一类高阶超双曲型方程的中量定理及其逆定理

Asgeirsson中量定理表明超双曲型方程的Cauchy问题一般是不适定的,对Asgeirsson中量定理的推广就有重要意义。目前关于高阶方程解的中量只有初步探讨,尚未得到具体结果,本文直接利用Asgeirsson中量定理结果和积分、微分的性质与关系,得到了高阶方程解的中量满足广义双轴对称位势方程,同时还证明了其逆定理。利用关于广义双轴对称位势方程正则解的表达式及雅可比多项式的特殊性质,得到了高阶方程解的中量公式,从而使得关于解的拓展性和适定性的讨论将有可能。     

超双曲型方程定性研究的一些问题

本文是超双曲型方程定性研究的一个综合报导，它包括L.Asgeirsson中量定理的一个新证法；中量定理的推广；广义势解的非解析性；解的拓展性，境界值问题；基本解问题。     

Huyghens原理推广到超双曲型方程

本文研究超双曲型方程首先建立了广义Asgeirsson中量定理,作为其应用,我们指出在一类特征流型上既推广了著名的Asgeirsson公式,也包含了Hans Lewy的结果接着Huyghens原理被推广到一类超双曲型方程。     

Huyghens原理推广到超双曲型方程

本文研究超双曲型方程首先建立了广义 Asgeirsson 中量定理,作为其应用,我们指出在一类特征流型上既推广了著名的 Asgeirsson 公式,也包含了 Hans Lewy 的结果接着 Huyghens 原理被推广到一类超双曲型方程.     

一类高阶方程的中量定理及其逆定理

本文对于高阶方程(?)在文［2］中量定义的基础上,利用Asgeirsson中量定理,得到了中量所满足的方程,即M(r,s)+M(s,r)满足E-P-D方程,同时也证明了其逆定理也成立.     

超双曲型方程定性研究的一些问题

本文是超双曲型方程定性研究的一个综合报导,它包括 L.Asgeirsson 的中量定理的一个新证法;中量定理的推广;广义势解的非解析性;解的拓展性,境界值问题;基本解问题。     

多偏差变元中立型Rayleigh方程周期解问题

作者研究了一类多偏差变元中立型Rayleigh方程周期解问题,利用Mawhin重合度拓展定理得到了周期解存在性的新结果.     

关于一类三阶混合型方程和变系数双曲型方程的边值问题

本文先扼要介绍一类三阶双曲抛物混合型偏微分方程的边值问题，然后讨论了与之有关的一类变系数三阶双曲型方程以特征线为支柱的边值问题，得到了有唯一解时此系数应满足的充分必要条件。     

关于—类三阶混合型方程和变系数双曲型方程的边值问题

本文先扼要介绍一类三阶双曲抛物混合型偏微分方程的边值问题，然后讨论了与之有关的一类变系数三阶双曲型方程以特征线为支柱的边值问题，得到了有唯一解时此变系数应满足的充分必要条件。     

一类非线性耦合双曲型方程组的研究

研究了一类非线性耦合双曲型方程组初边值问题,讨论了初边值条件以及非线性项中指数满足的条件,利用Galerkin方法和紧致性的结果,以及椭圆型方程解的正则性定理,经过严格的推导证明了该问题解的存在唯一性的若干结果.     

Unique continuation for solutions to pde's; between hörmander's theorem and holmgren' theorem

We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hörmander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hörmander's Theorem (which applies to problems with C¹ coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hörmander, namely that for the wave equation with C¹ but time independent coefficients one has unique continutaion across any noncharacteristic surface.     

关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.     

微分中值定理的另类证明与推广

通常教科书中,微分中值定理的证明建立在罗尔(Rolle)定理之上.本文以实数连续性中的重要定理———区间套定理为依据,给出了拉格朗日微分中值定理的另类证明.此外,还给出了中值定理的若干推广形式.     

On a Kipriyanov problem for a singular ultrahyperbolic equation

We obtain a differential equation for the spherical means generated by a multidimensional generalized shift of an arbitrary smooth “even” function. We study the Asgeirsson property of solutions of a singular ultrahyperbolic equation that includes singular differential operators Δ _B acting in Euclidean spaces, in general, of distinct dimensions. We represent the structure of a “radial” solution of the considered equation. A theorem similar to the Asgeirsson inverse theorem is proved.     

一类具多偏差变元的p-Laplacian方程周期解的存在性

研究了一类具多偏差变元的p-Laplacian方程,为了得到该方程周期解的先验估计,运用分析技巧建立了一些新的不等式,进而利用Mawhin连续定理得到了这类方程周期解的存在性,得到了一些新的结果,推广和改进了已有结果.     

Continuation theorem of fractional order evolutionary integral equations

The fractional order evolutionary integral equations have been considered by the first author in [6], the existence, uniqueness and some other properties of the solution have been proved. Here we study the continuation of the solution and its fractional order derivative. Also we study the generality of this problem and prove that the fractional order diffusion problem, the fractional order wave problem and the initial value problem of the equation of evolution are special cases of it. The abstract diffusion-wave problem will be given also as an application.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

A full multigrid method for semilinear elliptic equation

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.     

All General Solutions of Post Equations

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known （Theorem 6）. As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations （Theorem 9）.