Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 875057 17 pages doi 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 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 Correspondence should be addressed to Kohtaro Watanabe wata@ Received 14 August 2010 Accepted 10 February 2011 Academic Editor Irena Rachunkova 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. Green s function G x y of the clamped boundary value problem for the differential operator -1 M d dx 2M on the interval -s s is obtained. The best constant of corresponding Sobolev inequality is given by max y sG 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 . 1. Introduction For M 1 2 3 . s 0 let H HM -s s be a Sobolev Hilbert space associated with the inner product - M H H M ịu u M e L2 -s s u i s 0 0 i M - 1 Ị u v m u M x v M x dx uhM u u m. -s 2 Boundary Value Problems The fact that - M induces the equivalent norm to the standard norm of the Sobolev Hilbert space of Mth order follows from Poincare inequality. Let us introduce the functional S u as follows S u
