THE RIEMANN-HILBERT BOUNDARY VALUE PROBLEM FOR THE MOISIL-THEODORSCO SYSTEM

This article studies the inhomogeneous Moisil-Theodorsco system in the space R3, gives the integral expression of its solution, proves the Holder continuity of the solution. Moreover the author studies the Riemann-Hilbert boundary value problem for the Moisil-Theodorsco system in a cylindrical domain of R3, and gives the solvability conditions and the integral expressions of solutions. The Holder continuity of the solutions is proved.     

有界凸域上拟共形映射的Holder连续性

该文利用曲线族的模,得到了n维空间中的有界凸域 D 到单位球 Bⁿ上的 K -拟共形映射f的全局Holder连续性, 且Holder指数α=K^1/(1-n)是最佳的.     

On the Interior Smoothness of Solutions to Second-Order Elliptic Equations

We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space W _{2, loc} ¹ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in W _2,loc ¹ .     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

参数广义向量拟平衡问题的Hlder连续性

通过研究Hausdoff距离的性质,给出了在度量空间下广义向量拟平衡问题的解集是集值情况下的解集的Hlder连续性.     

非线性抛物椭圆方程组的正则解和奇异解

该文用单模方法在Lorentz空间研究了抛物椭圆方程组奇异解和正则解的存在性, 其中初值属于Lorentz空间L^{n/2,∞ ₍}Rⁿ), n≥ 3. 利用时间加权的Lorentz空间, 还得到了其正则解. 此外, 如果初值满足自相似结构, 也得到了自相似解的存在性.     

Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript>([<Emphasis Type="Italic">a,b</Emphasis>])

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C ^α([a, b]).     

On the Connectedness of Self-Affine Tiles

Let T be a self-affine tile in Rⁿ defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R², for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given.     

On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable

This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rⁿ, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption ‘strongly’ and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case.     

Rn上带奇异性的非线性双调和方程的正整解

该文以Schauder-Tychonoff不动点定理为工具,建立了一类Rⁿ上带奇异性的非线性双调和方程 Δ²u=f(|x|, u,| u|)u^-β (x ∈ Rⁿ, n ≥ 3, β > 0) 正的径向对称整体解的存在性定理,并给出了解的有关性质,所得的结果丰富和发展了文献[1--5]的结果.     

拟线性退化抛物型方程组解的整体存在性和爆破

该文研究光滑有界区域Ω( R^N (N≥ 1) 上具有齐次Dirichlet边界条件的拟线性退化抛物型方程组 u_t-div(|▽u|^p-2 ▽u) =av^α, v_t-div(|▽v|^q-2 ▽v) =bu^β 的非负解的性质, 其中p, q>2, α, β ≥ 1, a, b> 0是常数. 该文指出上述方程组的解是否在有限时刻爆破依赖于初值、系数 a 与 b以及 αβ 和 (p-1)(q-1)之间的关系.     

On the Ginzburg-Landau Wave Equation

This note studies the global existence of strong solutions ofthe initial value problem of the Ginzburg-Landau wave equationon Rⁿ (n = 1, 2, 3). Current address: Department of Mathematics and Statistics, Universityof New Mexico, Albuquerque, NM 87131, USA     

无界域上一类高阶奇异T算子的Holder连续性及其应用

首先定义了Rⁿ上一类高阶奇异T算子,然后将该算子分为两部分分别讨论得到了该算子在Rⁿ上的Holder连续性,最后给出了这个算子的广义导数并通过导数和正则函数的性质给出了高维空间中的一类广义H_n方程组的解.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Linear transmission problem for a Bitsadze system with a supersingular circle

For a second-order elliptic system with a supersingular circle whose leading part is the Bitsadze operator, we obtain an integral representation of the solution and the corresponding inversion formulas in the finite and infinite domains. The obtained integral representations are used to study the behavior of solutions as r → R and analyze a problem of linear transmission type.     

Asymptotic Behavior of Solutions of Differential Equations with Variable Delays

The authors consider the system of forced differential equationswith variable delays whereB_j(t) is a continuous n x n matrix on R⁺, F C(R⁺, Rⁿ) and C(R⁺, R⁺). Using Razumikhin-type techniques and Liapunov'sdirect method, they establish conditions to ensure the ultimateboundedness and the global attractivity of solutions of (*),and when F(t) = 0, the asymptotic stability of the zero solution.Under those same conditions, they also show that is a necessary and sufficient condition for allof the above properties to hold. 1991 Mathematics Subject Classification:34K15, 34C10.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Periodic Solutions of an Integrodifferential Equation Modelling Resonant Sloshing

This paper considers the problem of finding solutions of which have period 2 and mean 0. The equation models periodicgravity waves on the surface of a shallow rectangular tank whichis oscillating longitudinally near the primary resonance frequency.The singular integral memory term is well behaved, and the Lyapunov-Schmidtprocedure produces bifurcation equations to which the analysisof Hale & Rodrigues on the damped Duffing equation is directlyapplicable. The multiplicity of all small solutions of the equationis found as (µ₁µ₂µ₃) varies in a full neighbourhoodof 0 in R³.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.