The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold

Let M be an n-dimensional noncompact complete Riemannian manifold, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(M). First, we give a criterion of non-existence of eigenvalue by the heat kernel. Applying the criterion yields that the Laplacian on noncompact constant curvature space form has no eigenvalue. Then, we give a geometric condition of M under which the Laplacian of M has eigenvalues. It implies that changing the metric on a compact domain of constant negative curvature space form may yield eigenvalues.     

一个椭圆特征问题解对参数的连续性

用变分方法证明了一个限制在球面上的椭圆特征问题解对参数（球面半径）的连续性，从而得到相应的不带限制的椭圆特征问题的解分枝。     

Heisenberg群上Folland-Stein算子的Dirichlet特征值问题

该文研究Heisenberg群上Folland－Stein算子■（α为复数）的特征值问题，证明了当|Reα|＜n时，■具离散分布的特征值．当α∈（-n,n）时，特征值均为正的．然后给出了相邻特征值之差的估计．     

Weighted Composition Operators on Weighted Dirichlet Spaces

We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Carleson measure.     

Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A_∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ⊂ Rⁿ⁺¹ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in L^p(∂Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which L^p solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A_∞ property, and hence of solvability of the L^p Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.     

带权流形上的纽曼特征值估计

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.     

Weighted composition operators between Dirichlet spaces

In this article, we study the boundedness of weighted composition operators between different vector-valued Dirichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued Dirichlet spaces are different counterparts of the classical scalar-valued Dirichlet space and characterize the boundedness of multiplication operators between these different spaces.     

连续时间LQ控制主要本征对的算法

本文首先提出了离散时间LQ控制的本征值方程当△t→0时怎样退化成为连续时间LQ控制的本征值方程.在建立了分离出的n阶连续时间的本征值方程,并保证了其本征值必定都在左半平面后,本文提出计算其最靠近于虚轴的若干个本征对,可以通过A_e=eA的矩阵变换.A_e的本征值全在单位圆之内.本征向量不变,至于本征值则只要做一次对数运算就可以求得原阵的本征值.A_e阵的最接近于单位圆的若干个本征对的算法,可以通过共轭子空间迭代解解决之.     

Weighted norm inequalities for the Dirichlet transform

The Dirichlet transform is defined for suitable functions by

     

Interpolating Sequences in Harmonically Weighted Dirichlet Spaces

In this article we show that interpolating sequences on certain harmonically weighted Dirichlet spaces can be characterized in terms of a separation condition and a Carleson-measure condition. This is the first example of a space with Nevanlinna–Pick kernel with non-radially symmetric weights in which this characterization remains true.     

On 2n—th—Order Eigenvalue Problem

In [1], Lu discussed the existence of positive solution for the BVPwhere f: [0, 1] × [0, ∞) → R is continuous and satisfies the following conditions(H1) There exists an M > 0 such that(H2) limx→∞ f(t, x)/x =∞uniformly for t [α,β] (0, 1) for some α < β and λ > 0 is a parameter.In this paper, the result of [1] is generalized to the following 2n-th-order BVPThe result of this paper isTheorem 1 Assume (H1) and (H2) hold. Then the BVP (3)-(4) has a positive solutionifλ > 0 is sm…     

On a Fourth-order Eigenvalue Problem

WeconsidertheexistenceofpositivesolutionsfortheequationwithoneofthefollowingsetsofboundaryvalueconditionsBVP(1)-(2)describesthedeformationsofallelasticbeambothofwhoseendssimplysupported,andBVP(1)-(3)describesthedeformationsofanelasticbeamwithoneendsimplysupportedandtheotherendclampedbyslidingclamps[1].SeveralresultsontheexistenceofsolutionsfornonlinearbeamequationshavebeenestablishedbyseveralauthorsbyusingtheLeray-SchauderContinuationTheorem,seetilereferencesof[1,2].Recently,MaRuyunandWan…     

Volterra Type Operators on Weighted Dirichlet*

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Pel′aez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D^p_α to D^q_β (?1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.     

树上生灭过程的Dirichlet特征值估计

本文给出树上两类非常返的生灭过程的第一Dirichlet特征值的变分公式．一类是配称测度有限时,给出以根为吸收点的Dirichlet特征值的变分公式;另一类是配称测度无限时,给出树上生灭过程的Dirichlet主特征值的变分公式．     

一类广义特征值反问题

本文提出了一个实对称带状矩阵的广义特征值反问题,并且证明了对于Jacobi矩阵和一般对称矩阵,问题的存在性.     

Maximum Eigenvalue Problem for Escherization

This paper proposes a new and efficient method for “Escherization”, that is, for generating a tile which is close to a given shape and whose copies cover the plane without gaps or overlaps except at their boundaries. In this method, the Escherization problem is reduced to a maximum eigenvalue problem, which can be solved easily, while the existing method requires time consuming heuristic search. Furthermore, we show that the optimal shape corresponds to the orthogonal projection of the vector representing the given shape to the “space of tilable shapes”.     

Holder Continuity of the Dirichlet Solution for a General Domain

Let D be a bounded domain in Rⁿ. For a function f on the boundaryD, the Dirichlet solution of f over D is denoted by H_Df, providedthat such a solution exists. Conditions on D for H_D to transforma Hölder continuous function on D to a Hölder continuousfunction on D with the same Hölder exponent are studied.In particular, it is demonstrated here that there is no boundeddomain that preserves the Hölder continuity with exponent1. It is also also proved that a bounded regular domain D preservesthe Hölder continuity with some exponent , 0<<1,if and only if D satisfies the capacity density condition, whichis equivalent to the uniform perfectness of D if n = 2. 2000Mathematics Subject Classification 31A05, 31A20, 31B05, 31B25.