tailieunhanh - Báo cáo sinh học: " Research Article Asymptotic Behavior for a Class of Modified α-Potentials in a Half Space"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Asymptotic Behavior for a Class of Modified α-Potentials in a Half Space | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 647627 14 pages doi 2010 647627 Research Article Asymptotic Behavior for a Class of Modified a-Potentials in a Half Space Lei Qiao1 and Guantie Deng2 1 Department of Mathematics and Information Science Henan University of Finance and Economics Zhengzhou 450002 China 2 Laboratory of Mathematics and Complex Systems School of Mathematical Science Beijing Normal University MOE Beijing 100875 China Correspondence should be addressed to Guantie Deng denggt@ Received 13 March 2010 Accepted 21 June 2010 Academic Editor Shusen Ding Copyright 2010 L. Qiao and G. Deng. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A class of a-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates Ra i i x o Xn x 1 -2 2h x 1 at infinity in the upper-half space of the n-dimensional Euclidean space where the function h x is a positive nondecreasing function on the interval 0 to satisfying certain conditions. This result generalizes the growth properties of analytic functions harmonic functions and superharmonic functions. 1. Introduction and Main Results Let Rn n 2 denote the n-dimensional Euclidean space with points x x1 x2 . xn-1 xn x xn where x e Rn-1 and xn e R. The boundary and closure of an open Q of Rn are denoted by dQ and Q respectively. The upper half-space is the set H x x xn e Rn xn 0 whose boundary is dH. We identify Rn with Rn-1 X R and Rn-1 with Rn-1 X 0 writing typical points x y e Rn as x x xn y y yn where x x1 x2 . xn-1 y y1 y2 . yn-1 e Rn-1 and putting x y n 1 xy x y xnyn x Vx x x -f x x . For x e Rn and r 0 let Bn x r denote the open ball with center at x and radius r in Rn. It is well known that see . 1 Chapter 6 the .