EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A THREE-POINT BOUNDARY VALUE PROBLEM

In this paper, we are concerned with the existence and nonexistence of positive solutions to a three-point boundary value problems. By Krasnoselskii’s fixed point theorem in Banach space, we obtain sufficient conditions for the existence and non-existence of positive solutions to the above three-point boundary value problems.     

一阶混合型方程组的间断边值问题

本文论讨单连通区域上一阶线性混合型(椭圆—双曲型)方程组的间断边值问题.我们首先给出混合型方程组特别是最简单的混合型方程组解的表示式,然后使用逐次逼近法,证明上述边值问题解的存在性与唯一性.由以上结果,可导出A.V.Bitsadze所得的Lavrent’ev-Bitsadze方程u     

Oblique derivative problems for linear mixed equations of second order

Oblique derivative boundary value problems for the linear mixed (elliptic-hyperbolic) equation of the second order, i.e. the generalized Lavrent’ev-Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved, and a priori estimates of the solutions of the above problem are obtained. Project supported by the National Natural Science Foundation of China.     

INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS

This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.     

AN IRREGULAR OBLIQUE DERIVATIVE PROBLEM FOR SOME NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

1FormulationofirregularObliqueDerivativeProblemLetDbean(N 1)-connecteddomainwithboundaryr=UU=,r,6Cj(0相似文献   

二阶拟线性混合型方程的间断斜微商问题

本文处理二阶拟线性混合（椭圆-抛物）型方程在单连通区域上的间断斜微商问题。我们首先导出最简单的混合型方程上述边值问题解的表示式,并证明此边值问题解的唯一性,然后用逐次迭代法证明上述问题解的存在性。本文获得了此间断边值问题的可解性结果,包括有关文献的结果作为特殊情形。     

Oblique derivative problems for second order nonlinear equations of mixed type in multiply connected domains

In this paper, the unique solvability of oblique derivative boundary value problems for second order nonlinear equations of mixed (elliptic-hyperbolic) type in multiply connected domains is proved, which mainly is based on the representation of solutions for the above boundary value problem, and the uniqueness and existence of solutions of the above problem for the equation u_xx + sgn y u_yy = 0.     

含有广义p-Laplace算子的非线性边值问题解的存在性的研究

该文研究了两类含有广义p-Laplace算子的非线性边值问题. 首先, 利用变分不等式解的存在性的结果, 证明了含有广义p-Laplace算子的非线性Dirichlet边值问题解的存在性. 然后, 提出了一类含有广义p-Laplace算子的非线性Neumann边值问题. 通过深入挖掘这两类非线性边值问题间的关系, 借助于极大单调算子值域的一个扰动结果, 证明了含有广义p-Laplace算子的非线性Neumann边值问题解的存在性. 文中采用了一些新的证明技巧,推广和补充了作者以往的一些研究工作.     

二阶非线性无穷边值问题的奇摄动(Ⅱ)

本文讨论了含小参数ε>0的二阶非线性奇摄动无穷边值问题(其中a_i,β为常数,i=0,1)之解的存在性,并且给出了解的渐近估计式.     

含一阶导数的二阶微分方程组多点边值问题正解的存在性

讨论了一类含一阶导数的二阶常微分方程组多点边值问题正解的存在性.利用一个新的不动点定理,得到上述问题具有一个正解的充分条件.     

带有可测系数的二阶非线性抛物型方程组的初—混合边值问题

在多连通区域上研究带有可测系数的二阶非线性抛物型方程组的初－混合边值问题，首先我们将其化为复形式的方程组，并给出在一定条件下的上述初－这值问题解的先验估计，然后利用解的估计和列紧性原理，证明了这种初－边值问题解的存在性。在论证过程中，我们始终用复分析方法讨论文中所提出的问题，没有看到国外有人使用这种方法处理此类问题。     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

This is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

双解析函数的Haseman边值问题

本文利用双解析函数的Cauchy型积分和带位移的奇异积分方程方法,研究并得到了双解析函数的Haseman边值问题的一般解的表示式和可解条件以及线性无关解的个数与指标之间的关系.     

The generalized Thomas–Fermi singular boundary value problems for neutral atoms

This paper presents an upper and lower solution theory for singular boundary value problems modelling the Thomas–Fermi equation, subject to a boundary condition corresponding to the neutral atom with Bohr radius equal to its existence interval. Furthermore, we derive sufficient conditions for the existence–construction of the above‐mentioned upper–lower solutions. Copyright © 2005 John Wiley & Sons, Ltd.