tailieunhanh - Báo cáo hóa học: "Research Article Sharp Hardy-Sobolev Inequalities with General Weights and Remainder Terms"

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 Sharp Hardy-Sobolev Inequalities with General Weights and Remainder Terms | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 419845 24 pages doi 2009 419845 Research Article Sharp Hardy-Sobolev Inequalities with General Weights and Remainder Terms Yaotian Shen and Zhihui Chen Department of Mathematics South China University of Technology Guangzhou 510640 China Correspondence should be addressed to Zhihui Chen mazhchen@ Received 18 December 2008 Accepted 10 August 2009 Recommended by Panayiotis Siafarikas We consider a class of sharp Hardy-Sobolev inequality where the weights are functions of the distance from a surface. It is proved that the Hardy-Sobolev inequality can be successively improved by adding to the right-hand side a lower-order term with optimal weight and constant. Copyright 2009 Y. Shen and Z. Chen. 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 The classical Hardy inequality says lùfé Fdx mTt L l updx u e C0 rN J RN x N - p J RN where the constant lp N - p p is optimal but never attained see for example 1-4 . This suggests that one might look for an error term. Brezis and Vazques 5 showed that if Q is a bounded domain in RN N 3 with 0 e Q then there exists a positive constant ÃQ such that Vu 2dx N - 2 C2dx 1Q u2dx Q 4 Q x 2 Q u e H 1 Q . This result was extended to the Lp setting by Gazzola et al. 6 . Adimurthi et al. proved that Hardy s inequality can be successively improved by adding lower order terms see 7 8 for details. Abdellaoui et al. 9 obtained Hardy s inequality with the type of weight x -pY. See 10 11 for the case of general weight x . 2 Journal of Inequalities and Applications Another type of Hardy s inequality contains weight involving the distant to the boundary of the domain. For a convex domain Q c RN with smooth boundary the Hardy inequality dx -P-X Vu pdx u