带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

In the present paper, we consider a class of inverse spectral problem of fourth‐order boundary value problems. Under the so‐called “Atkinson type” conditions, the problem has finite spectrum and corresponding matrix representations. By using the method of inverse matrix eigenvalue problems of two‐banded matrix, the leading coefficient and potential functions are reconstructed from the given three sets of interlacing real numbers satisfying certain conditions.     

Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

We study matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions. Meanwhile, given any matrix eigenvalue problem with coupled eigenparameter‐dependent boundary conditions and transmission conditions, we construct a class of Sturm‐Liouville problems with given boundary conditions and transmission conditions such that they have the same eigenvalues.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Principal Eigenvalues for Generalised Indefinite Robin Problems

We consider the principal eigenvalue of generalised Robin boundary value problems on non-smooth domains, where the zero order coefficient of the boundary operator is negative or changes sign. We provide conditions so that the related eigenvalue problem has a principal eigenvalue. We work with the framework involving measure data on the boundary due to Arendt and Warma (Potential Anal 19:341–363, 2003). Examples of simple domains with cusps are used to illustrate all possible phenomena.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Existence of an extremal ground state energy of a nanostructured quantum dot

This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.     

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.     

Equivalence of Fourth Order Boundary Value Problems and Matrix Eigenvalue Problems

We show that a class of regular self-adjoint fourth order boundary value problems (BVPs) is equivalent to a certain class of matrix problems. Conversely, for any given matrix problem in this class, there exist fourth order self-adjoint BVPs which are equivalent to the given matrix problem. Equivalent here means that they have exactly the same spectrum.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.     

Analytic equivalence of the boundary eigenvalue operator function and its characteristic matrix function

The boundary eigenvalue operator function associated to an analytic family of boundary value problems is shown to be analytically equivalent to a simple extension of its characteristic matrix function. Explicit formulas for the operator functions that establish the equivalence are given.     

带有p-Laplace算子的分数阶边值问题的特征区间

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

Das Differenzenverfahren für singuläre Rand-Eigenwertaufgaben gewöhnlicher Differentialgleichungen

Summary The boundary value problem for a class of singular second order differential operators is defined. Using the standard three point discretisation for the differential equation and taking care of the limits involved in the boundary conditions in a natural way, finite difference approximations to the boundary value problems are defined and their convergence properties are investigated. The rate of convergence is given in terms of the data. It turns out that for problems of the first kind extrapolation is possible up to an arbitrary order after a suitable change of the independent variable, whereas for problems of the second kind neither theoretical nor numerical results indicate the possibility of extrapolation. Corresponding results hold for the eigenvalue problems. Some numerical examples show that the convergence rates given in the paper are best possible and demonstrate the effect of extrapolation.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

实轴上的双解析函数Riemann边值逆问题

提出了一类实轴上的双解析函数Riemann边值逆问题.先消去参变未知函数,再采用易于推广的矩阵形式记法,可把问题转化为两个实轴上的解析函数Riemann边值问题.利用经典的Riemann边值问题理论,讨论了该问题正则型情况的解法,得到了它的可解性定理.     

The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems

For domains with concave corners, the solutions to elliptic boundary values have the typical r^α‐singularity. The so‐called singularity exponents α are the eigenvalues of an eigenvalue problem which is associated with the given boundary value problem. This paper is aimed at deriving the mentioned eigenvalue problems for two examples, the Laplace equation and the linear elasticity problem. We will show interesting properties of these eigenvalue problems. For the linear elasticity problem, we explain in addition why the classical symmetry and positivity assumptions of the material tensor have to be used with care. Copyright © 2006 John Wiley & Sons, Ltd.     

Root vectors for geometrically simple two-parameter eigenvalues

A class of two-parameter eigenvalue problems involving generally nonselfadjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.

     

Regular Sturm - Liouville Problems Whose Coefficients Depend Rationally on the Eigenvalue Parameter

In this note we consider regular Sturm-Liouville equations with a floating singularity of a special type: the coefficient of the second order derivative contains the eigenvalue parameter. We determine the form of the boundary conditions which make the problem selfadjoint after linearizing. In general the boundary conditions for the linearized system give rise to boundary conditions which involve the eigenvalue parameter in the original, non-linearized, problem. The boundary conditions give rise to a 2 × 2 matrix function, the so-called Titchmarsh-Weyl coefficient. The characteristic properties of this function are studied. The formal aspects of the theory of this class of equations turn out to be quite parallel to those for the usual situation when there is no floating singularity.