scholarly journals Modified Poisson Integral and Green Potential on a Half-Space

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Poisson Integral ◽  
Dimensional Euclidean Space ◽  
Superharmonic Functions ◽  
Growth Properties ◽  
Green Potential

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of then-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

scholarly journals Growth properties of modified α-potentials in the upper-half space

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 703-712 ◽  
Author(s):  
Lei Qiao ◽  
Guantie Deng
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Dimensional Euclidean Space ◽  
Superharmonic Functions ◽  
Growth Properties ◽  
Modified Kernels

The aim of this paper is to discuss the behavior at infinity of modified ?-potentials represented by the modified kernels in the upper-half space of the n-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

scholarly journals RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Yulian Zhang ◽  
Valery Piskarev
Keyword(s):  
Green Function ◽  
Euclidean Space ◽  
Half Space ◽  
Boundary Behavior ◽  
Indian Acad ◽  
Boundary Property ◽  
Dimensional Euclidean Space ◽  
Retracted Article ◽  
Green Potential

Abstract Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.

scholarly journals On α-Harmonic Functions

1966 ◽  
Vol 26 ◽  
pp. 205-221 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Dimensional Euclidean Space ◽  
Usual Sense ◽  
Superharmonic Functions

M. Riesz [8] introduced the notion of α-superharmonic functions in n(≥1)-dimensional Euclidean space Rn in connection with the potential of order α. In this paper, we shall first define the α-superharmonic and α-harmonic functions in a domain D. In case α = 2, they coincide with ones in the usual sense.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

A boundary estimate for harmonic functions

1982 ◽  
Vol 91 (1) ◽  
pp. 79-90 ◽  
Author(s):  
P. J. Rippon
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Conformal Mappings ◽  
Dimensional Euclidean Space ◽  
Boundary Behaviour ◽  
Boundary Estimate ◽  
Distortion Theorems

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

A Comparison Between thin Sets and Generalized Azarin Sets

1975 ◽  
Vol 18 (3) ◽  
pp. 335-346 ◽  
Author(s):  
M. Essén ◽  
H. L. Jackson
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Martin Boundary ◽  
Dimensional Euclidean Space ◽  
Thin Sets

Let Rp(p≥2) denote p-dimensional Euclidean space, D the half space defined by {P = (x1, x2, …, xp) ∊ Rp: xp > 0} and ∂D the frontier of D in Rp. The Martin boundary (see [2]) of D can be identified with ∂D∪{∞}.

scholarly journals Boundary behavior of positive harmonic functions in balls of Rn

1981 ◽  
Vol 84 ◽  
pp. 1-8
Author(s):  
Michael Von Renteln
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Boundary Behavior ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Ball ◽  
The Real ◽  
Positive Harmonic Functions

Let Rn be the real n-dimensional euclidean space. Elements of Rn are denoted by x = (xl • • •, xn), and ‖ x ‖ denotes the euclidean norm of x.The open ball B(x, r) with center x and radius r is defined by

scholarly journals Lower Estimates for Certain Harmonic Functions in the Half Space

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Gang Xu ◽  
Xiaoyu Zhou
Keyword(s):  
Half Space ◽  
Half Plane ◽  
Harmonic Functions ◽  
Growth Properties

We will give the growth properties of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Levin in a half plane.

scholarly journals A uniqueness theorem for harmonic functions on half-spaces

1989 ◽  
Vol 31 (2) ◽  
pp. 189-191 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Euclidean Space ◽  
Uniqueness Theorem ◽  
Half Space ◽  
Harmonic Functions ◽  
Arbitrary Point ◽  
Euclidean Norm ◽  
Image Position

An arbitrary point of the Euclidean space Rn+1, where n > 1, is denoted by (X, y), where X ∈ Rn and y ∈ R, and we denote the Euclidean norm on Rn by ∥·∥. If h is harmonic on the half-space Ω = {(X, y): y > 0}, then we define extended real-valued functions m and M as follows:and

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