# Lyapunov-type inequalities for 2*M* th order equations under clamped-free boundary conditions

- Kohtaro Watanabe
^{1}Email author, - Kazuo Takemura
^{2}, - Yoshinori Kametaka
^{3}, - Atsushi Nagai
^{2}and - Hiroyuki Yamagishi
^{4}

**2012**:242

https://doi.org/10.1186/1029-242X-2012-242

© Watanabe et al.; licensee Springer 2012

**Received: **29 May 2012

**Accepted: **9 October 2012

**Published: **23 October 2012

## Abstract

This paper generalizes the well-known Lyapunov-type inequalities for second-order linear differential equations to certain 2*M* 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.

## 1 Introduction

will be one important aspect. The first result for the high-order equation (3) is due to Levin [3], which states without proof:

**Theorem A**

*Let*$n=2M$,

*and a non*-

*trivial solution of*(3)

*satisfies the clamped boundary condition*, ${u}^{(i)}(\pm s)=0$ ($i=0,1,\dots ,M-1$).

*Then it holds that*

*M*th order linear differential equation

under yet another boundary condition:

**Clamped-free boundary condition**

The main result is as follows.

**Theorem 1**

*Suppose a non*-

*trivial solution*

*u*

*of*(4)

*exists under the clamped*-

*free boundary condition*,

*then it holds*

*Moreover*, *the estimate is sharp in the sense that there exists a function* $r(x)$, *and for this* $r(x)$, *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 [[15], Theorem 1], which computes the best constant of some kind of a Sobolev inequality. In Section 4, we give a concise proof for an ${L}^{p}$ extension of Theorem 1 of [15].

## 2 Proof of Theorem 1

*u*belongs to

$1<p$, *m* runs over the range $0\le m\le M-1$, and ${u}^{(i)}$ is the *i* th derivative of *u* in a distributional sense. We denote by ${C}_{CF}(M,m,p)$ the best constant of the above Sobolev inequality (6). Here, we note that in [15], Takemura obtained the best constant for $p=2$, $m=0$ by constructing the Green function of the clamped-free boundary value problem. Although, for the proof of Theorem 1, we simply need the value ${C}_{CF}(M,0,2)$, we would like to compute ${C}_{CF}(M,m,p)$ 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 [15]. Now, we have the following propositions.

**Proposition 1**

*The best constant of*(6)

*is*

*and it is attained by*

**Proposition 2**

*Suppose a*${C}^{2M}[-s,s]$

*solution of*(4)

*with the clamped*-

*free boundary condition exists*,

*then it holds that*

*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.

**Corollary 1**

*Suppose a non*-

*trivial solution*

*u*

*of the non*-

*linear equation*

*exists under the clamped*-

*free boundary condition*,

*where*

*m*

*satisfies*($1\le m\le M-1$),

*then it holds*

The following are the examples of Theorem 1 and Corollary 1.

**Example 1**The following example corresponds to the case $M=1$ and $r(x)=-6/(-11{s}^{2}+2sx+{x}^{2})$ of (4) with the clamped-free boundary condition

**Example 2**The following example corresponds to the case $M=2$, $m=1$ and $r(x)=(3(17{s}^{2}-6sx+{x}^{2}))/(2{(7{s}^{2}-4sx+{x}^{2})}^{2})$ of (10) with the clamped-free boundary condition

## 3 Proof of Proposition 2

Assuming Proposition 1, we first prove Proposition 2.

*Proof of Proposition 2*Let

*u*be a solution of equation (4). Since

*u*satisfies the clamped-free boundary condition, multiplying (4) by

*u*and integrating it over $[-s,s]$, we have

*u*satisfies the clamped boundary condition at $x=-s$, we have $u\equiv 0$. This contradicts the assumption that

*u*is a non-trivial solution of (4). So, canceling ${\parallel {u}^{(M)}\parallel}_{2}^{2}$, we obtain

*u*is a constant. But, again from the clamped boundary condition at $x=-s$, we have $u\equiv 0$. Thus, the inequality is strict. Finally, we see (5) is sharp. For this purpose, let us define the functional

*J*has the minimizer $u\in W(M,2)$ (see, for example, [[16], Lemma 3]),

*i.e.*,

*δ*sufficiently small, we see that the right-hand side of (15) can be arbitrarily close to the left-hand side,

*i.e.*, for a small positive ${\u03f5}_{1}$, ${\lambda}_{1}$ can be written as

*u*of

*r*satisfies

Hence, (5) is sharp. □

*Proof of Corollary 1*Integrating equation (10), we have

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 ${u}^{(m)}$ is a constant. Hence, from the clamped boundary condition at $x=-s$, we have ${u}^{(m)}\equiv 0$. So, there exists ($i=0,\dots ,m-1$) such that $u(x)={\sum}_{i=0}^{m-1}{a}_{i}{x}^{i}$. But, again from the clamped boundary condition at $x=-s$, we have $u\equiv 0$. Thus, inequality (11) is strict. □

## 4 Proof of Proposition 1

We prepare the following lemmas for the proof of Proposition 1.

**Lemma 1**

*Suppose there exists a function*${u}_{\ast}\in W(M,p)$

*which attains the best constant*$C(M,n,p)$

*of*(6),

*then it holds that*

*Proof*Suppose it holds that

This contradicts the assumption that $C(M,m,p)$ is the best constant of (6). □

**Lemma 2**

*Let*

*then for*$u\in W(M,p)$

*it holds that*

*Proof* Integrating by parts, we obtain the result. □

*Proof of Proposition 1*From Lemma 2, we see that if the function attains the best constant $C(M,m,p)$, it belongs to ${W}_{\ast}(M,m,p)\subset W(M,p)$:

*q*satisfies $1/p+1/q=1$. Hence, if there exists the function ${u}_{\ast}\in {W}_{\ast}(M,m,p)$ which attains the equality of (21), it holds that $C(M,m,p)={\parallel {H}_{m}\parallel}_{{L}^{q}(-s,s)}^{p}$. On the contrary, we see that the equality holds for (21) if and only if

*u*satisfies

This completes the proof. □

## Declarations

## Authors’ Affiliations

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