Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.     

Jacobi Polynomials,Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this paper, we give some polynomial approximation results in a class of weighted Sobolev spaces, which are related to the Jacobi operator. We further give some embeddings of those weighted Sobolev spaces into usual ones and into spaces of continuous functions, in order to use the above approximation results in the p‐version (or the spectral method) of some finite or boundary element methods. Finally, two typical examples of the polynomial approximation of some singularities of boundary value problems in polygonal or polyhedral domains are presented.     

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Morrey Regularity of Solutions to Degenerate Elliptic Equations in Rn

In this paper, we study the Morrey regularity of solutions to the de- generate elliptic equation -(a_{ij}u_{xi})_{xj} = -(f_j)_{xj} in R^n. For this purpose, we introduce four weighted Morrey spaces in R^n.     

Banach空间中二阶微分方程的周期边值问题

本文在Banach空间中研究了二阶非线性微分方程的周期边值问题:-u″=f(t,u),u(0)= u(2π),u′(0)=u′,(2π)在上下解反向给定时,利用半序理论和新的比较原理,证明了该周期边值问题最小解和最大解的存在性,解的唯—性,并给出了唯一解的近似迭代序列的误差估计式．     

抽象经济均衡问题解的存在性及其算法

本文首先研究一类新的向量均衡问题,利用截口定理与KKM定理两种不同的工具证明此类均衡问题解的存在性,接着,把这类向量均衡问题推广到更为一般的情形,随后讨论了具有上下界的均衡问题,它是由Isac,Sehgal和Singh于1999年提出的一个公开问题,本文在一定条件下获得了一个新的解的存在性定理,并构造了一个迭代算法,讨论了算法的收敛性。     

Singular problems motivated from classical upper and lower solutions

General existence criteria are presented for nonlinear singular boundary value problems. Our nonlinearity may be singular in both the dependent and independent variable. This revised version was published online in June 2006 with corrections to the Cover Date.