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.
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
Theorem AIfis a nonzero solution of the following even-order partial differential equation:
whereand , with the boundary conditions
then the following inequality holds:
Theorem BIfis 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.
Then there exists a positive constantDsuch that for any , the Sobolev inequality
holds. Moreover, the best constantis as follows:
We give the first seven values of , , and in Table 1.
The first seven values of,and
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
Now, by using Lemma 5, we get for any , ,
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 7For any , we have
Proof Similar to the proofs given in  and , 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 .
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).
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