Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 874631, 12 pages
Research Article

Symmetrization of Functions and the Best Constant of 1-DIM Lp Sobolev Inequality

1Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan
2Graduate School of Mathematical Sciences, Faculty of Engineering Science, Osaka University, 1–3 Matikaneyamatyo, Toyonaka 560-8531, Japan
3Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei, Narashino 275-8576, Japan
4Department of Monozukuri Engineering, Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan

Received 25 June 2009; Accepted 16 October 2009

Academic Editor: Kunquan Lan

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.


The best constants C(m,p) of Sobolev embedding of W0m,p(s,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.