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Open Access Open Badges Research Article

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

Kohtaro Watanabe1*, Yoshinori Kametaka2, Hiroyuki Yamagishi3, Atsushi Nagai4 and Kazuo Takemura4

Author affiliations

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

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Citation and License

Boundary Value Problems 2011, 2011:875057  doi:10.1155/2011/875057

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/875057

Received:14 August 2010
Accepted:10 February 2011
Published:7 March 2011

© 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 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

For , , let be a Sobolev (Hilbert) space associated with the inner product :


The fact that induces the equivalent norm to the standard norm of the Sobolev (Hilbert) space of th order follows from Poincaré inequality. Let us introduce the functional as follows:


To obtain the supremum of (i.e., the best constant of Sobolev inequality), we consider the following clamped boundary value problem:


Concerning the uniqueness and existence of the solution to , we have the following proposition. The result is expressed by the monomial :


Proposition 1.1.

For any bounded continuous function on an interval , has a unique classical solution expressed by


where Green's function is given by



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.

Theorem 1.2.

(i) The supremum (abbreviated as if there is no confusion) of the Sobolev functional is given by




(ii) is attained by , that is, .

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

Corollary 1.3.

Let , then the best constant of Sobolev inequality (corresponding to the embedding of into )


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

Next, we introduce a connection between the best constant of Sobolev- and Lyapunov-type inequalities. Let us consider the second-order differential equation


where . If the above equation has two points and in satisfying , then these points are said to be conjugate. It is wellknown that if there exists a pair of conjugate points in , then the classical Lyapunov inequality


holds, where . Various extensions and improvements for the above result have been attempted; see, for example, Ha [1], Yang [2], and references there in. Among these extensions, Levin [3] and Das and Vatsala [4] extended the result for higher order equation


For this case, we again call two distinct points and s2conjugate 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.

If there exists a pair of conjugate points on with respect to (1.12), then


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.



In the second inequality, the equality holds for the function which attains the Sobolev best constant, so especially it is not a constant function. Thus, for this function, the first inequality is strict, and hence we obtain




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 [79], 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:










For and , , we have from (1.5)


For , noting the fact , we have (1). Next, for and , we have from (2.5)


Since , we have


Note that subtracting the th row from th row, the second equality holds. Equation is shown by the same way. Hence, we have (2). For , we have


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.

For any , one has the reproducing property



For functions and with arbitrarily fixed in , we have


Integrating this with respect to on intervals and , we have


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.

Applying Schwarz inequality to (2.9), we have


Note that the last equality holds from (2.9); that is, substituting (2.9), . Let us assume that


holds (this will be proved in the next section). From definition of , we have


Substituting in to the above inequality, we have


Combining this and trivial inequality , we have


Hence, we have


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

In this section, we consider the diagonal value of Green's function, that is, . From Proposition 1.1, we have for


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

Proposition 4.1.





where satisfy .

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

Lemma 4.2.

Let , where


( satisfy ), then it holds that




Lemma 4.3.

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

Proof of Proposition 4.1.

From Lemmas 4.2 and 4.3, and satisfy BVP (in case of ). So we have



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

Proof of Lemma 4.2.



then differentiating    times we have


At first, for , we have


The first term vanishes because


The third term also vanishes because


Thus, we have


Hence, we have


by which we obtain (4.5). Next, for , we have


Since , we have . Thus, we have . For , we have


The first term vanishes because . Next, we show that the second term also vanishes. Let


Since , two rows, including the last row, coincide, and hence we have . Thus, we have . So we have obtained . By the same argument, we have . Hence, we have (4.6). Finally, we will show (4.7). For , noting , we have




Thus, we obtain . Hence we have


that is,


This completes the proof of Lemma 4.2.

Proof of Lemma 4.3.



Differentiating times, we have


For , noting , , and , we have


Thus, we have (4.5). If , then we have


Since , we have . Hence, we have (4.6). If , then we have


This proves Lemma 4.3.


A. Deduction of (1.5)

In this section, (1.5) in Proposition 1.1 is deduced. Suppose that has a classical solution . Introducing the following notations:


is rewritten as


Let the fundamental solution be expressed as , where


then satisfy . satisfies the initial value problem , . is a unit matrix. Solving (A.2), we have


or equivalently, for , we have


Employing the boundary conditions (A.2), we have


In particular, if , then we have


On the other hand, using the boundary conditions (A.2) again, we have


Solving the above linear system of equations with respect to , , we have


Substituting (A.9) into (A.7), we have


Taking an average of the above two expressions and noting , we obtain (1.4), where Green's function is given by


Using properties , we have


where is Kronecker's delta defined by . Inserting these three relations into (A.11), we have


Applying the relation


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

B. Deduction of (1.6)

To prove (1.6), we show


Let . If (B.1) holds, substituting it to (1.5), replacing with , with , then we obtain (1.6). The case is shown in a similar way. Let be fixed, and let . Then satisfies


On the other hand, let


Differentiating times with respect to , we have


For , noticing , we have . For , we have


where we used . Similarly, for , we have . So satisfies


which is the same equation as (B.2). Hence, we have .


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