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

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

**2011**:875057

**DOI: **10.1186/1687-2770-2011-875057

© Kohtaro Watanabe et al. 2011

**Received: **14 August 2010

**Accepted: **10 February 2011

**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).

## 1. Introduction

Proposition 1.1.

is the determinant of matrix , , and .

With the aid of Proposition 1.1, we obtain the following theorem. The proof of Proposition 1.1 is shown in Appendices A and B.

- (i)

(ii) is attained by , that is, .

Clearly, Theorem 1.2(i), (ii) is rewritten equivalently as follows.

Corollary 1.3.

is . Moreover the best constant is attained by , where is an arbitrary complex number.

*conjugate*. It is wellknown that if there exists a pair of conjugate points in , then the classical Lyapunov inequality

_{2}

*conjugate*if there exists a nontrivial solution of (1.12) satisfying

We point out that the constant which appears in the generalized Lyapunov inequality by Levin [3] and Das and Vatsala [4] is the reverse of the Sobolev best embedding constant.

Corollary 1.4.

where is the best constant of the Sobolev inequality (1.9).

Without introducing auxiliary equation and the existence result of conjugate points as [2, 4], we can prove this corollary directly through the Sobolev inequality (the idea of the proof origins to Brown and Hinton [5, page 5]).

Proof of Corollary 1.4.

we obtain the result.

Here, at the end of this section, we would like to mention some remarks about (1.12). The generalized Lyapunov inequality of the form (1.14) was firstly obtained by Levin [3] without proof; see Section 4 of Reid [6]. Later, Das and Vatsala [4] obtained the same inequality (1.14) by constructing Green's function for . The expression of the Green's function of Proposition 1.1 is different from that of [4]. The expression of [4, Theorem 2.1] is given by some finite series of and on the other hand, the expression of Proposition 1.1 is by the determinant. This complements the results of [7–9], where the concrete expressions of Green's functions for the equation but different boundary conditions are given, and all of them are expressed by determinants of certain matrices as Proposition 1.1.

## 2. Reproducing Kernel

First we enumerate the properties of Green's function of . has the following properties.

Lemma 2.1.

Consider the following:

Proof.

where we used the fact , . So we have (3), and (4) follows from (3).

Using Lemma 2.1, we prove that the functional space associated with inner norm is a reproducing kernel Hilbert space.

Lemma 2.2.

Proof.

Using (1), (2), and (4) in Lemma 2.1, we have (2.9).

## 3. Sobolev Inequality

In this section, we give a proof of Theorem 1.2 and Corollary 1.3.

Proof of Theorem 1.2 and Corollary 1.3.

which completes the proof of Theorem 1.2 and Corollary 1.3.

Thus, all we have to do is to prove (3.2).

## 4. Diagonal Value of Green's Function

Thus, we can expect that takes the form . Precisely, we have the following proposition.

Proposition 4.1.

To prove this proposition, we prepare the following two lemmas.

Lemma 4.2.

Lemma 4.3.

Let , where , then it holds that (4.6) and .

Proof of Proposition 4.1.

Inserting (4.9) into (4.8), we have Proposition 4.1.

Proof of Lemma 4.2.

This completes the proof of Lemma 4.2.

Proof of Lemma 4.3.

This proves Lemma 4.3.

## Appendices

### A. Deduction of (1.5)

where is any regular matrix and and are any matrices, we have (1.5).

### B. Deduction of (1.6)

## Authors’ Affiliations

## References

- Ha C-W: Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type.
*Proceedings of the American Mathematical Society*1998, 126(12):3507–3511. 10.1090/S0002-9939-98-05010-2View ArticleMathSciNetMATHGoogle Scholar - Yang X: On inequalities of Lyapunov type.
*Applied Mathematics and Computation*2003, 134(2–3):293–300. 10.1016/S0096-3003(01)00283-1View ArticleMathSciNetMATHGoogle Scholar - Levin AJ: Distribution of the zeros of solutions of a linear differential equation.
*Soviet Mathematics*1964, 5: 818–821.MATHGoogle Scholar - Das KM, Vatsala AS: Green's function for
*n-n*boundary value problem and an analogue of Hartman's result.*Journal of Mathematical Analysis and Applications*1975, 51(3):670–677. 10.1016/0022-247X(75)90117-1View ArticleMathSciNetMATHGoogle Scholar - Brown RC, Hinton DB: Lyapunov inequalities and their applications. In
*Survey on Classical Inequalities, Math. Appl.*.*Volume 517*. Edited by: Rassias TM. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:1–25.View ArticleGoogle Scholar - Reid WT: A generalized Liapunov inequality.
*Journal of Differential Equations*1973, 13: 182–196. 10.1016/0022-0396(73)90040-5View ArticleMathSciNetMATHGoogle Scholar - Kametaka Y, Yamagishi H, Watanabe K, Nagai A, Takemura K: Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality.
*Scientiae Mathematicae Japonicae*2007, 65(3):333–359.MathSciNetMATHGoogle Scholar - Nagai A, Takemura K, Kametaka Y, Watanabe K, Yamagishi H: Green function for boundary value problem of
*2M*-th order linear ordinary differential equations with free boundary condition.*Far East Journal of Applied Mathematics*2007, 26(3):393–406.MathSciNetMATHGoogle Scholar - Kametaka Y, Watanabe K, Nagai A, Pyatkov S: The best constant of Sobolev inequality in an
*n*dimensional Euclidean space.*Scientiae Mathematicae Japonicae*2005, 61(1):15–23.MathSciNetMATHGoogle Scholar

## Copyright

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