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

  • Kohtaro Watanabe1Email author,

    Affiliated with

    • Yoshinori Kametaka2,

      Affiliated with

      • Hiroyuki Yamagishi3,

        Affiliated with

        • Atsushi Nagai4 and

          Affiliated with

          • Kazuo Takemura4

            Affiliated with

            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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq1_HTML.gif of the clamped boundary value problem for the differential operator http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq2_HTML.gif on the interval http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq3_HTML.gif is obtained. The best constant of corresponding Sobolev inequality is given by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq4_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq7_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq8_HTML.gif be a Sobolev (Hilbert) space associated with the inner product http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq9_HTML.gif :
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ1_HTML.gif
            (1.1)
            The fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq10_HTML.gif induces the equivalent norm to the standard norm of the Sobolev (Hilbert) space of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq11_HTML.gif th order follows from Poincaré inequality. Let us introduce the functional http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq12_HTML.gif as follows:
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ2_HTML.gif
            (1.2)
            To obtain the supremum of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq13_HTML.gif (i.e., the best constant of Sobolev inequality), we consider the following clamped boundary value problem:
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ3_HTML.gif
            Concerning the uniqueness and existence of the solution to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq176_HTML.gif , we have the following proposition. The result is expressed by the monomial http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq15_HTML.gif :
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ4_HTML.gif
            (1.3)

            Proposition 1.1.

            For any bounded continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq16_HTML.gif on an interval http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq176_HTML.gif has a unique classical solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq19_HTML.gif expressed by
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ5_HTML.gif
            (1.4)
            where Green's function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq20_HTML.gif is given by
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ6_HTML.gif
            (1.5)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ7_HTML.gif
            (1.6)

            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq21_HTML.gif is the determinant of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq22_HTML.gif matrix http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq23_HTML.gif    http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq25_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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.

            (i) The supremum http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq27_HTML.gif (abbreviated as http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq28_HTML.gif if there is no confusion) of the Sobolev functional http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq29_HTML.gif is given by
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ8_HTML.gif
            (1.7)
            Concretely,
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ9_HTML.gif
            (1.8)

            (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq30_HTML.gif is attained by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq31_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq32_HTML.gif .

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

            Corollary 1.3.

            Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq33_HTML.gif , then the best constant of Sobolev inequality (corresponding to the embedding of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq34_HTML.gif into http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq35_HTML.gif )
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ10_HTML.gif
            (1.9)

            is http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq36_HTML.gif . Moreover the best constant http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq37_HTML.gif is attained by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq38_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ11_HTML.gif
            (1.10)
            where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq40_HTML.gif . If the above equation has two points http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq42_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq43_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq44_HTML.gif , then these points are said to be conjugate. It is wellknown that if there exists a pair of conjugate points in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq45_HTML.gif , then the classical Lyapunov inequality
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ12_HTML.gif
            (1.11)
            holds, where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ13_HTML.gif
            (1.12)
            For this case, we again call two distinct points http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq47_HTML.gif and s2conjugate if there exists a nontrivial http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq48_HTML.gif solution of (1.12) satisfying
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq49_HTML.gif with respect to (1.12), then
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ15_HTML.gif
            (1.14)

            where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq50_HTML.gif is the best constant of the Sobolev inequality (1.9).

            Without introducing auxiliary equation http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ17_HTML.gif
            (1.16)
            Since
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq176_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq56_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq176_HTML.gif . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq58_HTML.gif has the following properties.

            Lemma 2.1.

            Consider the following:

            (1)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ19_HTML.gif
            (2.1)
            (2)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ20_HTML.gif
            (2.2)
            (3)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ21_HTML.gif
            (2.3)
            (4)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ22_HTML.gif
            (2.4)

            Proof.

            For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq61_HTML.gif , we have from (1.5)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ23_HTML.gif
            (2.5)
            For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq62_HTML.gif , noting the fact http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq63_HTML.gif , we have (1). Next, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq65_HTML.gif , we have from (2.5)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ24_HTML.gif
            (2.6)
            Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq66_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ25_HTML.gif
            (2.7)
            Note that subtracting the http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq67_HTML.gif th row from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq68_HTML.gif th row, the second equality holds. Equation http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq69_HTML.gif is shown by the same way. Hence, we have (2). For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq70_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ26_HTML.gif
            (2.8)

            where we used the fact http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq72_HTML.gif . So we have (3), and (4) follows from (3).

            Using Lemma 2.1, we prove that the functional space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq73_HTML.gif associated with inner norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq74_HTML.gif is a reproducing kernel Hilbert space.

            Lemma 2.2.

            For any http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq75_HTML.gif , one has the reproducing property
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ27_HTML.gif
            (2.9)

            Proof.

            For functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq77_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq78_HTML.gif arbitrarily fixed in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq79_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ28_HTML.gif
            (2.10)
            Integrating this with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq80_HTML.gif on intervals http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq82_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ30_HTML.gif
            (3.1)
            Note that the last equality holds from (2.9); that is, substituting (2.9), http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq83_HTML.gif . Let us assume that
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ31_HTML.gif
            (3.2)
            holds (this will be proved in the next section). From definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq84_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ32_HTML.gif
            (3.3)
            Substituting http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq85_HTML.gif in to the above inequality, we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ33_HTML.gif
            (3.4)
            Combining this and trivial inequality http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq86_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ34_HTML.gif
            (3.5)
            Hence, we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq87_HTML.gif . From Proposition 1.1, we have for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq88_HTML.gif
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ36_HTML.gif
            (4.1)

            Thus, we can expect that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq89_HTML.gif takes the form http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq90_HTML.gif . Precisely, we have the following proposition.

            Proposition 4.1.

            Consider
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ37_HTML.gif
            (4.2)
            Hence,
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ38_HTML.gif
            (4.3)

            where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq91_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq92_HTML.gif .

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

            Lemma 4.2.

            Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq93_HTML.gif , where
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ39_HTML.gif
            (4.4)
            ( http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq94_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq95_HTML.gif ), then it holds that
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ40_HTML.gif
            (4.5)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ41_HTML.gif
            (4.6)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ42_HTML.gif
            (4.7)

            Lemma 4.3.

            Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq96_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq97_HTML.gif , then it holds that (4.6) and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq98_HTML.gif .

            Proof of Proposition 4.1.

            From Lemmas 4.2 and 4.3, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq100_HTML.gif satisfy BVP http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq101_HTML.gif (in case of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq102_HTML.gif ). So we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ43_HTML.gif
            (4.8)
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ44_HTML.gif
            (4.9)

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

            Proof of Lemma 4.2.

            Let
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ45_HTML.gif
            (4.10)
            then differentiating http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq103_HTML.gif    http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq104_HTML.gif times we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ46_HTML.gif
            (4.11)
            At first, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq105_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ47_HTML.gif
            (4.12)
            The first term vanishes because
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ48_HTML.gif
            (4.13)
            The third term also vanishes because
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ49_HTML.gif
            (4.14)
            Thus, we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ50_HTML.gif
            (4.15)
            Hence, we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ51_HTML.gif
            (4.16)
            by which we obtain (4.5). Next, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq106_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ52_HTML.gif
            (4.17)
            Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq107_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq108_HTML.gif . Thus, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq109_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq110_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ53_HTML.gif
            (4.18)
            The first term vanishes because http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq111_HTML.gif . Next, we show that the second term also vanishes. Let
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ54_HTML.gif
            (4.19)
            Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq112_HTML.gif , two rows, including the last row, coincide, and hence we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq113_HTML.gif . Thus, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq114_HTML.gif . So we have obtained http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq115_HTML.gif . By the same argument, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq116_HTML.gif . Hence, we have (4.6). Finally, we will show (4.7). For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq117_HTML.gif , noting http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq118_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ55_HTML.gif
            (4.20)
            where
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ56_HTML.gif
            (4.21)
            Thus, we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq119_HTML.gif . Hence we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ57_HTML.gif
            (4.22)
            that is,
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ58_HTML.gif
            (4.23)

            This completes the proof of Lemma 4.2.

            Proof of Lemma 4.3.

            Let
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ59_HTML.gif
            (4.24)
            Differentiating http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq120_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq121_HTML.gif times, we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ60_HTML.gif
            (4.25)
            For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq122_HTML.gif , noting http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq124_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq125_HTML.gif , we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ61_HTML.gif
            (4.26)
            Thus, we have (4.5). If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq126_HTML.gif , then we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ62_HTML.gif
            (4.27)
            Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq127_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq128_HTML.gif . Hence, we have (4.6). If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_IEq129_HTML.gif , then we have
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-875057/MediaObjects/13661_2010_63_Article_Equ63_HTML.gif
            (4.28)

            This proves Lemma 4.3.

            Declarations

            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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            © KohtaroWatanabe et al 2011

            This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.