Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.     

Symmetric Stable Processes on Unbounded Domains

Let be a symmetric -stable process in , , . We give necessary and sufficient condition under which the expectation of a very general function of the exit time from horns is finite. These domains include the symmetric domains given by increasing functions studied earlier by various authors. Our methods differ from those in earlier papers in that we obtain our results from estimates on the transition densities instead of harmonic measure. Some of this estimates are of independent interest.Supported in part by NSF grant #9700585-DMS and RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP.     

有界凸平衡域上的双全纯凸映照的判别准则

本文讨论Cn中有界强凸平衡域和凸平衡域上局部双全纯映照成为双全纯凸映照的充要条件,从而得到Reinhardt域Dp= 上双全纯凸映照的充要条件,其中Pj≥2(j=1,2,…,n).     

Large Deviations for Additive Functionals of Symmetric Stable Processes

We establish the large deviation principle for additive functionals of symmetric α-stable processes employing the Gärtner-Ellis theorem.     

两平面凸域的对称混合等周不等式

设K_k(k=i,j)为欧氏平面R~2中面积为A_k,周长为P_k的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit)为σ(K_i,K_j)=P_i~2P_j~2-16π~2A_iA_j.根据周家足,任德麟(2010)和Zhou,Yue(2009)中的思想,用积分几何方法,得到了两平面凸域的Bonnesen型对称混合不等式及对称混合等周不等式,给出了两域的对称混合等周亏格的一个上界估计.还得到了两平面凸域的离散Bonnesen型对称混合不等式及两凸域的对称混合等周亏格的一个上界估计,并应用这些对称混合(等周)不等式估计第二类完全椭圆积分.     

基于谱隙的依全变差指数式收敛速度估计

利用与谱隙作比较的手法，本文研究可逆马氏过程依全变差的指数式收敛速度．我们证明在相当一般的情况下，只要对初分布略加限制，则此速度以谱隙为下界．进而，对于紧空间情形或生灭过程或半直线上的扩散，我们证得此速度等于谱隙．     

Symmetric stable processes in parabola-shaped regions

We identify the critical exponent of integrability of the first exit time of the rotation invariant stable Lévy process from a parabola-shaped region.

     

Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

Based on the generalized Laguerre and Hermite functions, we construct two types of Birkhoff-type interpolation basis functions. The explicit expressions are derived, and fast and stable algorithms are provided for computing these basis functions. As applications, some well-conditioned collocation methods are proposed for solving various second-order differential equations in unbounded domains. Numerical experiments illustrate that our collocation methods are more efficient than the standard Laguerre/Hermite collocation approaches.     

有界凸圆型域上的准凸映照

本文研究星形映照的一类子族——有界凸圆型城上的准凸映照,并讨论了它与星形映照及凸映照的关系,建立其与凸映照同形的增长定理和掩盖定理．     

紧流形上的Schrodinger算子的谱间隙估计

M是一个n维紧黎曼流形,具有严格凸边界,且Ricci曲率不小于(n-1)K(其中K≥0为某个常数).假定Schrodinger算子的Dirichlet (或Robin)特征值问题的第一特征函数f1在M上是对数凹的,该文得到了此类Schrodinger算子的前两个Dirichlet(或Robin)特征值之差的下界估计,这推广了最近Andrews等人在R^n中有界凸区域上关于Laplace算子的一个相应结果[4].     

凸区域上椭圆方程弱解的边界唯一延拓性和B_p权特性

本文研究非光滑凸区域上的散度型二阶椭圆方程 ｉ（ａｉｊ（Ｘ）　ｊｕ（Ｘ））＝０的非零弱解的近边无穷次消失性和双倍性质，刻划弱解梯度在区域边界上的Ｂｐ权特性，建立弱解和弱解的梯度在凸区域边界的任意开子集上不可能同时消失的边界唯一延拓性定理．     

On a Class of Generalized Curve Flows for Planar Convex Curves

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., limt→T A(t) = 0, or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.     

Spectral gap for the Kac model with hard sphere collisions

We prove the analog of the Kac conjecture for hard sphere collisions, giving a computable, close estimate on the spectral gap that is independent of the number of particles. Previous work has focused on the case in which the collision rates are independent of the particle velocities, the case of so-called Maxwellian molecules. The new methods introduced here allow us to deal with collision rates that are not bounded from below. We also obtain information on the structure of the gap eigenfunction.     

Some Problems in Operator Theory on Bounded Symmetric Domains

We discuss some problems concerning the operators K _m (Z,Z ^*), where K _m is the reproducing kernel of the Peter–Weyl space with signature m and Z stands for the commuting tuple of operators of multiplication by the coordinate variables on a bounded symmetric domain. These problems have applications for construction of operator models, as well as for a possible generalization of the recent dilation theory for row contractions due to Arveson.     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

Symmetric Stable Processes in Cones

We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the spherical fractional Laplacian and derive some of its spectral properties.     

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.     

马尔科夫过程几类收敛速度的比较

本文讨论可逆Ｍａｒｋｏｖ过程的几类指数收敛速度——谱隙,各种对数Ｓｏ－ｂｅｌｅｖ常数,Ｂ－Ｓ 熵指数收敛速度之间的关系,并通过例子说明它们有着本质的差别．