- Research Article
- Open Access

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

**2009**:874631

https://doi.org/10.1155/2009/874631

© KohtaroWatanabe et al. 2009

**Received:**25 June 2009**Accepted:**16 October 2009**Published:**11 November 2009

## Abstract

## Keywords

- Unique Solution
- Riemannian Manifold
- Arbitrary Element
- Sobolev Inequality
- Distributional Sense

## 1. Introduction

Let be a Sobolev space which consists of the functions whose derivatives up to vanish at , that is,

where denotes th derivative of in a distributional sense. The purpose of this paper is to investigate the best constant of Sobolev inequality

where and . The result is as follows.

Theorem 1.1.

For special values of , the solution of (1.6) can be written explicitly, and is expressed as follows.

Corollary 1.2.

The best constants and were recently obtained by Oshime [1]. This paper gives an alternative proof which simplifies the derivation process of and and computes further constant . To compute these constants, the following lemma with respect to the symmetrization of functions plays an important role.

Lemma 1.3.

Remark 1.4.

may fail to hold for (see (3.4)–(3.6)). Hence the problem to obtain for (essentially) still remains.

The existence of the maximizer of can be seen in the proof of Theorem 1.1, where we construct it concretely, but here we would like to see this briefly though the proof of the following lemma.

Lemma 1.5.

Assume that the assertion of Lemma 1.3 holds, then the maximizer of exists.

Proof.

From Lemma 1.3, we see that if the maximizer exists, it is the element of . So, we can restrict the definition domain of to . Since is convex and strongly closed (by Sobolev inequality) in , it is weakly closed. In addition, is weakly compact, so is also weakly compact. Moreover, is weakly lower-semicontinuous in , and hence attains its minimum in . This proves the lemma.

Finally, we introduce some studies related to the present paper. When (Hilbertian Sobolev space case), the best constants for the embeddings of into for various conditions were treated in Richardson [2], Kalyabin [3], and [4–8]; see also references of these literatures. On the other hand, for the case , few literature seems to be available. In [9], Kametaka, Oshime, Watanabe, Yamagishi, Nagai, and Takemura obtained the best constant of (1.2) when belongs to a subspace of which consists of periodic functions

as

where is a Bernoulli polynomial, and is an unique solution of the equation

in the interval . Moreover, in [1], Oshime obtained the best constant and . Other topics on this subject, especially the best constant of Sobolev inequalities on Riemannian manifolds, are seen in Hebey [10].

## 2. Proof of Theorem 1.1

First, we prepare the following lemma.

Lemma 2.1.

where is an arbitrary constant (later, in Lemma 2.3, one fixes the value of to satisfy (1.6)).

Proof.

By integration by parts, we obtain the result.

From Lemma 1.3, to obtain the best constant of (1.2), we can restrict the definition domain of the functional to the nonzero element of . Now, let , then from Lemma 2.1 and Hölder's inequality, we have for ,

So, we have

and the equality holds in (2.4) if and only if there exists satisfying

To confirm the existence of such , we use the following lemmas.

Lemma 2.2.

Proof.

Moreover, from the assumption, it holds that .

Lemma 2.3.

The solution of (1.6) uniquely exists.

Proof.

the assertion is proved.

Using Lemma 2.2 and 5, we obtain the following lemma.

Lemma 2.4.

Let be a solution of (1.6) (when ), then the solution of (2.5) belongs to for .

Proof.

First, we prove the case and . For simplicity, let us put . Note that in these cases is a continuous function on .

holds, so the integration of over the interval also vanishes. Hence, by Lemma 2.2, the solution of (2.5) belongs to . Properties and follow from the fact that is an even function and (2.12). So, we have proven .

Therefore, . This proves the case .

Proof of Theorem 1.1.

From Lemma 1.3, and the argument of this section, especially Lemma 2.4, . This proves Theorem 1.1.

Proof.

So, we can explicitly solve this equation with respect to . Substituting to (1.5), we obtain the result.

## 3. Proof of Lemma 1.3

Now, all we have to do is to prove Lemma 1.3.

Proof of Lemma 1.3.

- (ii)
There is the case

Then, again as case (i), we have and by (3.33), . In addition, . So, we have proven the case (ii)-(b). This completes the proof.

## Authors’ Affiliations

## References

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## Copyright

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