The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem
© Kohtaro Watanabe et al. 2011
Received: 14 August 2010
Accepted: 10 February 2011
Published: 7 March 2011
Green's function of the clamped boundary value problem for the differential operator on the interval is obtained. The best constant of corresponding Sobolev inequality is given by . In addition, it is shown that a reverse of the Sobolev best constant is the one which appears in the generalized Lyapunov inequality by Das and Vatsala (1975).
is the determinant of matrix , , and .
With the aid of Proposition 1.1, we obtain the following theorem. The proof of Proposition 1.1 is shown in Appendices A and B.
- (i)The supremum (abbreviated as if there is no confusion) of the Sobolev functional is given by(1.7)
(ii) is attained by , that is, .
Clearly, Theorem 1.2(i), (ii) is rewritten equivalently as follows.
is . Moreover the best constant is attained by , where is an arbitrary complex number.
where is the best constant of the Sobolev inequality (1.9).
Without introducing auxiliary equation and the existence result of conjugate points as [2, 4], we can prove this corollary directly through the Sobolev inequality (the idea of the proof origins to Brown and Hinton [5, page 5]).
Proof of Corollary 1.4.
we obtain the result.
Here, at the end of this section, we would like to mention some remarks about (1.12). The generalized Lyapunov inequality of the form (1.14) was firstly obtained by Levin  without proof; see Section 4 of Reid . Later, Das and Vatsala  obtained the same inequality (1.14) by constructing Green's function for . The expression of the Green's function of Proposition 1.1 is different from that of . The expression of [4, Theorem 2.1] is given by some finite series of and on the other hand, the expression of Proposition 1.1 is by the determinant. This complements the results of [7–9], where the concrete expressions of Green's functions for the equation but different boundary conditions are given, and all of them are expressed by determinants of certain matrices as Proposition 1.1.
2. Reproducing Kernel
First we enumerate the properties of Green's function of . has the following properties.
Consider the following:
where we used the fact , . So we have (3), and (4) follows from (3).
Using Lemma 2.1, we prove that the functional space associated with inner norm is a reproducing kernel Hilbert space.
Using (1), (2), and (4) in Lemma 2.1, we have (2.9).
3. Sobolev Inequality
In this section, we give a proof of Theorem 1.2 and Corollary 1.3.
Proof of Theorem 1.2 and Corollary 1.3.
which completes the proof of Theorem 1.2 and Corollary 1.3.
Thus, all we have to do is to prove (3.2).
4. Diagonal Value of Green's Function
Thus, we can expect that takes the form . Precisely, we have the following proposition.
where satisfy .
To prove this proposition, we prepare the following two lemmas.
Let , where , then it holds that (4.6) and .
Proof of Proposition 4.1.
Inserting (4.9) into (4.8), we have Proposition 4.1.
Proof of Lemma 4.2.
This completes the proof of Lemma 4.2.
Proof of Lemma 4.3.
This proves Lemma 4.3.
A. Deduction of (1.5)
where is any regular matrix and and are any matrices, we have (1.5).
B. Deduction of (1.6)
which is the same equation as (B.2). Hence, we have .
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