一类非线性抛物型方程组解的整体存在及爆破

研究了一类带有非线性边界条件的非线性抛物型方程组解的整体存在及解在有限时刻爆破问题.通过构造方程组的上、下解.得到了解整体存在及解在有限时刻爆破的充分条件.对指数型反应项和边界流采用了常微分方程方法构造其上下解,而其它例如第一特征值等方法运用于该方程就比较困难.     

带跳跃非线性边界的p-Laplacian问题(英文)

利用Fadell与Rabinowitz提出的同调指标理论,研究了一类带不定权非线性边界的p-Laplacian特征值问题;更进一步,利用环绕定理,证明了一类带跳跃非线性边界的p-Laplacian问题非平凡解的存在性.     

谱变形的非线性双曲型方程

文[1]研究了一个非线性双曲型方程 u_(xt) [(uu_xu_t)/(1-u~2)]-u(1-u~2)=0.方程(*)可以用来描述沿类脂膜扩张波的传播,它所对应的特征值问题的谱是不变的,随着科技的发展,非均匀介质中的传播问题日益受到重视。类问题相应的特征值问题是非保谱的,因此,在[2]中研究了谱变形的特征值问题和发展方程.这本文研究谱变形的非线性双曲型方程     

一个Bargmann系统与TD族解的对合表示

§1.引言 孤子方程的Lax系统的非线性化方法已成功的用于寻找新的有限维完全可积的系统。本文基于以前的讨论研究与TD特征值问题相联系的非线性保谱演化方程族(TD族)。 TD特征值问题可写为下面等价的形式:     

Heisenberg群上四阶非线性方程特征值问题的正解

本文通过Leray-Schauder度,给出Heisenberg群上四阶非线性次椭圆方程特征值问题的正解的一些存在性结果.     

R2中含临界位势的非线性椭圆问题

考虑R²中的含临界位势的非线性椭圆方程齐次Dirichlet问题. 通过建立一常数为最佳的含权不等式, 确定了临界位势, 并讨论了含临界位势的Laplace方程特征值问题. 通过建立含一个奇点的解的Pohozaev型恒等式并结合Cauchy-Kovalevskaya定理得到了含临界位势非线性椭圆型方程有奇点的解的不存在性结果. 此外, 利用山路定理和特征值的性质得到了这一问题多重解的存在性的一系列结果.     

具混合边界的双曲型微分方程非平凡解的存在性[英文]

本文将具混合边界的一类双曲型微分方程分解为两个线性算子和三个非线性算子.证明了这些算子具有单调性质,由此得到一类算子方程存在解的结论,进而证明具混合边界的双曲型非线性微分方程存在唯一非退化解的结论.此文是对含有p-Laplacian算子的非线性椭圆和非线性抛物方程相关研究工作的推广,并采用了一些新的证明技巧.     

高精度特征解及其外推求解位势方程

根据位势理论,基本边界特征值问题可转化为具有对数奇性的边界积分方程.利用机械求积方法求解特征值和特征向量,以及利用这些特征解求解Laplace方程.特征解和Laplace方程的解具有高精度和低的计算复杂度.利用Anselone聚紧和渐近紧理论,证明了方法的收敛性和稳定性.此外,还给出了误差的奇数阶渐近展开.利用h3-Richardson外推,不仅误差近似的精度阶大为提高,而且,得到的后验误差估计可以构造自适应算法.具体的数值例子说明了算法的有效性.     

三族保谱方程的换位表示

基于曹的框架,本文提供三族新的保谱方程的换位表示;并且阐明与同一个特征值问题可联系着两族不同的非线性保谱演化方程.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

R2 中带不定权与临界位势的非线性椭圆问题

该文讨论了R²中一类带不定权且含临界位势的二阶椭圆方程的特征值问题,然后,借此特征值问题的第一特征值性质,利用临界点理论及Trudinger-Moser不等式,证明了R² 中一类带不定权且含临界位势的二阶非线性椭圆方程非平凡解的存在性.     

EIGENVXLUE PROBLEMS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH INDEFINITE WEIGHT

1Introductionit'ic'?lll,ly[;IJcoTlsi(lerodl,tl(3big(~nvalll~problemfortifieolliljticeqllat,iollwithinderillit(3w'3iglltfllflcl,iononRe'an(l,triIJarticlllar,showedillal,positivepllillcipaleigenvalllesdonotexistifn=1or2and.l,,.g(x)dx>0.Oneresultof[1]laterpointedo[ltthefttilepl.obleTnilasa,[)riTI(:il)aleigcrlvallloirn=2andg(x)sansfriessomesuitabl(3col,l(litiolls.W(3willsl,lldyiii1,ills[)al)(3rtheProblemwflcnn=l,i.e.,l,ha-tofhIleaf,sc'toll(]or'1()ror(lirlary(hfi'(?l.()[ltialeq[lationalldgiveex…     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

Periodic-parabolic eigenvalue problems with indefinite weight functions

We improve a result on the existence and uniqueness of a positive principal eigenvalue of a periodic parabolic equation with respect to an indefinite weight function due to Beltramo and Hess. We remove the regularity conditions on the domain and weaken considerably the regularity assumptions on the weight and the coefficients of the parabolic operator. Further we give a perturbation theorem for the principal eigenvalue which allows to perturb the domain, the coefficients of the parabolic operator and the weight simultaneously.     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e. those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations obtained.     

Heisenberg群上带不确定权的p-次Laplacian特征值问题(英文)

本文讨论Heisenberg群上带不确定权的p-次Laplacian特征值问题解的存在性,并且证明相应的非负特征函数的第一特征值是单重的,孤立的和唯一的.     

A Hardy Type Inequality and Inde Nite Eigenvalue Problems on the Homogeneous Group

In this paper we present a Hardy type inequality and a Picone type identity for the real sub-Laplacian on the homogeneous group. The existence of the indefinite eigenvalue problem and the simplicity of the principal eigenvalue are proved.     

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi-level incomplete LDL^T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi-level preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)