关于半线性椭圆共振问题的注记

本文应用Ｍｏｒｓｅ理论和惩罚性技巧研究了一类半线性椭圆方程在无穷远处和在原点处都共振情形下非平凡解的存在性．     

Spline Techniques for the Numerical Solution of Singular Perturbation Problems

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.     

解非线性椭圆型方程奇异摄动问题的一致收敛差分格式

给出了求解非线性椭圆型偏微分方程奇异摄动问题的广义OCI差分格式．证明了这种格式的解关于摄动参数一致收敛于连续问题的解．给出了数值例子．     

含积分算子拟线性系统边值问题的奇摄动

研究奇摄动积分微分方程组边值问题εy"＝f(x,y,Ty,ε)y′++g(x,y,Ty,ε);y(0,ε)＝A(ε),y(1,ε)=B(ε)其中y、g、A和B均为n维向量函数,f是n×n矩阵函数,(Ty)(x)=∫xK(x,s,y(sε),ε)ds在一定假设条件下,利用对角化技巧和逐步逼近法证明解的存在,并给出解的直到0(εN+1)的渐近展开式.     

On Singular Phenomena in Certain Time-Optimal Problem

The purpose of this paper is to present the solution of time-optimal problem ofthe controlled object the dynamics of which is given by: , , where and motion resistance function if ,f(x)=-A if x > 0 where . That model describes dynamicsof a very important class of industrial installations. As the time-optimalproblem will be understood a transfer of the initial state to the target state in a minimumtime . There has been shown that in the formula defining resistancefunction f(x)there exists a value that plays an essentialrole in time-optimal structure formation. Namely, if then thetime-optimal control process is typical, analogous as in classical case , i.e. there exists a switching curve formed by thetrajectories of time-optimal solutions reaching the target state and thetime-optimal process is formed by at most one switching operation. For the caseA>A_bwe will examine two following singular phenomena.(a) If the target state z₁=(0, 0) then there exists theswitching curve, dividing the state plane into two sets, however only one itsbranch is formed by the time-optimal solution reaching the target z₁=(0, 0) and generated by the control u=-1. None of solution formsthe second branch of switching curve. It is formed by a state-locus dependingon the value of Aonly. In dependency of the starting state z₀ thetime-optimal control process is generated by bang-bang control with none,one or two switching operations. This is the first singular phenomenon,because any small decrease of the value Aover A _b requires to change thestructure which would be able to generate the time-optimal process.(b) The paper shows, that if the target state z ₁(x_1, 0), x₁>0then there exists a set of the starting states from which there start twotrajectories reaching the target in the same minimum time. This is thesecond phenomenon.Finally, some suggestions as to practical applications have been given too.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.     

Nonradial solutions of a nonhomogeneous semilinear elliptic problem with linear growth

In this paper we consider the problem

     

一类奇异边值问题解的存在性

在非线性项系数无界的情形下，得到了一类半线性椭圆型方程爆破解的存在性。     

关于奇异摄动问题一致有效渐近解的构造方法

本文指出文［１］的不足，提出构造在整个区域内一致有效渐近解的方法，解决了长期以来未能解决的问题．     

Singular quasilinear equations with quadratic growth in the gradient without sign condition

Given a bounded domain Ω in RN, and a function a∈Lq(Ω) with q>N/2, we study the existence of a positive solution for the quasilinear problem

     

Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

We consider the system of Fredholm integral equations

高阶椭圆型偏微分方程奇异摄动问题一致收敛差分格式

本文,我们讨论了一类高阶椭圆型偏微分方程奇异摄动问题。给出了连续问题解的先验估计。另外,我们还提供了一种数值求解该类问题的指数型差分格式。最后,证明了差分问题的解在能量范数意义下关于小参数一致收敛到连续问题的解。     

Singular Perturbation Elliptic Boundary-Value Problems in the Case of a Nonisolated Root of the Degenerate Equation

For a singularly perturbed elliptic equation (the Neumann boundary-value problem), we prove a theorem on the passage to the limit for the case in which the degenerate equation has a nonisolated root.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 26–36.Original Russian Text Copyright © 2005 by V. F. Butuzov, M. A. Terent’ev.     

一类非线性向量微分方程无穷边值问题的奇摄动

讨论一类非线性向量微分方程无穷边值问题的奇摄动,虽然此类边值问题在有限区间上曾被广泛地研究过,但在无限区间上还未曾采用对角化的方法去研究它.在适当的条件下,该文采用新的方法去证明解的存在性和任意阶的一致有效的渐近展开式,同时也给出误差估计.     

具有转向点的一类奇摄动边值问题解的多重边界层现象

本文讨论如下边值问题：x=0是转向点（c（0）=0），而在x=－1处出现多重边界层现象．对不同层次采用不同的伸长变量，构造具有不同量级的边界层校正项，得到关于解的一致有效的渐近展开式和有关的余项估计．