue测度。定理设u∈H~(1,2)(Ω,R~N)是泛函f的极小函数,即对任意的φ∈H_0~(1,2)(Ω,R~N),都有■而f(x,u,p)满足下列假设 (H1) f满足严格拟凸性,即(2)成立, (H2) f关于p的二阶导数存在,且存在常数L>0,使得■对任意的(x,u,p)∈Ω×R~N×R~(nN),都有■ (3)|f_(pp)(x,u,p)|≤L_0 (H3) 存在[0,∞]上的连续、有界、凹的函数∞(t),使得(4)■(5)■(6)■且ω(t)≤At~α,其中A,α是正常数。那么存在常数δ∈(0,1)和开集Ω_0Ω,使得|Ω-Ω_0|=0,Du∈C~6(Ω_0,R(nN))。     

关于Schwarz交替法的拟边界松弛方法

§1.松弛方法 我们讨论二阶自共轭椭圆型方程的Dirichlet问题.设Ω?R~2为一多边形区域. a(u,v)=(f,v),v∈H_0~1(Ω),f∈H~(-1)(Ω), u∈H_0~1(Ω)是定义在其上的边值问题的变分形式,这里取齐次边界条件仅为叙述问题方便.双线性型a(·,·)满足:     

周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移北大核心CSCD

该文解析研究了有限个周期排列的抛物形沟槽激发的水波Bragg共振.首先,利用变量替换,先将系数为隐函数的修正缓坡方程(MMSE)转化为系数为显函数的显式方程.然后,构造了修正缓坡方程的Frobenius级数解,并给出了级数解的收敛条件.最后,利用质量守恒的耦合条件,建立了反射系数的解析公式.根据反射系数的解析公式,分析了沟槽个数、沟槽深度与宽度对Bragg共振峰值、共振相位和共振带宽的影响.当沟槽深度和宽度固定而沟槽个数增加时,共振峰值逐渐增大并趋向于1,而共振带宽则逐渐变窄并趋于固定值.当沟槽个数和宽度固定时,Bragg共振峰值随沟槽深度增加而增加.当沟槽个数和深度固定时,Bragg共振反射峰值随沟槽宽度增加而先增后减,预示了沟槽存在某个宽度使得共振峰值达到最大,为Bragg共振反射针对沟槽宽度的优化奠定了理论基础.特别地,前不久在有限个周期排列旋轮线形沟槽上刚刚观察到的Bragg共振反射峰值相位的上移现象,再次在该文考虑的抛物形沟槽上得到确认,表明针对有限周期排列的沟槽地形,Bragg共振反射峰值的相位上移是一个普遍现象.也因此说明,凡是正弦沙纹和周期人工沙坝所激发的Bragg共振反射,其主振相位将会下移,而凡是周期系列沟槽所激发的Bragg共振反射,无论沟槽形状如何,其主振相位都将上移.另外,我们从Bragg共振的原始定义出发,定量地解释了相位上移发生的数学机理.     

Integrals of semiintegrable k-tuples of germs of vector fields in the plane

An ordered collection of vector fields in the plane is called a semiintegrable collection if each succeeding element of the collection lies in the stationary subalgebra (in the algebra of smooth vector fields) of the phase portraits of all the preceding elements of the collection. The orbital equivalence of semiintegrable collections means the possibility of transferring, with the aid of a diffeomorphism of the plane, the phase portraits of the elements of one collection into those of the other one, respectively, with the preservation of the order. A complete classification of finite-modal collections relative to orbital equivalence is obtained. The invariants of the normal forms, namely the generators of the algebra of the first integrals, the invariant measures, the stationary subalgebras of the phase portraits of the collection, are computed in terms of elementary functions. Integer-valued invariants are obtained, like the number of elements in the collection, the dimension of the stationary subalgebra, etc.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 70–105, 1992.     

Studying the longitudinal flow of polymers with periodic deformation

Conclusions 1. At low rates of elongation the state of a polymer melt depends neither on the strain nor on the strain rate. The frequency dependence of the components of the complex longitudinal modulus, measured by superposition of a periodic deformation on a steady longitudinal flow, resemble, at low elongation rates, the frequency characteristics of the components of the complex shear modulus.2. Relaxation processes due to steady longitudinal flow affect the components of the complex longitudinal modulus measured by periodic deformation.3. A melt of a polydisperse polymer under conditions of longitudinal flow begins, at some definite strain rate, to respond to periodic deformation as a rubber-like body and continues to do so as the amount of elongation increases, which is manifested by the trend of the frequency dependence of the modulus component E', this trend becoming similar to that for a solid body.4. As the strain increases during elongation, the upper boundary of the flow range, which is determined by the periodic deformation, shifts toward low frequencies.5. The viscoelasticity characteristics of a material subject to elongation can be accurately enough described by nonlinear equations with the aid of the linear shear-relaxation spectrum.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 507–513, May–June, 1978.     

Quality rating of a metal matrix-diamondcomposite from its thermal conductivity and electric resistance

The efficiency of hot-pressed diamond-containing composite materials (DCM) for various tool applications is greatly affected by microdefects, namely, the residual porosity of the metal matrix, damaged diamond grains, and imperfect diamond-matrix interfaces. An instrumental evaluation of these microdefects, predetermining the quality of a tool equipped with DCM, is rather difficult due to the small size, the nonstandard shape, and the strong heterogeneity of specimens. Proposed here is an alternative, nondestructive technique of DCM quality rating, which includes the measurement of electric resistance and thermal conductivity of diamond-containing composites and processing the obtained data by the methods of composite mechanics. It exploits the fact that diamond, being a dielectric, possesses an extremely high thermal conductivity, which allows estimating the residual porosity of a sintered metal matrix from the ratio of specific electric resistances, one being measured and another predicted by a theory. These data, in turn, are utilized to predict the thermal conductivity ofDCMwith an imperfect matrix. Matching with experiments, after solving the inverse problem gives the thermal resistance of diamond-matrix interface, which, within the frame work of the given model, simulates the damage of both the diamond grains and their bonds with the matrix. Thus, the numerical rating of quality is given in terms of two dimensionless parameters. The first one, 0 < K < 1, reflects the quality of the sintered metal matrix, whereas the second one, 0 < R <1, is an aggregate measure of the integrity of diamond grains and the perfection degree of composite interfaces. The quite satisfactory agreement observed between the theory and experiment confirms the efficiency of the technique and the reliability of the data obtained. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 361–374, May–June, 2006.     

Convolution equations in multidimensional spaces

Author's abstract of a dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on Nov. 10, 1977, at the session of the Specialized Council D 002.38.03 at the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. Official opponents: Corresponding Member of the Academy of Sciences of the USSR, Doctor of Physicomathematical Sciences, Professor A. G. Vitushkin; Corresponding Member of the Academy of Sciences of the USSR, Doctor of Physicomathematical Sciences, Professor V. K. Ivanov; Doctor of Physicomathematical Sciences, Professor Yu. F. Korobeinik.Translated from Matematicheskie Zametki, Vol. 25, No. 5, pp. 761–774, May, 1979.     

A non-equilibrium model of a porous medium saturated with immiscible fluids

A model of a thermoelastic porous medium, saturated with two immiscible fluids, is considered. It is assumed that there are no phase transitions, the contribution of pulsations to the stress and kinetic energy is small, and that the components of the medium are in thermodynamic equilibrium. The non-equilibrium of the state, related to the finite time of redistribution of the fluids among the pores of the channels due to the presence of surface forces, is taken into account. A general form of the governing relations, necessary and sufficient to satisfy the principles of thermodynamic compatibility and independence of the choice of system of coordinates, is obtained. It is shown that the establishment of equilibrium is accompanied by dissipation due to capillary forces, which does not lead to seepage dissipation or thermal dissipation. For the case when the deformation of the skeleton and the deviation of the mean porous pressure and the temperature from the initial values are small, while the saturation and the non-equilibrium parameter undergo finite changes, an approximation of the potential of the skeleton is proposed in the form of a quadratic expansion in small parameters. A feature of the expansion is the presence of an initial value of the potential, which depends on the saturation and non-equilibrium. The relationship between the thermodynamic potential and the non-equilibrium kinetics, related to the requirement that the dissipation by the capillary forces should be non-negative, is determined. A generalized Darcy's law is formulated, which takes cross terms into account. It is shown that the proposed approximations enable key effects, which accompany the motion of immiscible fluids in a porous medium, to be described.     

Discrete symmetries in quantum scattering

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrödinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 60–79, January, 1994.     

Rheology and molecular structure of polyethylene melts

Conclusions 1. The molecular structure indices of low-density polyethylene (LDPE), i.e., the weighted-average (wt.-av.) molecular weight, the number of branch points of long-chain branchings per molecule, and the polydispersity index, were calculated from the data of gel-permeation chromatography, assuming that the Mendelson—Drott hypothesis holds. They show a fairly close correlational relationship to the index of the mechanical characteristics of this melt.2. The correlation between the highest Newtonian viscosity and the molecular characteristics, taking into account the polydispersity index, is improved if the low-molecular-weight part of the molecular weight distribution MWD (M < 800) is neglected.3. The index of the exponential equation, which determines the dependence of the highest Newtonian viscosity of LDPE on the wt.-av. molecular weight, increases when the influence of the polydispersity is taken into account.4. Commercial samples of LDPE synthesized under similar technological conditions show a fairly good correlation between the polydispersity and the degree of branching of the molecule.5. From the highest Newtonian viscosity, the specific relaxation time of the LDPE, and the dependence of the degree of branching of the molecules on the polydispersity of the molecular weights, common for the given technology of the synthesis, it is possible to calculate the molecular structure indices of LDPE with satisfactory accuracy.For communication 3, see [3].The Walter Ulbricht Enterprise "Leunawerke," German Democratic Republic. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. The Okhtinsk Scientific and Production Union "Plastpolimer," Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 283–293, March–April, 1977.     

Numerical identification of properties of particle-reinforced composite materials

The paper deals with numerical identification of the average elastic properties of particle-reinforced composite materials. The finite element method for the determination of deformation energy of the characteristic volume element was used. In earlier analytical investigations, an approximation function of the averaged elastic properties of the composite was derived. An identification procedure allows the estimation of the unknown approximation parameters from numerical experiments. The obtained functions describe precisely the numerical data for any relationships between constituents of the material.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Institute of Computer Analysis of Structures, Riga Technical University, Riga PDP-1658, Latvia. Institute of Materials Science, Department of Materials Science, Martin-Luther-University Halle-Wittenberg, D-06099 Halle, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 383–390, May–June, 1998.     

上市公司财务绩效的改进集成评价研究——以26家运输公司为例

为解决以往上市公司集成评价模型中各方法评价结果不一致问题,本文构建了改进集成评价模型。该模型先采用层次分析法、灰色关联度法、因子分析法进行评价,并运用KENDALL-W协和系数法对各评价结果进行事前一致性检验;通过检验后,再分别运用算术平均组合评价模型、Borda组合评价模型和Copeland组合评价模型进行组合评价。为了衡量组合评价与各评价方法的评价结果是否吻合,应用Spearman等级相关系数进行组合评价方法的事后检验,并根据Spearman等级相关系数的大小,选出最优的组合评价方法。最后,对中国26家上市运输公司财务绩效进行了集成评价的实证研究,并以算术平均组合评价模型的标准得分为聚类指标,采用欧氏距离法对26家上市运输公司进行聚类。结果显示:铁路、水路、公路运输各存在一个典型绩优企业,分别是:铁龙物流、北京传媒、海峡股份,四类上市运输公司中铁路上市公司财务绩效最优。