RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space
- PDF / 1,441,390 Bytes
- 7 Pages / 595.276 x 793.701 pts Page_size
- 73 Downloads / 171 Views
RESEARCH
Open Access
A note on the boundary behavior for a modified Green function in the upper-half space Yulian Zhang1 and Valery Piskarev2* *
Correspondence: [email protected] 2 Faculty of Science and Technology, University of Wollongong, Wollongong, NSW 2522, Australia Full list of author information is available at the end of the article
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. Keywords: modified Green function; capacity; upper-half space
1 Introduction and main results Let Rn (n ≥ ) denote the n-dimensional Euclidean space. The upper half-space H is the set H = {x = (x , x , . . . , xn ) ∈ Rn : xn > }, whose boundary and closure are ∂H and H respectively. For x ∈ Rn and r > , let B(x, r) denote the open ball with center at x and radius r. Set Eα (x) =
– log |x| if α = n = , if < α < n. |x|α–n
Let Gα be the Green function of order α for H, that is, Gα (x, y) = Eα (x – y) – Eα x – y∗ ,
x, y ∈ H, x = y, < α ≤ n,
where ∗ denotes reflection in the boundary plane ∂H just as y∗ = (y , y , . . . , –yn ). In case α = n = , we consider the modified kernel function, which is defined by En,m (x – y) =
En (x – y) En (x – y) + (log y –
m–
if |y| < , k
x k= ( kyk )) if |y| ≥ .
In case < α < n, we define Eα,m (x – y) =
Eα (x – y) Eα (x – y) –
m–
n–α
|x|k k= |y|n–α+k Ck
if |y| < , x·y ( |x||y| )
if |y| ≥ ,
© 2015 Zhang and Piskarev. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Zhang and Piskarev Boundary Value Problems (2015) 2015:114
Page 2 of 7
where m is a non-negative integer, Ckω (t) (ω = n–α ) is the ultraspherical (or Gegenbauer) polynomial (see []). The expression arises from the generating function for Gegenbauer polynomials
– tr + r
–ω
=
∞
Ckω (t)rk ,
(.)
k=
where |r| < , |t| ≤ and ω > . The coefficient Ckω (t) is called the ultraspherical (or Gegenbauer) polynomial of degree k associated with ω, the function Ckω (t) is a polynomial of degree k in t. Then we define the modified Green function Gα,m (x, y) by Gα,m (x, y) =
En,m+ (x – y) – En,m+ (x – y∗ ) if α = n = , Eα,m+ (x – y) – Eα,m+ (x – y∗ ) if < α < n,
where x, y ∈ H and x = y. We remark that this modified Green function is also used to give unique solutions of the Neumann and Dirichlet problem in the upper-half space [–]. Write Gα,m (x, μ) = Gα,m (x, y) dμ(y), H
where μ is a non-negative measure on H. Here note that G, (x,
Data Loading...
