锥中一类调和函数的增长估计

给出了锥中一类调和函数在无穷远点处的增长估计,推广了Siegel和Talvila,张和邓在半空间的相关结果.     

Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057, and is also part of the programme of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).     

On Domains in Which Harmonic Functions Satisfy Generalized Mean Value Properties

Potential Analysis - Assume that a bounded domain Ω??N (N ≥ 2) has the property that there exists a signed measure µ with compact support in Ω such that, for every...     

Approximation in the Cone of Positive Harmonic Functions

In the course of studying quadrature domains Gustafsson, Sakai and Shapiro were led to the question of whether it is the case that the positive integrable harmonic functions on a bounded domain are dense among all positive harmonic functions (w.r.t. uniform convergence on compact subsets). In this article we will show how such approximation problems are related to representing measures on the Martin boundary, and then we use these results to give a counterexample to the question posed above. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057.     

Boundary Integrals and Approximations of Harmonic Functions

Steklov expansions for a harmonic function on a rectangle are derived and studied with a view to determining an analog of the mean value theorem for harmonic functions. It is found that the value of a harmonic function at the center of a rectangle is well approximated by the mean value of the function on the boundary plus a very small number (often 3 or fewer) of specific further boundary integrals. These integrals are coefficients in the Steklov representation of the function. Similar approximations are found for the central values of solutions of Robin and Neumann boundary value problems. The results follow from analyses of the explicit expressions for the Steklov eigenvalues and eigenfunctions.     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计，同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

Support Points of Harmonic Typically-Real Functions

ＳｕｐｐｏｒｔＰｏｉｎｔｓｏｆＨａｒｍｏｎｉｃＴｙｐｉｃａｌｌｙ－ＲｅａｌＦｕｎｃｔｉｏｎｓ宫贵福ＳｕｐｐｏｒｔＰｏｉｎｔｓｏｆＨａｒｍｏｎｉｃＴｙｐｉｃａｌｌｙ－ＲｅａｌＦｕｎｃｔｉｏｎｓ￥ＧｏｎｇＧｕｉｆｕ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａ...     

Existence of Harmonic Functions with Finite Energy on Complete Riemannian Manifolds

Let M be a noncompact complete Riemannian manifold. We consider the existence of harmonic functions with |∇u| ∈ L^p(M).     

A Note on the Maximal Functions on Weighted Harmonic AN Groups

In this paper, the authors point out that the methods used by Li (2004, 2005, 2007) can be applied to study maximal functions on weighted harmonic AN groups.     

Busemann Functions in a Harmonic Manifold

For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces.     

A Remark on the Level Sets of the Graph of Harmonic Functions Bounded by Two Circles in Parallel Planes

In this paper,we find two auxiliary functions andmake use of themaximum principle to study the level sets of harmonic function defined on a convex ring with homogeneous Dirichlet boundary conditions in $\mathbb{R}^2$. In higher dimensions, we also have a similar result to Jagy's.     

插值逼近具边界数据的调和函数

该文考虑光滑闭Jondan曲线Γ围成的单连区域D,证明了在Γ上具有已知导数数据的D内调和函数u(x,y)的存在性.继而构造了一个调和插值多项式序列在(?)=D∪Γ上一致收敛于u(x,y),且具理想的收敛速度.此外,以往同类研究工作中的边界Γ是解析曲线,而在该文中已减少边界限制为Γ∈J_0.     

On Harmonic Majorization of the Martin Function at Infinity in a Cone

This paper shows that some characterizations of the harmonic majorization of the Martin function for domains having smooth boundaries also hold for cones.     

修改的Poisson核和调和函数的积分表示

在本文中,对于半平面中的调和函数u(z),利用半平面中修改的Poisson核,证明了如果它的正部u~ (z)=max{u(z),0}满足某些限制增长条件,则它可以用半平面边界上的积分表示出来,并且它的负部u~-(z)=max{-u(z),0}也被类似的增长条件所控制,这一结果改进了在半平面中调和函数的某些经典结果。     

锥内特定调和函数的渐近状态

给出了锥内特定调和函数在无穷远点处的渐近状态,推广了Siegel-Talvila在半空间的相关结果.同时,也得到了锥内Dirichlet问题的解.     

A Note on Boundary Unique Contination for Harmonic Functions in Non-Smooth Domains

In this note, we prove the boundary unique continuation property for harmonic function dfined on convex domain     

On the Cone of Bounded Lower Semicontinuous Functions

We prove that the cone of bounded lower semicontinuous functions defined on a Tychonoff space X is algebraically and structurally isomorphic and isometric to a convex cone contained in the cone of all bounded lower semicontinuous functions defined on the Stone-Cech compactification βX if and only if the space X is normal. We apply this theorem to the study of relationship between a class of multivalued maps and sublinear operators. Using these results, we obtain new proofs of theorems about continuous selections.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 886–902.Original Russian Text Copyright ©2005 by Yu. E. Linke.     

Harmonic Functions for a Class of Integro-differential Operators

We consider the operator defined on functions by

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.     

Boundary Behavior of Harmonic Functions on Manifolds

In this paper, we mainly set up a kind of representation theorem of harmonic functions on manifolds with Ricci curvature bounded below and study non-tangential limits of harmonic functions.