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Open Access Research Article

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

Kohtaro Watanabe1*, Yoshinori Kametaka2, Hiroyuki Yamagishi3, Atsushi Nagai4 and Kazuo Takemura4

Author Affiliations

1 Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan

2 Division of Mathematical Sciences, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan

3 Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan

4 Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei, Narashino 275-8576, Japan

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Boundary Value Problems 2011, 2011:875057  doi:10.1155/2011/875057

Published: 7 March 2011

Abstract

Green's function of the clamped boundary value problem for the differential operator on the interval is obtained. The best constant of corresponding Sobolev inequality is given by . 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).