- Research Article
- Open Access
© KohtaroWatanabe et al. 2009
- Received: 25 June 2009
- Accepted: 16 October 2009
- Published: 11 November 2009
- Unique Solution
- Riemannian Manifold
- Arbitrary Element
- Sobolev Inequality
- Distributional Sense
The best constants and were recently obtained by Oshime . 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.
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 , Kalyabin , and [4–8]; see also references of these literatures. On the other hand, for the case , few literature seems to be available. In , 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
in the interval . Moreover, in , 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 .
First, we prepare the following lemma.
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
the assertion is proved.
Using Lemma 2.2 and 5, we obtain the following lemma.
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 .
Proof of Theorem 1.1.
Now, all we have to do is to prove Lemma 1.3.
Proof of Lemma 1.3.
There is the case
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