For

,

, let

be a Sobolev (Hilbert) space associated with the inner product

:

The fact that

induces the equivalent norm to the standard norm of the Sobolev (Hilbert) space of

th order follows from Poincaré inequality. Let us introduce the functional

as follows:

To obtain the supremum of

(i.e., the best constant of Sobolev inequality), we consider the following clamped boundary value problem:

Concerning the uniqueness and existence of the solution to

, we have the following proposition. The result is expressed by the monomial

:

Proposition 1.1.

For any bounded continuous function

on an interval

,

has a unique classical solution

expressed by

where Green's function

is given by

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.

Theorem 1.2.

(i) The supremum

(abbreviated as

if there is no confusion) of the Sobolev functional

is given by

(ii)
is attained by
, that is,
.

Clearly, Theorem 1.2(i), (ii) is rewritten equivalently as follows.

Corollary 1.3.

Let

, then the best constant of Sobolev inequality (corresponding to the embedding of

into

)

is
. Moreover the best constant
is attained by
, where
is an arbitrary complex number.

Next, we introduce a connection between the best constant of Sobolev- and Lyapunov-type inequalities. Let us consider the second-order differential equation

where

. If the above equation has two points

and

in

satisfying

, then these points are said to be

*conjugate*. It is wellknown that if there exists a pair of conjugate points in

, then the classical Lyapunov inequality

holds, where

. Various extensions and improvements for the above result have been attempted; see, for example, Ha [

1], Yang [

2], and references there in. Among these extensions, Levin [

3] and Das and Vatsala [

4] extended the result for higher order equation

For this case, we again call two distinct points

and s

_{2}*conjugate* if there exists a nontrivial

solution of (1.12) satisfying

We point out that the constant which appears in the generalized Lyapunov inequality by Levin [3] and Das and Vatsala [4] is the reverse of the Sobolev best embedding constant.

Corollary 1.4.

If there exists a pair of conjugate points on

with respect to (1.12), then

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.

In the second inequality, the equality holds for the function which attains the Sobolev best constant, so especially it is not a constant function. Thus, for this function, the first inequality is strict, and hence we obtain

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 [3] without proof; see Section 4 of Reid [6]. Later, Das and Vatsala [4] 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 [4]. 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.