无穷维Hamilton 算子广义本征函数系的完备性及其在弹性力学中的应用

本文利用无穷维Hamilton 算子的结构特性, 得到由算子的基本本征函数和若当型本征函数构成的广义本征函数系在Cauchy 主值意义下完备的充分必要条件. 进而将结果应用于弹性力学中的板弯曲问题. 相应结论为Hamilton 体系下的分离变量法(弹性力学求解新体系) 提供了理论保证.     

Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

对角无穷维Hamilton算子点谱关于实轴的对称性

在不同条件下得到对角无穷维Hamilton算子点谱的两个组成部分σp（A）与σp（-A^＊）关于实轴对称的充分必要条件.以此为基础,完全刻画了对角无穷维Hamilton算子点谱关于实轴的对称性.     

一类无穷维Hamilton算子本征函数系的完备性

研究了Sturm-Liouvile偏微分方程导出的无穷维Hamilton算子的本征值问题.证明了导出的无穷维Hamilton算子族本征函数系的完备性,为对此类方程应用基于Hamilton体系的分离变量法提供了理论基础.最后举例说明了结果的有效性.     

一类无穷维Hamilton算子的半群生成定理

研究了无穷维H am ilton算子生成C0半群的问题,得到了类无穷维H am ilton算子生成C0半群的一个充分条件.把结果应用在一类双曲型混合问题生成的无穷维H am ilton算子上,证明此类算子生成C0半群,并利用H ille-Y osida定理进一步说明了结果的正确性和有效性.另外,还给出了波动方程相应的无穷维H am ilton算子所生成的C0半群的具体表达式.     

斜对角无穷维Hamilton算子的点谱和特征函数系辛结构的非退化性

本文研究斜对角无穷维Hamilton算子$H=\begin{pmatrix}0&B\\C&0\end{pmatrix}$的点谱和特征函数系辛结构的非退化性, 给出斜对角无穷维Hamilton算子$H$的特征函数系具有非退化辛结构的充分必要条件. 基于此, 进一步刻画了斜对角无穷维Hamilton算子$H$的点谱分别包含于实轴、虚轴以及其它区域的充分必要条件. 最后, 以板弯曲问题和弦振动问题中导出的斜对角无穷维Hamilton算子为例, 验证了所得结论的正确性.     

On the impulsive boundary value problems for nonlinear Hamiltonian systems

In this work, we deal with two‐point boundary value problems for nonlinear impulsive Hamiltonian systems with sub‐linear or linear growth. A theorem based on the Schauder fixed point theorem is established, which gives a result that yields existence of solutions without implications that solutions must be unique. An upper bound for the solution is also established. Examples are given to illustrate the main result. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

弹性理论中上三角无穷维Hamilton算子根向量组的完备性

考虑弹性力学中一类上三角无穷维 Hamilton 算子．首先,给出此类Hamilton算子特征值的几何重数和代数指标,进而得到代数重数．其次,根据Hamilton算子特征值的代数重数确定其特征(根)向量组完备的形式,得到此类Hamilton算子特征(根)向量组的完备性是由内部算子特征向量组决定．最后,将所得结果应用到弹性力学问题中．     

Symmetry of the point spectrum of infinite dimensional hamiltonian operators and its applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪σp1(-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.     

Asymptotics of eigenvalues of the Dirichlet boundary value problem for the Lame operator in a three-dimensional domain with a small cavity

A boundary value problem for the Lame operator in a bounded three-dimensional domain with a small cavity is studied. The domain is filled with an elastic homogeneous isotropic medium that is clamped at the boundary, which corresponds to the Dirichlet boundary condition. The leading term of an asymptotic expansion for the eigenvalue is constructed in the case of the Dirichlet limit problem. The asymptotic expansion is constructed in powers of a small parameter ? that is the diameter of the cavity.     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

The RH boundary value problem of the k-monogenic functions

In this paper we study the Riemann and Hilbert problems of k-monogenic functions. By using Euler operator, we transform the boundary value problem of k-monogenic functions into the boundary value problems of monogenic functions. Then by the Almansi-type theorem of k-monogenic functions, we get the solutions of these problems.     

Asymptotic behavior of eigenvalues of the Laplace operator in infinite cylinders perturbed by transverse extensions

We present sufficient conditions for the existence of an eigenvalue of the Laplace operator with zero Dirichlet conditions in a weakly perturbed infinite cylinder in the case of localized perturbations which are extensions along the transverse coordinates with coefficients depending on the longitudinal coordinate. If such an eigenvalue exists, then, for this eigenvalue, we obtain an asymptotic formula with respect to a small parameter characterizing the values of extensions.     

Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator

In this paper, we study the existence of positive solutions for the nonlinear four-point singular boundary value problem for higher-order with p-Laplacian operator. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem with p-Laplacian operator are obtained.     

Nontrivial Solutions of Superquadratic Hamiltonian systems with Lagrangian Boundary Conditions and the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z（t） = J△↓H（t, z（t）） with Lagrangian boundary conditions, where ^H（t,z）=1/2（^B（t）z, z） ＋ ^H（t, z）,^B（t） is a semipositive symmetric continuous matrix and ^H（t, z） = satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u² and the derivatives of u.     

The Fredholm index of a second order elliptic system on an infinite cylinder

This paper investigates the Fredholmness of a second order elliptic system on an infinite cylinder in various weighted or non-weighted Sobolev spaces, assuming homogeneous Dirichlet boundary conditions and suitable asymptotic behavior of the coefficients. It is shown that the index coincides with the spectral flow of an associated parametrized family of elliptic systems on the cross-section of the cylinder. Dependence or independence of the index upon the functional setting is discussed along with their impact on spectral invariance and elliptic regularity.