Lyapunov-type inequalities for 2M th order equations under clamped-free boundary conditions
© Watanabe et al.; licensee Springer 2012
Received: 29 May 2012
Accepted: 9 October 2012
Published: 23 October 2012
This paper generalizes the well-known Lyapunov-type inequalities for second-order linear differential equations to certain 2M th order linear differential equations
under clamped-free boundary conditions. The usage of the best constant of some kind of a Sobolev inequality helps clarify the process for obtaining the result.
under yet another boundary condition:
The main result is as follows.
Moreover, the estimate is sharp in the sense that there exists a function , and for this , the solution u of (4) exits such that the right-hand side is arbitrarily close to the left-hand side.
The result is obtained using Takemura [, Theorem 1], which computes the best constant of some kind of a Sobolev inequality. In Section 4, we give a concise proof for an extension of Theorem 1 of .
2 Proof of Theorem 1
, m runs over the range , and is the i th derivative of u in a distributional sense. We denote by the best constant of the above Sobolev inequality (6). Here, we note that in , Takemura obtained the best constant for , by constructing the Green function of the clamped-free boundary value problem. Although, for the proof of Theorem 1, we simply need the value , we would like to compute for general p and m since the proof presented in Section 4 does not depend on special values of p and m and quite simplifies the proof of Theorem 1 of . Now, we have the following propositions.
Moreover, the estimate is sharp.
Proof of Theorem 1 Clearly, Theorem 1 is obtained from Propositions 1 and 2. □
Thus, all we have to do is to show Propositions 1 and 2. Before proceeding with the proof of these propositions, we would like to show a corollary obtained from Proposition 1.
The following are the examples of Theorem 1 and Corollary 1.
3 Proof of Proposition 2
Assuming Proposition 1, we first prove Proposition 2.
Hence, (5) is sharp. □
Next, we show that the inequality (11) is strict. To see this, we note that in (19), the equality holds for the first inequality if and only if is a constant. Hence, from the clamped boundary condition at , we have . So, there exists () such that . But, again from the clamped boundary condition at , we have . Thus, inequality (11) is strict. □
4 Proof of Proposition 1
We prepare the following lemmas for the proof of Proposition 1.
This contradicts the assumption that is the best constant of (6). □
Proof Integrating by parts, we obtain the result. □
This completes the proof. □
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