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Báo cáo hóa học: " Research Article Representation of Multivariate Functions via the Potential Theory and Applications to Inequalities"

Tham khảo luận văn - đề án 'báo cáo hóa học: " research article representation of multivariate functions via the potential theory and applications to inequalities"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 475957 15 pages doi 10.1155 2008 475957 Research Article Representation of Multivariate Functions via the Potential Theory and Applications to Inequalities Florica C. Cirstea1 and Sever S. Dragomir2 1 Department of Mathematics Australian National University Canberra ACT 0200 Australia 2 School of Computer Science and Mathematics Victoria University P.O. Box 14428 Melbourne City Victoria 8001 Australia Correspondence should be addressed to Sever S. Dragomir sever.dragomir@vu.edu.au Received 12 February 2007 Revised 2 August 2007 Accepted 9 November 2007 Recommended by Siegfried Carl We use the potential theory to give integral representations of functions in the Sobolev spaces W1 p Q where p 1 and Q is a smooth bounded domain in RN N 2 . As a byproduct we obtain sharp inequalities of Ostrowski type. Copyright 2008 F. C. Cĩrstea and S. S. Dragomir. 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 and main results Let N 2 and let denote the canonical inner product on RN X RN. If CO stands for the area of the surface of the N - 1 -dimensional unit sphere then CO 2nN 2 r N 2 where r is the gamma function defined by r s 0 e-tts-1 dt for s 0 see 1 Proposition 0.7 . Let E denote the normalized fundamental solution of Laplace equation E x - ln Ix 2n 1 2 - 0N x N-2 x 0 if N 2 x 0 if N 3. 1.1 Unless otherwise stated we assume throughout that Q c RN is a bounded domain with C2 boundary dQ. Let V denote the unit outward normal to dQ and let dơ indicate the N-1 -dimensional area element in dQ. The Green-Riemann formula says that any function 2 Journal of Inequalities and Applications f G C2 Q a C1 Q satisfying Af G C Q can be represented in Q as follows see 2 Section 2.4 c ĩ dE òf i f y l fx dv x - y - dv WE x