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Research Article Symmetrization of Functions and the Best Constant of 1-DIM Lp Sobolev Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 874631 12 pages doi 2009 874631 Research Article Symmetrization of Functions and the Best Constant of 1-DIM Lp Sobolev Inequality Kohtaro Watanabe 1 Yoshinori Kametaka 2 Atsushi Nagai 3 Hiroyuki Yamagishi 4 and Kazuo Takemura3 1 Department of Computer Science National Defense Academy 1-10-20 Hashirimizu Yokosuka 239-8686 Japan 2 Graduate School of Mathematical Sciences Faculty of Engineering Science Osaka University 1-3 Matikaneyamatyo Toyonaka 560-8531 Japan 3 Liberal Arts and Basic Sciences College of Industrial Technology Nihon University 2-11-1 Shinei Narashino 275-8576 Japan 4 Department of Monozukuri Engineering Tokyo Metropolitan College of Industrial Technology 1-10-40 Higashi-ooi Shinagawa Tokyo 140-0011 Japan Correspondence should be addressed to Kohtaro Watanabe wata@ Received 25 June 2009 Accepted 16 October 2009 Recommended by Kunquan Lan The best constants C m p of Sobolev embedding of wm -5 s into L -s s for m 1 2 3 and 1 p are obtained. A lemma concerning the symmetrization of functions plays an important role in the proof. Copyright 2009 Kohtaro Watanabe et al. 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 Let W0m p -s s be a Sobolev space which consists of the functions whose derivatives up to m - 1 vanish at x s that is w -s s u u i e Lp -s s i 0 . . m uj s 0 j 0 . .m - 1 j where u denotes ith derivative of u in a distributional sense. The purpose of this paper is to investigate the best constant C m p of Lp Sobolev inequality sup u x C u x Tdx -s x s -s where u e W0m p -s s and 1 p. The result is as follows. 2 Journal of Inequalities and Applications Theorem . The best constant of inequality is C 1 p 2- q-V qs1 q s q 1 q C 2 p -777 2 .