Open Access

The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem

  • Kohtaro Watanabe1Email author,
  • Yoshinori Kametaka2,
  • Hiroyuki Yamagishi3,
  • Atsushi Nagai4 and
  • Kazuo Takemura4
Boundary Value Problems20112011:875057

DOI: 10.1186/1687-2770-2011-875057

Received: 14 August 2010

Accepted: 10 February 2011

Published: 7 March 2011

Abstract

Green's function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq1_HTML.gif of the clamped boundary value problem for the differential operator https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq2_HTML.gif on the interval https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq3_HTML.gif is obtained. The best constant of corresponding Sobolev inequality is given by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq4_HTML.gif https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq5_HTML.gif . 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).

1. Introduction

For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq7_HTML.gif , let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq8_HTML.gif be a Sobolev (Hilbert) space associated with the inner product https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq9_HTML.gif :
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ1_HTML.gif
(1.1)
The fact that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq10_HTML.gif induces the equivalent norm to the standard norm of the Sobolev (Hilbert) space of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq11_HTML.gif th order follows from Poincaré inequality. Let us introduce the functional https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq12_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ2_HTML.gif
(1.2)
To obtain the supremum of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq13_HTML.gif (i.e., the best constant of Sobolev inequality), we consider the following clamped boundary value problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ3_HTML.gif
(BVPM)
Concerning the uniqueness and existence of the solution to , we have the following proposition. The result is expressed by the monomial https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq15_HTML.gif :
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ4_HTML.gif
(1.3)

Proposition 1.1.

For any bounded continuous function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq16_HTML.gif on an interval https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq17_HTML.gif , has a unique classical solution https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq19_HTML.gif expressed by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ5_HTML.gif
(1.4)
where Green's function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq20_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ6_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ7_HTML.gif
(1.6)

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq21_HTML.gif is the determinant of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq22_HTML.gif matrix https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq23_HTML.gif    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq25_HTML.gif , and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq26_HTML.gif .

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.
  1. (i)
    The supremum https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq27_HTML.gif (abbreviated as https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq28_HTML.gif if there is no confusion) of the Sobolev functional https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq29_HTML.gif is given by
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ8_HTML.gif
    (1.7)
     
Concretely,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ9_HTML.gif
(1.8)

(ii) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq30_HTML.gif is attained by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq31_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq32_HTML.gif .

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

Corollary 1.3.

Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq33_HTML.gif , then the best constant of Sobolev inequality (corresponding to the embedding of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq34_HTML.gif into https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq35_HTML.gif )
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ10_HTML.gif
(1.9)

is https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq36_HTML.gif . Moreover the best constant https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq37_HTML.gif is attained by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq38_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq39_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ11_HTML.gif
(1.10)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq40_HTML.gif . If the above equation has two points https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq42_HTML.gif in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq43_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq44_HTML.gif , then these points are said to be conjugate. It is wellknown that if there exists a pair of conjugate points in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq45_HTML.gif , then the classical Lyapunov inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ12_HTML.gif
(1.11)
holds, where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq46_HTML.gif . 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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ13_HTML.gif
(1.12)
For this case, we again call two distinct points https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq47_HTML.gif and s2conjugate if there exists a nontrivial https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq48_HTML.gif solution of (1.12) satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ14_HTML.gif
(1.13)

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq49_HTML.gif with respect to (1.12), then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ15_HTML.gif
(1.14)

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq50_HTML.gif is the best constant of the Sobolev inequality (1.9).

Without introducing auxiliary equation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq51_HTML.gif 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.

Consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ16_HTML.gif
(1.15)
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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ17_HTML.gif
(1.16)
Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ18_HTML.gif
(1.17)

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq54_HTML.gif on the other hand, the expression of Proposition 1.1 is by the determinant. This complements the results of [79], where the concrete expressions of Green's functions for the equation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq55_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq56_HTML.gif of . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq58_HTML.gif has the following properties.

Lemma 2.1.

Consider the following:

(1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ19_HTML.gif
(2.1)
(2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ20_HTML.gif
(2.2)
(3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ21_HTML.gif
(2.3)
(4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ22_HTML.gif
(2.4)

Proof.

For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq61_HTML.gif , we have from (1.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ23_HTML.gif
(2.5)
For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq62_HTML.gif , noting the fact https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq63_HTML.gif , we have (1). Next, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq65_HTML.gif , we have from (2.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ24_HTML.gif
(2.6)
Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq66_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ25_HTML.gif
(2.7)
Note that subtracting the https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq67_HTML.gif th row from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq68_HTML.gif th row, the second equality holds. Equation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq69_HTML.gif is shown by the same way. Hence, we have (2). For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq70_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ26_HTML.gif
(2.8)

where we used the fact https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq72_HTML.gif . So we have (3), and (4) follows from (3).

Using Lemma 2.1, we prove that the functional space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq73_HTML.gif associated with inner norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq74_HTML.gif is a reproducing kernel Hilbert space.

Lemma 2.2.

For any https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq75_HTML.gif , one has the reproducing property
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ27_HTML.gif
(2.9)

Proof.

For functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq77_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq78_HTML.gif arbitrarily fixed in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq79_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ28_HTML.gif
(2.10)
Integrating this with respect to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq80_HTML.gif on intervals https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq82_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ29_HTML.gif
(2.11)

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.

Applying Schwarz inequality to (2.9), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ30_HTML.gif
(3.1)
Note that the last equality holds from (2.9); that is, substituting (2.9), https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq83_HTML.gif . Let us assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ31_HTML.gif
(3.2)
holds (this will be proved in the next section). From definition of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq84_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ32_HTML.gif
(3.3)
Substituting https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq85_HTML.gif in to the above inequality, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ33_HTML.gif
(3.4)
Combining this and trivial inequality https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq86_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ34_HTML.gif
(3.5)
Hence, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ35_HTML.gif
(3.6)

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

In this section, we consider the diagonal value of Green's function, that is, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq87_HTML.gif . From Proposition 1.1, we have for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq88_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ36_HTML.gif
(4.1)

Thus, we can expect that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq89_HTML.gif takes the form https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq90_HTML.gif . Precisely, we have the following proposition.

Proposition 4.1.

Consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ37_HTML.gif
(4.2)
Hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ38_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq91_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq92_HTML.gif .

To prove this proposition, we prepare the following two lemmas.

Lemma 4.2.

Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq93_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ39_HTML.gif
(4.4)
( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq94_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq95_HTML.gif ), then it holds that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ40_HTML.gif
(4.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ41_HTML.gif
(4.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ42_HTML.gif
(4.7)

Lemma 4.3.

Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq96_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq97_HTML.gif , then it holds that (4.6) and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq98_HTML.gif .

Proof of Proposition 4.1.

From Lemmas 4.2 and 4.3, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq100_HTML.gif satisfy BVP https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq101_HTML.gif (in case of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq102_HTML.gif ). So we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ43_HTML.gif
(4.8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ44_HTML.gif
(4.9)

Inserting (4.9) into (4.8), we have Proposition 4.1.

Proof of Lemma 4.2.

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ45_HTML.gif
(4.10)
then differentiating https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq103_HTML.gif    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq104_HTML.gif times we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ46_HTML.gif
(4.11)
At first, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq105_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ47_HTML.gif
(4.12)
The first term vanishes because
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ48_HTML.gif
(4.13)
The third term also vanishes because
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ49_HTML.gif
(4.14)
Thus, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ50_HTML.gif
(4.15)
Hence, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ51_HTML.gif
(4.16)
by which we obtain (4.5). Next, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq106_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ52_HTML.gif
(4.17)
Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq107_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq108_HTML.gif . Thus, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq109_HTML.gif . For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq110_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ53_HTML.gif
(4.18)
The first term vanishes because https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq111_HTML.gif . Next, we show that the second term also vanishes. Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ54_HTML.gif
(4.19)
Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq112_HTML.gif , two rows, including the last row, coincide, and hence we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq113_HTML.gif . Thus, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq114_HTML.gif . So we have obtained https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq115_HTML.gif . By the same argument, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq116_HTML.gif . Hence, we have (4.6). Finally, we will show (4.7). For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq117_HTML.gif , noting https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq118_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ55_HTML.gif
(4.20)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ56_HTML.gif
(4.21)
Thus, we obtain https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq119_HTML.gif . Hence we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ57_HTML.gif
(4.22)
that is,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ58_HTML.gif
(4.23)

This completes the proof of Lemma 4.2.

Proof of Lemma 4.3.

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ59_HTML.gif
(4.24)
Differentiating https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq120_HTML.gif https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq121_HTML.gif times, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ60_HTML.gif
(4.25)
For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq122_HTML.gif , noting https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq123_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq124_HTML.gif , and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq125_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ61_HTML.gif
(4.26)
Thus, we have (4.5). If https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq126_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ62_HTML.gif
(4.27)
Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq127_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq128_HTML.gif . Hence, we have (4.6). If https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq129_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ63_HTML.gif
(4.28)

This proves Lemma 4.3.

Appendices

A. Deduction of (1.5)

In this section, (1.5) in Proposition 1.1 is deduced. Suppose that has a classical solution https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq131_HTML.gif . Introducing the following notations:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ64_HTML.gif
(A1)
is rewritten as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ65_HTML.gif
(A2)
Let the fundamental solution https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq133_HTML.gif be expressed as https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq134_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ66_HTML.gif
(A3)
then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq135_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq136_HTML.gif . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq137_HTML.gif satisfies the initial value problem https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq139_HTML.gif . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq140_HTML.gif is a unit matrix. Solving (A.2), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ67_HTML.gif
(A4)
or equivalently, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq141_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ68_HTML.gif
(A5)
Employing the boundary conditions (A.2), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ69_HTML.gif
(A6)
In particular, if https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq142_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ70_HTML.gif
(A7)
On the other hand, using the boundary conditions (A.2) again, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ71_HTML.gif
(A8)
Solving the above linear system of equations with respect to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq143_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq144_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ72_HTML.gif
(A9)
Substituting (A.9) into (A.7), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ73_HTML.gif
(A10)
Taking an average of the above two expressions and noting https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq145_HTML.gif , we obtain (1.4), where Green's function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq146_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ74_HTML.gif
(A11)
Using properties https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq147_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ75_HTML.gif
(A12)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq148_HTML.gif is Kronecker's delta defined by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq149_HTML.gif . Inserting these three relations into (A.11), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ76_HTML.gif
(A13)
Applying the relation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ77_HTML.gif
(A14)

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq150_HTML.gif is any https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq151_HTML.gif regular matrix and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq153_HTML.gif are any https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq154_HTML.gif matrices, we have (1.5).

B. Deduction of (1.6)

To prove (1.6), we show
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ78_HTML.gif
(B1)
Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq155_HTML.gif . If (B.1) holds, substituting it to (1.5), replacing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq156_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq157_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq158_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq159_HTML.gif , then we obtain (1.6). The case https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq160_HTML.gif is shown in a similar way. Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq161_HTML.gif https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq162_HTML.gif be fixed, and let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq163_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq164_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ79_HTML.gif
(B2)
On the other hand, let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ80_HTML.gif
(B3)
Differentiating https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq165_HTML.gif times with respect to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq166_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ81_HTML.gif
(B4)
For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq167_HTML.gif , noticing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq168_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq169_HTML.gif . For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq170_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ82_HTML.gif
(B5)
where we used https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq171_HTML.gif . Similarly, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq172_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq173_HTML.gif . So https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq174_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_Equ83_HTML.gif
(B6)

which is the same equation as (B.2). Hence, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_Article_63_IEq175_HTML.gif .

Authors’ Affiliations

(1)
Department of Computer Science, National Defense Academy
(2)
Division of Mathematical Sciences, Graduate School of Engineering Science, Osaka University
(3)
Tokyo Metropolitan College of Industrial Technology
(4)
Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University

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© Kohtaro Watanabe et al. 2011

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