半线性多调和方程解的分支

该文主要讨论一类多调和方程,通过研究此方程对应的线性化问题本征谱,得到了方程的全局分支结果.所讨论的方法主要依赖于P.M.Fitzpatrick,J.Pejsachowicz和P.J.Rabier C~2 Fredholm算子的度理论思想.     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

Translatable radii of an operator in the direction of another operator II

One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.     

A Quadratic Eigenproblem in the Analysis of a Time Delay System

In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problem where φ ₁, …, φ _{m –1} ∈ ℝ are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation, the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift-invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation. The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem

Summary. In this paper, we study the inverse eigenvalue problem of a specially structured Jacobi matrix, which arises from the discretization of the differential equation governing the axial of a rod with varying cross section (Ram and Elhay 1998 Commum. Numer. Methods Engng. 14 597-608). We give a sufficient and some necessary conditions for such inverse eigenvalue problem to have solutions. Based on these results, a simple method for the reconstruction of a Jacobi matrix from eigenvalues is developed. Numerical examples are given to demonstrate our results.Research supported in part by National Natural Science Foundation of ChinaResearch supported in part by RGC Grant Nos. 7130/02P and 7046/03P, and HKU CRCG Grant Nos 10203501, and 10204437     

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind. Postdoctoral researcher FWO-Flanders.     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

A correspondence between the AKNS hierarchy and the JM hierarchy

Reference [1] presented a gauge transformation between thex parts of the AKNS eigenvalue problem and those of the JM (Jaulent-Miodek) eigenvalue problem. In this paper we discuss the correspondence between thet parts of the AKNS eigenvalue problem and thet parts of the JM eigenvalue prohlem under the gauge transformation, and give a correspondence between the AKNS hierarchy and the JM hierarchy and also three types of Darboux transformation for the JM hierarchy.Project supported by the Science Fund of the Ministry of Education.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

Solutions and multiple solutions for problems with the p-Laplacian

In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ₂ = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory.     

New inequalities for the minimum eigenvalue of M-matrices

Some new inequalities for the minimum eigenvalue of M-matrices are established. These inequalities improve the results in [G. Tian and T. Huang, Inequalities for the minimum eigenvalue of M-matrices, Electr. J. Linear Algebra 20 (2010), pp. 291–302].     

A Perturbation Result for a Double Eigenvalue Hemivariational Inequality with Constraints and Applications

In this paper we prove a perturbation result for a new type of eigenvalue problem introduced by D. Motreanu and P.D. Panagiotopoulos (1998). The perturbation is made in the nonsmooth and nonconvex term of a double eigenvalue problem on a spherlike type manifold considered in Multiple solutions for a double eigenvalue hemivariational inequality on a spherelike type manifold (to appear in Nonlinear Analysis). For our aim we use some techniques related to the Lusternik-Schnirelman theory (including Krasnoselski's genus) and results proved by J.N. Corvellec et al. (1993), M. Degiovanni and S. Lancelotti (1995), and V.D. Rdulescu and P.D. Panagiotopoulos (1998). We apply these results in the study of some problems arising in Nonsmooth Mechanics.     

Solutions and multiple solutions for problems with the <Emphasis Type="Italic">p</Emphasis>-Laplacian

In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ₂ = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory. Second author is Corresponding author.     

On the existence of transmission eigenvalues in an inhomogeneous medium

We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783–753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough.     

A parallel eigenvalue problem solver for computational nanoelectronics

A new parallel eigenvalue solver for finding the interior eigenvalues of a standard Hermitian eigenvalue problem arising in atomistic simulations in nanoelectronics is presented. It is based on the Tracemin algorithm which finds the p smallest eigenpairs of a generalized Hermitian eigenvalue problem. The original problem is modified using spectrum folding or a quadratic mapping so that the interior eigenvalues are mapped onto the smallest or the largest, respectively. In the latter case the solution of systems in every iteration of Tracemin is avoided and Chebyshev polynomials are used to speedup convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On the higher eigenvalues for the \infty-eigenvalue problem

We study the higher eigenvalues and eigenfunctions for the so-called -eigenvalue problem. The problem arises as an asymptotic limit of the nonlinear eigenvalue problems for the p-Laplace operators and is very closely related to the geometry of the underlying domain. We are able to prove several properties that are known in the linear case p = 2 of the Laplacian, but are unknown for other values of p. In particular, we establish the validity of the Payne-Pólya-Weinberger conjecture regarding the ratio of the first two eigenvalues and the Payne nodal conjecture, which deals with the zero set of a second eigenfunction. The limit problem also exhibits phenomena that are not encountered for any .Received: 4 January 2004, Accepted: 9 June 2004, Published online: 3 September 2004Mathematics Subject Classification (2000): 35P30, 35J70, 49R50     

Sturm–Liouville properties for Atkinson's semi‐definite p‐Laplacian eigenvalue problems

We define Atkinson's semi‐definite p‐Laplacian eigenvalue problems, which include the regular p‐Laplacian eigenvalue problems with L ¹ coefficient functions. Then we show that the Sturm oscillation theorem also holds for this eigenvalue problem.     

Faber-Krahn inequality for robin problems involving <Emphasis Type="Italic">p</Emphasis>-Laplacian

The eigenvalue problem for the p-Laplace operator with Robin boundary conditions is considered in this paper.A Faber-Krahn type inequality is proved.More precisely,it is shown that amongst all the domains of fixed volume,the ball has the smallest first eigenvalue.