The Discrete and Classical Dirichlet Problem

Let W ì \mathbbR^d{\Omega \subset \mathbb{R}^d} be some bounded domain with reasonable boundary and let f be a continuous function on the complement Ω ^c . We can construct an unique continuous function u that is harmonique on Ω and u = f on Ω ^c . Similarly, u _d is the unique function on the lattice points such that for each lattice point of Ω satisfies the “average” property with respect to its nearest neighbours and u _d = f on Ω ^c . In this paper when ∂Ω is Lipschitz I give a “best possible” estimate of ||u − u _d ||_∞.     

Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals

It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V ⁽⁰⁾, which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V ⁽⁰⁾. In this paper, I prove that, provided an eigenform exists, even if the form on V ⁽⁰⁾ is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket.     

Zeros of functions with finite Dirichlet integral

In this paper, we refine a result of Nagel, Rudin, and Shapiro (1982) concerning the zeros of holomorphic functions on the unit disk with finite Dirichlet integral.

     

Concave univalent functions and Dirichlet finite integral

The article deals with the class consisting of non‐vanishing functions f that are analytic and univalent in such that the complement is a convex set, and the angle at ∞ is less than or equal to for some . Related to this class is the class of concave univalent mappings in , but this differs from with the standard normalization A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for settled by Avkhadiev et al. 3 . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for is also presented.     

Discrete approximation of integral operators

A method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables. This area of application includes the approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.

     

Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

In this paper we are interested in the existence of infinitely many solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, we determine unbounded intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. Finally, the attained solutions are positive when the nonlinearity is supposed to be nonnegative thanks to a discrete maximum principle.     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

The integral modulus of continuity of the Dirichlet kernel and the conjugate Dirichlet kernel

Asymptotic estimates for the integral modulus of continuity of order s of the Dirichlet kernel and the conjugate Dirichlet kernel are obtained. For example, if k/2, then _s (D _k ,)=2 ^s +1/²sin ^s k/2 log(1+k/s)+O(2 ^s sin ^s k/2)holds uniformly with respect to all the parameters.Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 98–105, September, 1993.     

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. Partially supported by Lithuanian Foundation of Studies and Science     

Discrete Limit Theorems for General Dirichlet Series. II

We prove a discrete limit theorem for general Dirichlet series in the sense of weak convergence of probability measures in the space of analytic funtions. Examples are presented.     

An integral formula on the Heisenberg group

Let $\mathbb{H }^n$ denote the $(2n+1)$ -dimensional (sub-Riemannian) Heisenberg group. In this note, we shall prove an integral identity (see Theorem 1.2) that generalizes a formula obtained in the Seventies by Reilly (Indiana Univ Math J 26(3):459–472, 1977). Some first applications will be given in Sect. 4.     

Non-tangential limits, fine limits and the Dirichlet integral

Let denote the unit ball in This paper characterizes the subsets of with the property that for all harmonic functions on which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on at points of the boundary.

     

伪脐点上流形的内蕴积分不等式

研究了拟常曲率黎充形上的伪脐点子流形,得到了这种子汉形的一个Ｓｉ－ｍｏｌｓ型内蕴积分不等式,从而推广改进了Ｂ．Ｙ．Ｃｈｅｎ关于常曲率黎曼流形中的脐点子流形的一个相应结果。