scholarly journals An Lp-Comparison, $p\in (1,\infty )$, on the Finite Differences of a Discrete Harmonic Function at the Boundary of a Discrete Box

2020 ◽  
Author(s):  
Tuan Anh Nguyen
Keyword(s):  
Harmonic Function ◽  
Euclidean Space ◽  
Unit Ball ◽  
Finite Differences ◽  
Harmonic Functions ◽  
Normal Component ◽  
Discrete Analogue ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Discrete Harmonic Functions

AbstractIt is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇u on the sphere are comparable by means of the Lp-norms, $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) , up to multiplicative constants that depend only on d,p. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the d-dimensional lattice with multiplicative constants that do not depend on the size of the box.

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

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

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 group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (1) ◽  
pp. 199-204
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn. Bn will be the open unit ball {z ∈ Cn: |z| < 1}, and Un will be the unit polydisc in Cn. For 1 ≤p<∞, p≠2, Gp(Bn) (resp., Gp (Un)) will denote the group of all isometries of Hp (Bn) (resp., Hp (Un)) onto itself, where Hp (Bn) and Hp (Un) are the usual Hardy spaces.

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 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 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.

scholarly journals Decomposition of the n-Dimensional Lattice-Graph into Hamiltonian Lines

1961 ◽  
Vol 12 (3) ◽  
pp. 123-131 ◽  
Author(s):  
C. ST.J. A. Nash-Williams
Keyword(s):  
Euclidean Space ◽  
Unit Length ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Edge Disjoint ◽  
Unordered Pair ◽  
Spanning Subgraph ◽  
Lattice Graph ◽  
Disjoint Sets

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

Polynomial expansions of positive harmonic functions in the unit ball

1991 ◽  
Vol 110 (3) ◽  
pp. 533-544
Author(s):  
D. H. Armitage
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Partial Sums ◽  
Open Unit ◽  
Harmonic Polynomials ◽  
Open Unit Ball ◽  
The Individual ◽  
Polynomial Expansions

It is well known that if h is harmonic in the open unit ball B of the Euclidean space N (where N ≥ 2), then there exist homogeneous harmonic polynomials Hn of degree n in N such that converges absolutely and locally uniformly to h in B (see e.g. Brelot[2], appendice). Further, it is easy to show that this series is unique and that each polynomial Hn is the sum of all the monomial terms of degree n in the multiple Taylor series of h about the origin 0. We call the polynomial expansion of h. Our aim is to obtain sharp upper and lower bounds for the individual terms Hn and for the partial sums of this expansion in the case where h > 0 in B.

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.

Sign in / Sign up

Export Citation Format

Share Document