The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem
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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, [email protected] Received 14 August 2010; Accepted 10 February 2011 Academic Editor: Irena Rachunkov´ a ˚ Copyright q 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 Gx, y of the clamped boundary value problem for the differential operator −1M d/dx2M on the interval −s, s is obtained. The best constant of corresponding Sobolev inequality is given by max|y|≤s Gy, 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 H0M −s, s be a Sobolev Hilbert space associated with the inner product ·, ·M : H HM
u | uM ∈ L2 −s, s, ui ±s 0 0 ≤ i ≤ M − 1 ,
u, vM
s −s
u
M
xv
M
1.1 xdx,
u2M
u, uM .
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 Poincar´e inequality. Let us introduce the functional Su as follows: Su
2 sup|y|≤s u y u2M
1.2
.
To obtain the supremum of S i.e., the best constant of Sobolev inequality, we consider the following clamped boundary value problem: −1M u2M fx −s < x < s, BVPM
i
u ±s 0 0 ≤ i ≤ M − 1. Concerning the uniqueness and existence of the solution to BVPM, we have the following proposition. The result is expressed by the monomial Kj x: ⎧ x2M−1−j ⎪ ⎪ ⎨ 2M − 1 − j ! Kj x Kj M; x ⎪ ⎪ ⎩ 0
0 ≤ j ≤ 2M − 1 , 2M ≤ j .
1.3
Proposition 1.1. For any bounded continuous function fx on an interval −s < x < s, BVPM has a unique classical solution ux expressed by ux
s −s
G x, y f y dy
−s < x < s,
1.4
where Green’s function Gx, y GM; x, y −s < x, y < s is given by G x, y −1M 2
Kij 2s Ki s − y −1 K0 x − y D Kj s x 0
Kij 2s Ki s y Kj s − x 0
1.5
Kij 2s s x ∧ y K i −1 D−1 Kj s − x ∨ y 0 M
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