一般形式的一阶椭圆型偏微分方程组拟线性Riemann-Hilbert问题

在文[1,2,3]中,E.Wegert和L.V.Wolfersdorf等人讨论了一类全纯函数的拟线性Riemann-Hilbert问题在Hardy空间中的可解性,在文[4]中,讨论了广义解析函数的拟线性Riemann-Hilbert问题,同样得到该边值问题在H2类解空间中的可解性.本文在前面研究工作的基础上,对一般形式的一阶椭圆型偏微分方程组拟线性Riemannn-Hilbert问题作了更深入的讨论,在适当的假设条件下,应用积分算子理论,函数论方法及不动点原理,证明了该边值问题在相应的泛函空间中同样是可解的.     

一阶拟线性偏微分方程Cauchy问题的整体光滑解

本文应用整体隐函数存在定理研究了一阶拟线性偏微分方程及主部相同的方程组的整体光滑解问题。     

一阶拟线性偏微分方程广义Cauchy问题的整体光滑解

本文研究了下列一阶拟线性偏微分方程的广义Ｃａｕｃｈｙ问题：ｕ_ｔ＋λ（ｕ）ｕ_x＝０，ｕ｜Γ＝φ（ｘ），Γ：ｘ＝ｒ（σ），ｔ＝ｓ（σ）．证明了该问题在一定条件下，于上半平面Ω＝｛－∞＜ｘ＜＋∞，ｔ≥０｝上存在整体光滑解．     

一阶拟线性偏微分方程组局部几何解的实现定理

用奇点理论研究一阶拟线性偏微分方程组,得到局部几何解的实现定理等结果.     

一阶拟线性偏微分方程组局部几何解的实现定理

用奇点理论研究一阶拟线性偏微分方程组,得到局部几何解的实现定理等结果.     

一类拟线性椭圆型偏微分方程的先验界的估计

近几年对边值问题－ｄｉｖ（｜Ｄｕ｜^p-2Ｄｕ）＝λｆ（ｕ）｝在Ω上ｕ｜(?)Ω＝０正解方面已经得到了许多结果．这里λ＞０，Ω是有界区域和对ｓ≥０，ｆ（ｓ）≥０．在本文中在条件Ｎ≥ｐ＞１，Ω＝Ｂ_１＝｛ｘ∈Ｒ^N，｜ｘ｜＜１｝和ｆ∈Ｃ¹（０，∞）∩Ｃ⁰（［０，∞）），ｆ（０）＝０，研究了这类问题的正对称解的先验界估计．     

一阶拟线性偏微分方程组稳定几何解的局部分类

在切触流形框架中, 研究一阶拟线性偏微分方程组. 利用奇点理论的通用形变及Σ¹-型稳定映射芽的分类,对方程组的所有稳定局部几何解进行分类.     

一类高阶拟线性椭圆型方程奇摄动边值问题

本文讨论了一类拟线性椭圆型方程奇摄动Dirichlet边值问题．在适当的条件下，利用不动点定理，研究了边值问题解的存在唯一性及其渐近性态．     

一类拟线性偏微分方程组的Laplace空间解的形式相似性

本作研究了一类拟线性偏微分方程组在不同的外边界条件(无穷大外边界，封闭外边界，定值外边界)和随机时间变化的内边界条件下的初值问题在Laplace空间的解的形式相似性，它能很好地帮助我们认识模型遵从的内在规律及设计相应的应用软件．     

拟线性滞后型微分方程正解的存在性

本文研究了下面这种拟线性滞后型微分方程(g(u′)′+a(t) f (ut) =0 ,　　 0 1 ,满足非线性边界条件 .并且通过应用锥不动定理与阿尔采拉 -阿斯卡里定理 ,证明了上述方程至少存在一个正解 .     

THE GENERALIZED RIEMANN-HILBERT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER QUASI-LINEAR ELLIPTIC EQUATIONS

In this paper, we consider the generalized Riemann-Hilberij problem for second order quasi-linear elliptic complex equation \[\begin{array}{l} \frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + {q_1}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial {z^2}}} + {q_2}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {q_3}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + {q_4}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma (z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}}),z \in G \end{array}\] satifying the boundary condition \[{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_1}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _1}(z),{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_2}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _2}(z),z \in \gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} (2)\] Many authors (see that papers 1, 4-6) have studied the Diriohlet problem and Riemann-Hilbert problem for linear elliptic complex equation. In our papers 2, 3 we also considered the generalized Riemann-Hilbert problem of the general second order linear elliptic complex equation. We obtained the existence theorem, the explicit form of generalized solution and the sufficient and necessary conditions for the solvability of the above mentioned boundary value problem. Based on these results and applying the property of the introduced integral operators and Schauder's fixed-point principle, it can be proved that the analogous deductions in 3 also hold for the generalized Riemann-Hilber problem (1), (2) of the quasi-linear complex equation, i, e., we have the following theorem: Theorem, If the coefficients of second order quasi-linear elliptic complex equation (1) satifies some conditions then i) When index \({n_1} \ge 0,{n_2} \ge 0\), the boundary value problem (1), (2) is always solvable and the solution depends on 2 \(2({n_1} + {n_2} + 1)\) arbitrary real constants. ii) When index \({n_1} \ge 0,{n_2} < 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} {\kern 1pt} {n_1} < 0,{n_2} \ge 0{\kern 1pt} )\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1),(2) consists of \( - 2{n_2} - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} ( - 2n, - 1)\) real equalities, if and only if the equalities are satisfied, the boundary value problem is solvable and the solution depends on \(2{n_1} + 1{\kern 1pt} {\kern 1pt} (2{n_2} + 1)\) arbitrary real constants. iii)When index \({n_1} < 0,{n_2} < 0\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1) , (2) consists of \( - 2({n_1} + {n_2} + 1)\) real equalities, if and only if the equalitieis are satisfied, the boundary-value problem is solvable. Finally, in the similar way, we may farther extend the result to the case of the nonlinear uniform elliptic complex equation.     

一类高阶椭圆型方程Dirichlet问题

本文讨论了一类奇摄动高阶椭圆型方程Dirichlet问题，利用伸长变量和变界层校正法，得到了问题解的形式渐近展开式．再用微分不等式理论，证明了解的一致有效性．     

NUMERICAL SOLUTIONS OF DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER

1Formulati0nofDiscontinuousBoundaryValueProblemsLetDbeanN 1-connectedb0undeddomaininthez=x iy-planeCwiththeboundaryFEC:(0相似文献   

BOUNDARY VALUE PROBLEMS FOR THIRD ORDER DIFFERENTIAL EQUATIONS

ＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ王金枝内蒙古大学，邮编：０１００２１ＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩ...     

一类二阶退化半线性椭圆型方程边值问题的适定性及解的正则性

本文考虑一类二阶退化半线性椭圆型方程边值问题.由椭圆正则化方法建立能量不等式,利用紧性推理,Banach—Saks定理,弱解与强解一致性,解常微分方程,椭圆型方程正则性定理,迭代方法.极值原理和Fredholm—Riesz-Schauder理论,可得相应线性问题适定性及解的高阶正则性;再由Moser引理和Banach不动点定理可得半线性问题解的存在性.这类问题与几何中无穷小等距形变刚性问题密切相关,其高阶正则性解的存在性对几何应用尤为重要.     

一类二阶退化半线性椭圆型方程边值问题的适定性及解的正则性

本文考虑一类二阶退化半线性椭圆型方程边值问题.由椭圆正则化方法建立能量不等式,利用紧性推理,Banach-Saks定理,弱解与强解一致性,解常微分方程,椭圆型方程正则性定理,迭代方法,极值原理和Fredholm-Riesz-Schauder理论,可得相应线性问题适定性及解的高阶正则性;再由Moser引理和Banach不动点定理可得半线性问题解的存在性.这类问题与几何中无穷小等距形变刚性问题密切相关,其高阶正则性解的存在性对几何应用尤为重要.     

一类具有非局部边界条件的半线性椭圆型方程奇摄动问题

本文研究了一类具有非局部边界条件的奇摄动半线性椭圆型方程边值问题。在适当的条件下，利用比较定理讨论了问题解的渐近性态。     

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, we study the multiplicity results of positive solutions for a class of quasi-linear elliptic equations involving critical Sobolev exponent. With the help of Nehari manifold and a mini-max principle, we prove that problem admits at least two or three positive solutions under different conditions.     

一阶带参数的时滞微分方程的边值问题

本文利用上下解和单调迭代法，讨论了带参数的一阶时滞微分方程的边值问题，获得了这类问题极值解的存在性定理．