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Báo cáo toán học: "Polar Coordinates on H-type Groups  and Applications"

Trong bài báo này chúng ta xây dựng các tọa độ cực vào các nhóm H-. Khi các ứng dụng, chúng tôi tính toán một cách rõ ràng khối lượng của quả bóng trong ý nghĩa của khoảng cách và liên tục trong các giải pháp cơ bản của p-sub-Laplacian vào nhóm H-loại. Ngoài ra, chúng tôi chứng minh một số kết quả không tồn tại của các giải pháp yếu cho một thoái hóa. | Vietnam Journal of Mathematics 34 3 2006 307-316 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 Polar Coordinates on H-type Groups and Applications Junqiang Han and Pengcheng Niu Department of Applied Math. Northwestern Polytechnical University Xi an Shaanxi 710072 China Received August 11 2005 Revised November 14 2005 Abstract. In this paper we construct polar coordinates on H-type groups. As applications we explicitly compute the volume of the ball in the sense of the distance and the constant in the fundamental solution of p-sub-Laplacian on the H-type group. Also we prove some nonexistence results of weak solutions for a degenerate elliptic inequality on the H-type group. 2000 Mathematics Subject Classification 35R45 35J70. Keywords H-type group polar coordinate nonexistence degenerate elliptic inequality. 1. Introduction The polar coordinates for the Heisenberg group H1 and Hn were defined by Greiner 8 and D Ambrosio 3 respectively. Using their introduction as in 3 we can explicitly compute the volume of the Heisenberg ball see 6 and the constant in the fundamental solution of AHn see 4 5 . In this paper we will construct polar coordinates on H-type groups. In 1 the polar coordinates were given in Carnot groups and groups of H-type but the expression here is slightly different. As an application we will explicitly calculate the volume of the ball in the sense of the distance and the constant in the fundamental solution of The project was supported by National Natural Science Foundation of China Grant No. 10371099. 308 Junqiang Han and Pengcheng Niu p-sub-Laplacian on the H-type groups. Nonexistence results of weak solutions for some degenerate and singular elliptic parabolic and hyperbolic inequalities on the Euclidean space Rn have been largely considered see 13 14 and their references. The singular sub-Laplace inequality and related evolution inequalities on the Heisenberg group Hn were studied in 3 6 . In this paper we will discuss the nonexistence of .