Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article An Improved Hardy-Rellich Inequality with Optimal Constant | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 610530 10 pages doi 2009 610530 Research Article An Improved Hardy-Rellich Inequality with Optimal Constant Ying-Xiong Xiao1 and Qiao-Hua Yang2 1 School of Mathematics and Statistics Xiaogan University Xiaogan Hubei 432000 China 2 School of Mathematics and Statistics Wuhan University Wuhan 430072 China Correspondence should be addressed to Ying-Xiong Xiao xyx21cn@ Received 25 May 2009 Accepted 11 September 2009 Recommended by Siegfried Carl We show that a Hardy-Rellich inequality with optimal constants on a bounded domain can be refined by adding remainder terms. The procedure is based on decomposition into spherical harmonics. Copyright 2009 . Xiao and . Yang. 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. 1. Introduction Hardy inequality in RN reads for all u e Cf RN and N 3 f Vu 2dx N - 2 rn 4 u2 RN x 2 and N - 2 2 4 is the best constant in and is never achieved. A similar inequality with the same best constant holds if RN is replaced by an arbitrary domain Q c RN and Q contains the origin. Moreover Brezis and Vazquez 1 have improved it by establishing that for u e Cf Q Vu 2dx N - 2 f - dx A -A 2 f N f u2dx Q 4 Q x 2 Q Q 2 Journal of Inequalities and Applications where WN and Q denote the volume of the unit ball B1 and Q respectively and A -A 2 is the first eigenvalue of the Dirichlet Laplacian of the unit disc in R2. In case Q is a ball centered at zero the constant A - A 2 in is sharp. Similar improved inequalities have been recently proved if instead of one considers the corresponding Lp Hardy inequalities. In all these cases a correction term is added on the right-hand side see . 2-4 . On the other hand the classical Rellich inequality states that for N 5 Au
