- Research Article
- Open Access
Symmetrization of Functions and the Best Constant of 1-DIM Sobolev Inequality
© KohtaroWatanabe et al. 2009
- Received: 25 June 2009
- Accepted: 16 October 2009
- Published: 11 November 2009
The best constants of Sobolev embedding of into for and are obtained. A lemma concerning the symmetrization of functions plays an important role in the proof.
- Unique Solution
- Riemannian Manifold
- Arbitrary Element
- Sobolev Inequality
- Distributional Sense
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.
For special values of , the solution of (1.6) can be written explicitly, and is expressed as follows.
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.
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.
Assume that the assertion of Lemma 1.3 holds, then the maximizer of exists.
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
where is a Bernoulli polynomial, and is an unique solution of the equation
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.
where is an arbitrary constant (later, in Lemma 2.3, one fixes the value of to satisfy (1.6)).
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.
exists in .
Moreover, from the assumption, it holds that .
The solution of (1.6) uniquely exists.
the assertion is proved.
Using Lemma 2.2 and 5, we obtain the following lemma.
Let be a solution of (1.6) (when ), then the solution of (2.5) belongs to for .
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.
So, we can explicitly solve this equation with respect to . Substituting to (1.5), we obtain the result.
Now, all we have to do is to prove Lemma 1.3.
Proof of Lemma 1.3.
There is the case
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.
- Oshime Y: On the best constant for Sobolev inequalities. Scientiae Mathematicae Japonicae 2008,68(3):333–344.MATHMathSciNetGoogle Scholar
- Richardson W: Steepest descent and the least for Sobolev's inequality. The Bulletin of the London Mathematical Society 1986,18(5):478–484. 10.1112/blms/18.5.478MATHMathSciNetView ArticleGoogle Scholar
- Kalyabin GA: Sharp constants in inequalities for intermediate derivatives (the Gabushin case). Functional Analysis and Its Applications 2004,38(3):184–191.MATHMathSciNetView ArticleGoogle Scholar
- Kametaka Y, Watanabe K, Nagai A, Pyatkov S: The best constant of Sobolev inequality in an dimensional Euclidean space. Scientiae Mathematicae Japonicae 2005,61(1):15–23.MATHMathSciNetGoogle 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.MATHMathSciNetGoogle Scholar
- Nagai A, Takemura K, Kametaka Y, Watanabe K, Yamagishi H: Green function for boundary value problem of -th order linear ordinary differential equations with free boundary condition. Far East Journal of Applied Mathematics 2007,26(3):393–406.MATHMathSciNetGoogle Scholar
- Kametaka Y, Yamagishi H, Watanabe K, Nagai A, Takemura K: The best constant of Sobolev inequality corresponding to Dirichlet boundary value problem for . Scientiae Mathematicae Japonicae 2008,68(3):299–311.MATHMathSciNetGoogle Scholar
- Watanabe K, Kametaka Y, Nagai A, Takemura K, Yamagishi H: The best constant of Sobolev inequality on a bounded interval. Journal of Mathematical Analysis and Applications 2008,340(1):699–706. 10.1016/j.jmaa.2007.08.054MATHMathSciNetView ArticleGoogle Scholar
- Kametaka Y, Oshime Y, Watanabe K, Yamagishi H, Nagai A, Takemura K: The best constant of Sobolev inequality corresponding to the periodic boundary value problem for Scientiae Mathematicae Japonicae 2007,66(2):169–181.MATHMathSciNetGoogle Scholar
- Hebey E: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics. Volume 5. New York University Courant Institute of Mathematical Sciences, New York, NY, USA; 1999:x+309.Google Scholar
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