Boundary Value Problems
Volume 2011 (2011), Article ID 875057, 17 pages
doi:10.1155/2011/875057
Research Article

The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem

Kohtaro Watanabe,1 Yoshinori Kametaka,2 Hiroyuki Yamagishi,3 Atsushi Nagai,4 and Kazuo Takemura4

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

Received 14 August 2010; Accepted 10 February 2011

Academic Editor: Irena Rachůnková

Copyright © 2011 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.

Abstract

Green's function 𝐺 ( 𝑥 , 𝑦 ) of the clamped boundary value problem for the differential operator ( − 1 ) 𝑀 ( 𝑑 / 𝑑 𝑥 ) 2 𝑀 on the interval ( − 𝑠 , 𝑠 ) is obtained. The best constant of corresponding Sobolev inequality is given by m a x | 𝑦 | ≤ 𝑠 𝐺 ( 𝑦 , 𝑦 ) . In addition, it is shown that a reverse of the Sobolev best constant is the one which appears in the generalized Lyapunov inequality by Das and Vatsala (1975).