Abstract
In this paper, by using the best Sobolev constant method, we obtain some new Lyapunov-type inequalities for a class of even-order partial differential equations; the results of this paper are new which generalize and improve some early results in the literature.
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1 Introduction
It is well known that the Lyapunov inequality for the second-order linear differential equation
states that if , is a nonzero solution of (1) such that , then the following inequality holds:
and the constant 4 is sharp.
There have been many proofs and generalizations as well as improvements on this inequality. For example, the authors in [1–3] generalized the Lyapunov-type inequality to the partial differential equations or systems.
First let us recall some background and notations which are introduced in [1, 2].
Let A be a spherical shell for , i.e. for , where for and is the Euclidean norm. Denote , the unit sphere in with surface area
where is the gamma function. Then every has a unique representation of the form , where and . Therefore, for any , we have
In [1], Aktaş obtained the following results.
Theorem A If is a nonzero solution of the following even-order partial differential equation:
where and , with the boundary conditions
then the following inequality holds:
Theorem B If is a nonzero solution of (4) with the boundary conditions
then the following inequality holds:
In this paper, we generalize Theorem A and Theorem B to a more general class of even order partial differential equations. Moreover, as we shall see by the end of this paper, Theorem 1 improves Theorem A significantly.
2 Main results
Let us consider the following even-order partial differential equation:
where , , , and .
The main results of this paper are the following theorems.
Theorem 1 If is a nonzero solution of (9) satisfying boundary conditions (5), then the following inequality holds:
where is the Riemann zeta function.
Theorem 2 If is a nonzero solution of (9) satisfying boundary conditions (7), then the following inequality holds:
3 Proofs of theorems
For the proofs of Theorem 1 and Theorem 2, let us consider first the following ordinary even-order linear ordinary differential equation:
where , .
Proposition 3 If (12) has a nonzero solution satisfying the following boundary value conditions:
then the following inequality holds:
where is the Riemann zeta function: , .
Proposition 4 If (12) has a nonzero solution satisfying the following boundary value conditions:
then we have the following inequality:
In order to prove the above propositions, we need the following lemmas.
Lemma 5 ([[4], Proposition 2.1])
Let and
Then there exists a positive constant C such that, for any , the Sobolev inequality
holds. Moreover, the best constant is as follows:
Lemma 6 ([[5], Theorem 1.2 and Corollary 1.3])
Let and
Then there exists a positive constant D such that for any , the Sobolev inequality
holds. Moreover, the best constant is as follows:
We give the first seven values of , , and in Table 1.
Since the proof of Proposition 4 is similar to that of Proposition 3, we give only the proof of Proposition 3 below.
Proof of Proposition 3 Multiplying both sides of (12) by and integrating from a to b by parts and using the boundary value condition (13), we can obtain
This yields
Now, by using Lemma 5, we get for any , ,
and
Substituting (19) and (20) into (18), we obtain
Now by applying Hölder’s inequality, we get
Substituting (22) into (21) and by using the fact that is not a constant function, we obtain the following strict inequality:
Dividing both sides of (23) by , which can be proved to be positive by using the boundary value condition (13) and the assumption that , we obtain
This is equivalent to (14). Thus we finished the proof of Proposition 3. □
Lemma 7 For any , we have
Proof Similar to the proofs given in [1] and [2], we have
which implies that
□
Proof of Theorem 1 It follows from (14) and Lemma 7 that for any fixed , we have
which is (10). This finishes the proof of Theorem 1. □
The proof of Theorem 2 is similar to that of Theorem 1, so we omit it for simplicity.
Let us compare Theorem 1 and Theorem 2 with Theorem A and Theorem B. It is evident that Theorem 2 is a natural generalization of Theorem B. If we let , , , then (10) reduces to the following inequality:
Let us compare the right sides of inequalities (6) and (25): if we denote , then we have
since as . Table 2 gives the first eight values of .
From Table 2 we see that increases very quickly, so Theorem 1 improves Theorem A significantly even in the special case of (4).
References
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Watanabe K, Kametaka Y, Yamagishi H, Nagai A, Takemura K: The best constant of Sobolev inequality corresponding to clamped boundary value problem. Bound. Value Probl. 2011., 2011: Article ID 875057 10.1155/2011/875057
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Ji, T., Fan, J. On multivariate higher order Lyapunov-type inequalities. J Inequal Appl 2014, 503 (2014). https://doi.org/10.1186/1029-242X-2014-503
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DOI: https://doi.org/10.1186/1029-242X-2014-503