Sturdy Harmonic Functions and their Integral Representations

Sturdy harmonic functions constitute all but the least tractable of the positive harmonic functions in potential-theoretic settings. They are the uniform limits on compact sets of positive, bounded harmonic functions and are also produced by a simple integral representation on the boundary of a natural compactification of the space on which they are defined. The boundary of that compactification is metrizable, and more regular for the Dirichlet problem, in general, than is the Martin boundary if that boundary is even defined in the setting.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.     

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions

This paper presents a new boundary integral method for the solution of Laplace’s equation on both bounded and unbounded multiply connected regions, with either the Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical results are presented to illustrate the efficiency of the proposed method.     

Integral Representations for Correlation Functions of the XXZ Heisenberg Chain

We consider new integral representations for two-point correlation functions of local spins 1/2 in the XXZ Heisenberg chain.     

Sharp Growth Estimates for Modified Poisson Integrals in a Half Space

For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in Rⁿ. Pointwise growth estimates for these integrals are given and the estimates are proved sharp in a strong sense. For decaying data, a new type of modified Poisson integral is introduced and used to develop asymptotic expansions for solutions of these half space problems.     

积分算子作用下的亚纯多叶函数

令∑_p表示形如f(z)=z~(-p)+∑m=1∞(p∈N={1,2,3…})且在去心单位开圆盘D=U\{0}={z∶z∈C且0|z|1}上解析的亚纯多叶函数类.利用一个作用在∑_p上的乘积算子定义了几个新的亚纯函数的子类,并考虑这些函数类在积分算子作用下的性质.     

A Random Walk on Rectangles Algorithm

In this article, we introduce an algorithm that simulates efficiently the first exit time and position from a rectangle (or a parallelepiped) for a Brownian motion that starts at any point inside. This method provides an exact way to simulate the first exit time and position from any polygonal domain and then to solve some Dirichlet problems, whatever the dimension. This method can be used as a replacement or complement of the method of the random walk on spheres and can be easily adapted to deal with Neumann boundary conditions or Brownian motion with a constant drift. AMS 2000 Subject Classification 60C05, 65N