Degenerate Anisotropic Differential Operators and Applications

  • Ravi Agarwal1,

    Affiliated with

    • Donal O'Regan2 and

      Affiliated with

      • Veli Shakhmurov3Email author

        Affiliated with

        Boundary Value Problems20112011:268032

        DOI: 10.1155/2011/268032

        Received: 2 December 2010

        Accepted: 18 January 2011

        Published: 23 February 2011

        Abstract

        The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banach-valued http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq1_HTML.gif spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. In the last section, some applications of the main results are given.

        1. Introduction and Notations

        It is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed as differential-operator equations (DOEs). As a result, many authors investigated PDEs as a result of single DOEs. DOEs in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq2_HTML.gif -valued (Hilbert space valued) function spaces have been studied extensively in the literature (see [114] and the references therein). Maximal regularity properties for higher-order degenerate anisotropic DOEs with constant coefficients and nondegenerate equations with variable coefficients were studied in [15, 16].

        The main aim of the present paper is to discuss the separability properties of BVPs for higher-order degenerate DOEs; that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq4_HTML.gif are weighted functions, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq6_HTML.gif are linear operators in a Banach Space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq7_HTML.gif . The above DOE is a generalized form of an elliptic equation. In fact, the special case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq9_HTML.gif reduces (1.1) to elliptic form.

        Note, the principal part of the corresponding differential operator is nonself-adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness, discreetness of the spectrum, and completeness of root elements of this operator are established.

        We prove that the corresponding differential operator is separable in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq10_HTML.gif ; that is, it has a bounded inverse from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq11_HTML.gif to the anisotropic weighted space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq12_HTML.gif . This fact allows us to derive some significant spectral properties of the differential operator. For the exposition of differential equations with bounded or unbounded operator coefficients in Banach-valued function spaces, we refer the reader to [8, 1525].

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq13_HTML.gif be a positive measurable weighted function on the region http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq14_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq15_HTML.gif denote the space of all strongly measurable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq16_HTML.gif -valued functions that are defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq17_HTML.gif with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ2_HTML.gif
        (1.2)

        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq18_HTML.gif , the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq19_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq20_HTML.gif .

        The weight http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq21_HTML.gif we will consider satisfies an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq22_HTML.gif condition; that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq24_HTML.gif if there is a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq25_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ3_HTML.gif
        (1.3)

        for all cubes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq26_HTML.gif .

        The Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq27_HTML.gif is called a UMD space if the Hilbert operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq28_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq30_HTML.gif (see, e.g., [26]). UMD spaces include, for example, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq32_HTML.gif spaces, and Lorentz spaces http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq35_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq36_HTML.gif be the set of complex numbers and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ4_HTML.gif
        (1.4)
        A linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq37_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq38_HTML.gif -positive in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq39_HTML.gif with bound http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq40_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq41_HTML.gif is dense on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq42_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ5_HTML.gif
        (1.5)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq44_HTML.gif is an identity operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq45_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq46_HTML.gif is the space of bounded linear operators in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq47_HTML.gif . Sometimes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq48_HTML.gif will be written as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq49_HTML.gif and denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq50_HTML.gif . It is known [27, Section 1.15.1] that there exists fractional powers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq51_HTML.gif of the sectorial operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq52_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq53_HTML.gif denote the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq54_HTML.gif with graphical norm
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ6_HTML.gif
        (1.6)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq56_HTML.gif be two Banach spaces. Now, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq58_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq59_HTML.gif will denote interpolation spaces obtained from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq60_HTML.gif by the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq61_HTML.gif method [27, Section 1.3.1].

        A set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq62_HTML.gif is called http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq63_HTML.gif -bounded (see [3, 25, 26]) if there is a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq64_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq67_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ7_HTML.gif
        (1.7)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq68_HTML.gif is a sequence of independent symmetric http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq69_HTML.gif -valued random variables on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq70_HTML.gif .

        The smallest http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq71_HTML.gif for which the above estimate holds is called an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq72_HTML.gif -bound of the collection http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq73_HTML.gif and is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq74_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq75_HTML.gif denote the Schwartz class, that is, the space of all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq76_HTML.gif -valued rapidly decreasing smooth functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq77_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq78_HTML.gif be the Fourier transformation. A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq79_HTML.gif is called a Fourier multiplier in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq80_HTML.gif if the map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq82_HTML.gif is well defined and extends to a bounded linear operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq83_HTML.gif . The set of all multipliers in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq84_HTML.gif will denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq85_HTML.gif .

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ8_HTML.gif
        (1.8)

        Definition 1.1.

        A Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq86_HTML.gif is said to be a space satisfying a multiplier condition if, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq87_HTML.gif , the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq88_HTML.gif -boundedness of the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq89_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq90_HTML.gif is a Fourier multiplier in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq91_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq92_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq93_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq94_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq95_HTML.gif be a multiplier function dependent on the parameter http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq96_HTML.gif . The uniform http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq97_HTML.gif -boundedness of the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq98_HTML.gif ; that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ9_HTML.gif
        (1.9)

        implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq99_HTML.gif is a uniform collection of Fourier multipliers.

        Definition 1.2.

        The http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq100_HTML.gif -positive operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq101_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq102_HTML.gif -positive in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq103_HTML.gif if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq104_HTML.gif such that the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq105_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq106_HTML.gif -bounded.

        A linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq107_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq108_HTML.gif -positive in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq109_HTML.gif uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq110_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq111_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq112_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq113_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq115_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq117_HTML.gif .

        The http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq118_HTML.gif -positive operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq120_HTML.gif is said to be uniformly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq121_HTML.gif -positive in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq122_HTML.gif if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq123_HTML.gif such that the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq124_HTML.gif is uniformly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq125_HTML.gif -bounded; that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ10_HTML.gif
        (1.10)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq126_HTML.gif denote the space of all compact operators from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq127_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq128_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq129_HTML.gif , it is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq130_HTML.gif .

        For two sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq132_HTML.gif of positive numbers, the expression http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq133_HTML.gif means that there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq135_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ11_HTML.gif
        (1.11)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq136_HTML.gif denote the space of all compact operators from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq137_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq138_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq139_HTML.gif , it is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq140_HTML.gif .

        Now, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq141_HTML.gif denotes the approximation numbers of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq142_HTML.gif (see, e.g., [27, Section 1.16.1]). Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ12_HTML.gif
        (1.12)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq144_HTML.gif be two Banach spaces and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq145_HTML.gif continuously and densely embedded into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq147_HTML.gif .

        We let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq148_HTML.gif denote the space of all functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq149_HTML.gif possessing generalized derivatives http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq150_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq151_HTML.gif with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ13_HTML.gif
        (1.13)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq152_HTML.gif . Consider the following weighted spaces of functions:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ14_HTML.gif
        (1.14)

        2. Background

        The embedding theorems play a key role in the perturbation theory of DOEs. For estimating lower order derivatives, we use following embedding theorems from [24].

        Theorem A1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq154_HTML.gif and suppose that the following conditions are satisfied:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq155_HTML.gif is a Banach space satisfying the multiplier condition with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq157_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq158_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq159_HTML.gif -positive operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq160_HTML.gif ,

        (3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq162_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq163_HTML.gif -tuples of nonnegative integer such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ15_HTML.gif
        (2.1)

        (4) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq164_HTML.gif is a region such that there exists a bounded linear extension operator from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq165_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq166_HTML.gif .

        Then, the embedding http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq167_HTML.gif is continuous. Moreover, for all positive number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq169_HTML.gif , the following estimate holds
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ16_HTML.gif
        (2.2)

        Theorem A2.

        Suppose that all conditions of Theorem A1 are satisfied. Moreover, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq171_HTML.gif be a bounded region and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq172_HTML.gif . Then, the embedding
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ17_HTML.gif
        (2.3)

        is compact.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq173_HTML.gif denote the closure of the linear span of the root vectors of the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq174_HTML.gif .

        From [18, Theorem  3.4.1], we have the following.

        Theorem A3.

        Assume that

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq175_HTML.gif is an UMD space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq176_HTML.gif is an operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq178_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq179_HTML.gif are non overlapping, differentiable arcs in the complex plane starting at the origin. Suppose that each of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq180_HTML.gif regions into which the planes are divided by these arcs is contained in an angular sector of opening less then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq181_HTML.gif ,

        (3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq182_HTML.gif is an integer so that the resolvent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq183_HTML.gif satisfies the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ18_HTML.gif
        (2.4)

        as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq184_HTML.gif along any of the arcs http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq185_HTML.gif .

        Then, the subspace http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq186_HTML.gif contains the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq187_HTML.gif .

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ19_HTML.gif
        (2.5)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ20_HTML.gif
        (2.6)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq188_HTML.gif denote the embedding operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq189_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq190_HTML.gif .

        From [15, Theorem 2.8], we have the following.

        Theorem A4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq192_HTML.gif be two Banach spaces possessing bases. Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ21_HTML.gif
        (2.7)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ22_HTML.gif
        (2.8)

        3. Statement of the Problem

        Consider the BVPs for the degenerate anisotropic DOE
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ23_HTML.gif
        (3.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ24_HTML.gif
        (3.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ25_HTML.gif
        (3.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq195_HTML.gif are complex numbers, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq196_HTML.gif are complex-valued functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq198_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq199_HTML.gif are linear operators in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq200_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq201_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq202_HTML.gif are such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ26_HTML.gif
        (3.4)

        A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq203_HTML.gif and satisfying (3.1) a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq204_HTML.gif is said to be solution of the problem (3.1)-(3.2).

        We say the problem (3.1)-(3.2) is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq205_HTML.gif -separable if for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq206_HTML.gif , there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq207_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq208_HTML.gif of the problem (3.1)-(3.2) and a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq209_HTML.gif depending only http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq210_HTML.gif such that the coercive estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ27_HTML.gif
        (3.5)

        holds.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq211_HTML.gif be a differential operator generated by problem (3.1)-(3.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq212_HTML.gif ; that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ28_HTML.gif
        (3.6)

        We say the problem (3.1)-(3.2) is Fredholm in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq213_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq214_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq215_HTML.gif is a conjugate of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq216_HTML.gif .

        Remark 3.1.

        Under the substitutions
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ29_HTML.gif
        (3.7)
        the spaces http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq217_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq218_HTML.gif are mapped isomorphically onto the weighted spaces http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq220_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ30_HTML.gif
        (3.8)
        Moreover, under the substitution (3.7) the problem (3.1)-(3.2) reduces to the nondegenerate BVP
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ31_HTML.gif
        (3.9)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ32_HTML.gif
        (3.10)
        By denoting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq221_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq223_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq224_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq225_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq226_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq227_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq228_HTML.gif again by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq229_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq231_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq232_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq233_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq234_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq236_HTML.gif , respectively, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ33_HTML.gif
        (3.11)

        4. BVPs for Partial DOE

        Let us first consider the BVP for the anisotropic type DOE with constant coefficients
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ34_HTML.gif
        (4.1)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ35_HTML.gif
        (4.2)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq237_HTML.gif are boundary conditions defined by (3.2), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq238_HTML.gif are complex numbers, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq239_HTML.gif is a complex parameter, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq240_HTML.gif is a linear operator in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq241_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq242_HTML.gif be the roots of the characteristic equations
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ36_HTML.gif
        (4.3)
        Now, let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ37_HTML.gif
        (4.4)

        By applying the trace theorem [27, Section 1.8.2], we have the following.

        Theorem A5.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq244_HTML.gif be integer numbers, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq246_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq247_HTML.gif . Then, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq248_HTML.gif , the transformations http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq249_HTML.gif are bounded linear from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq250_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq251_HTML.gif , and the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ38_HTML.gif
        (4.5)

        Proof.

        It is clear that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ39_HTML.gif
        (4.6)

        Then, by applying the trace theorem [27, Section 1.8.2] to the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq252_HTML.gif , we obtain the assertion.

        Condition 1.

        Assume that the following conditions are satisfied:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq253_HTML.gif is a Banach space satisfying the multiplier condition with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq254_HTML.gif and the weight function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq256_HTML.gif ;

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq257_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq258_HTML.gif -positive operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq259_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq260_HTML.gif ;

        (3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq261_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ40_HTML.gif
        (4.7)

        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq262_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq263_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq264_HTML.gif denote the operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq265_HTML.gif generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.

        Theorem A6.

        Let Condition 1 be satisfied. Then,
        1. (a)
          the problem (4.1) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq266_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq267_HTML.gif with sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq268_HTML.gif has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq269_HTML.gif that belongs to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq270_HTML.gif and the following coercive uniform estimate holds:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ41_HTML.gif
          (4.8)
           

        (b)the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq271_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq272_HTML.gif -positive in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq273_HTML.gif .

        From Theorems A5 and A6 we have.

        Theorem A7.

        Suppose that Condition 1 is satisfied. Then, for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq274_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq275_HTML.gif the problem (4.1) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq276_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq277_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq278_HTML.gif . Moreover, the following uniform coercive estimate holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ42_HTML.gif
        (4.9)
        Consider BVP (3.11). Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq279_HTML.gif be roots of the characteristic equations
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ43_HTML.gif
        (4.10)

        Condition 2.

        Suppose the following conditions are satisfied:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq280_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ44_HTML.gif
        (4.11)
        for
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ45_HTML.gif
        (4.12)

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq281_HTML.gif is a Banach space satisfying the multiplier condition with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq282_HTML.gif and the weighted function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq283_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq284_HTML.gif .

        Remark 4.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq285_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq286_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq287_HTML.gif are real-valued positive functions. Then, Condition 2 is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq288_HTML.gif .

        Consider the inhomogenous BVP (3.1)-(3.2); that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ46_HTML.gif
        (4.13)

        Lemma 4.2.

        Assume that Condition 2 is satisfied and the following hold:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq289_HTML.gif is a uniformly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq290_HTML.gif -positive operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq291_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq292_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq293_HTML.gif are continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq294_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq295_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq297_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq298_HTML.gif .

        Then, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq299_HTML.gif and for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq300_HTML.gif the following coercive uniform estimate holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ47_HTML.gif
        (4.14)

        for the solution of problem (4.13).

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq301_HTML.gif be regions covering http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq302_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq303_HTML.gif be a corresponding partition of unity; that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq304_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq305_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq306_HTML.gif . Now, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq307_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq308_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ48_HTML.gif
        (4.15)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ49_HTML.gif
        (4.16)
        here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq309_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq310_HTML.gif are boundary operators which orders less than http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq311_HTML.gif . Freezing the coefficients of (4.15), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ50_HTML.gif
        (4.17)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ51_HTML.gif
        (4.18)
        It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq312_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq313_HTML.gif on neighborhoods of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq314_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ52_HTML.gif
        (4.19)
        on neighborhoods of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq315_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq316_HTML.gif on other parts of the domains http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq317_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq318_HTML.gif are positive constants. Hence, the problems (4.17) are generated locally only on parts of the boundary. Then, by Theorem A7 problem (4.17) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq319_HTML.gif and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq320_HTML.gif the following coercive estimate holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ53_HTML.gif
        (4.20)
        From the representation of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq321_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq322_HTML.gif and in view of the boundedness of the coefficients, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ54_HTML.gif
        (4.21)
        Now, applying Theorem A1 and by using the smoothness of the coefficients of (4.16), (4.18) and choosing the diameters of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq323_HTML.gif so small, we see there is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq324_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq325_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ55_HTML.gif
        (4.22)
        Then, using Theorem A5 and using the smoothness of the coefficients of (4.16), (4.18), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ56_HTML.gif
        (4.23)
        Now, using Theorem A1, we get that there is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq327_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ57_HTML.gif
        (4.24)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ58_HTML.gif
        (4.25)
        Using the above estimates, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ59_HTML.gif
        (4.26)
        Consequently, from (4.22)–(4.26), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ60_HTML.gif
        (4.27)
        Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq328_HTML.gif from the above inequality, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ61_HTML.gif
        (4.28)

        Then, by using the equality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq329_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq330_HTML.gif and the above estimates, we get (4.14).

        Condition 3.

        Suppose that part (1.1) of Condition 1 is satisfied and that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq331_HTML.gif is a Banach space satisfying the multiplier condition with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq332_HTML.gif and the weighted function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq333_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq334_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq335_HTML.gif .

        Consider the problem (3.11). Reasoning as in the proof of Lemma 4.2, we obtain.

        Proposition 4.3.

        Assume Condition 3 hold and suppose that

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq336_HTML.gif is a uniformly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq337_HTML.gif -positive operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq338_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq339_HTML.gif , and that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq340_HTML.gif are continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq341_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq342_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq343_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq344_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq345_HTML.gif .

        Then, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq346_HTML.gif and for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq347_HTML.gif , the following coercive uniform estimate holds
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ62_HTML.gif
        (4.29)

        for the solution of problem (3.11).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq348_HTML.gif denote the operator generated by problem (3.11) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq349_HTML.gif ; that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ63_HTML.gif
        (4.30)

        Theorem 4.4.

        Assume that Condition 3 is satisfied and that the following hold:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq350_HTML.gif is a uniformly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq351_HTML.gif -positive operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq352_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq353_HTML.gif are continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq354_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq355_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq356_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq357_HTML.gif .

        Then, problem (3.11) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq358_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq359_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq360_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq361_HTML.gif with large enough http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq362_HTML.gif . Moreover, the following coercive uniform estimate holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ64_HTML.gif
        (4.31)

        Proof.

        By Proposition 4.3 for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq363_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ65_HTML.gif
        (4.32)
        It is clear that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ66_HTML.gif
        (4.33)
        Hence, by using the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq364_HTML.gif and applying Theorem A1, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ67_HTML.gif
        (4.34)
        From the above estimate, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ68_HTML.gif
        (4.35)
        The estimate (4.35) implies that problem (3.11) has a unique solution and that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq365_HTML.gif has a bounded inverse in its rank space. We need to show that this rank space coincides with the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq366_HTML.gif ; that is, we have to show that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq367_HTML.gif , there is a unique solution of the problem (3.11). We consider the smooth functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq368_HTML.gif with respect to a partition of unity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq369_HTML.gif on the region http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq370_HTML.gif that equals one on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq371_HTML.gif , where supp http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq372_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq373_HTML.gif . Let us construct for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq374_HTML.gif the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq375_HTML.gif that are defined on the regions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq376_HTML.gif and satisfying problem (3.11). The problem (3.11) can be expressed as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ69_HTML.gif
        (4.36)
        Consider operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq377_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq378_HTML.gif that are generated by the BVPs (4.17); that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ70_HTML.gif
        (4.37)
        By virtue of Theorem A6, the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq379_HTML.gif have inverses http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq380_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq381_HTML.gif and for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq382_HTML.gif . Moreover, the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq383_HTML.gif are bounded from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq384_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq385_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq386_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ71_HTML.gif
        (4.38)
        Extending http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq387_HTML.gif to zero outside of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq388_HTML.gif in the equalities (4.36), and using the substitutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq389_HTML.gif , we obtain the operator equations
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ72_HTML.gif
        (4.39)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq390_HTML.gif are bounded linear operators in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq391_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ73_HTML.gif
        (4.40)
        In fact, because of the smoothness of the coefficients of the expression http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq392_HTML.gif and from the estimate (4.38), for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq393_HTML.gif with sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq394_HTML.gif , there is a sufficiently small http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq395_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ74_HTML.gif
        (4.41)
        Moreover, from assumption (2.2) of Theorem 4.4 and Theorem A1 for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq396_HTML.gif , there is a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq397_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ75_HTML.gif
        (4.42)
        Hence, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq398_HTML.gif with sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq399_HTML.gif , there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq400_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq401_HTML.gif . Consequently, (4.39) for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq402_HTML.gif have a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq403_HTML.gif . Moreover,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ76_HTML.gif
        (4.43)
        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq404_HTML.gif are bounded linear operators from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq405_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq406_HTML.gif . Thus, the functions
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ77_HTML.gif
        (4.44)
        are solutions of (4.38). Consider the following linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq407_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq408_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ78_HTML.gif
        (4.45)
        It is clear from the constructions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq409_HTML.gif and from the estimate (4.39) that the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq410_HTML.gif are bounded linear from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq411_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq412_HTML.gif , and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq413_HTML.gif with sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq414_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ79_HTML.gif
        (4.46)
        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq415_HTML.gif is a bounded linear operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq416_HTML.gif . Since the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq417_HTML.gif coincide with the inverse of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq418_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq419_HTML.gif , then acting on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq420_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq421_HTML.gif gives
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ80_HTML.gif
        (4.47)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq422_HTML.gif are bounded linear operators defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ81_HTML.gif
        (4.48)
        Indeed, from Theorem A1 and estimate (4.46) and from the expression http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq423_HTML.gif , we obtain that the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq424_HTML.gif are bounded linear from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq425_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq426_HTML.gif , and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq427_HTML.gif with sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq428_HTML.gif , there is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq429_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq430_HTML.gif . Therefore, there exists a bounded linear invertible operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq431_HTML.gif ; that is, we infer for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq432_HTML.gif that the BVP (3.11) has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ82_HTML.gif
        (4.49)

        Result 1.

        Theorem 4.4 implies that the resolvent http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq433_HTML.gif satisfies the following anisotropic type sharp estimate:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ83_HTML.gif
        (4.50)

        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq434_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq435_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq436_HTML.gif denote the operator generated by BVP (3.1)-(3.2). From Theorem 4.4 and Remark 3.1, we get the following.

        Result 2.

        Assume all the conditions of Theorem 4.4 hold. Then,

        (a)the problem (3.1)-(3.2) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq437_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq438_HTML.gif and for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq439_HTML.gif has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq440_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq441_HTML.gif , and the following coercive uniform estimate holds
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ84_HTML.gif
        (4.51)

        (b)if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq442_HTML.gif , then the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq443_HTML.gif is Fredholm from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq444_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq445_HTML.gif .

        Example 4.5.

        Now, let us consider a special case of (3.1)-(3.2). Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq446_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq447_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq448_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq449_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq450_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq451_HTML.gif ; that is, consider the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ85_HTML.gif
        (4.52)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ86_HTML.gif
        (4.53)
        Theorem 4.4 implies that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq452_HTML.gif , problem (4.52) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq453_HTML.gif satisfying the following coercive estimate:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ87_HTML.gif
        (4.54)

        Example 4.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq454_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq455_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq456_HTML.gif are positive continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq457_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq458_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq459_HTML.gif is a diagonal matrix-function with continuous components http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq460_HTML.gif .

        Then, we obtain the separability of the following BVPs for the system of anisotropic PDEs with varying coefficients:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ88_HTML.gif
        (4.55)

        in the vector-valued space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq461_HTML.gif .

        5. The Spectral Properties of Anisotropic Differential Operators

        Consider the following degenerated BVP:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ89_HTML.gif
        (5.1)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ90_HTML.gif
        (5.2)

        Consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq462_HTML.gif generated by problem (5.1).

        Theorem 5.1.

        Let all the conditions of Theorem 4.4 hold for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq463_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq464_HTML.gif . Then, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq465_HTML.gif is Fredholm from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq466_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq467_HTML.gif .

        Proof.

        Theorem 4.4 implies that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq468_HTML.gif for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq469_HTML.gif has a bounded inverse http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq470_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq471_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq472_HTML.gif ; that is, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq473_HTML.gif is Fredholm from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq474_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq475_HTML.gif . Then, from Theorem A2 and the perturbation theory of linear operators, we obtain that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq476_HTML.gif is Fredholm from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq477_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq478_HTML.gif .

        Theorem 5.2.

        Suppose that all the conditions of Theorem 5.1 are satisfied with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq479_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq480_HTML.gif is a Banach space with a basis and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ91_HTML.gif
        (5.3)

        Then,

        (a)for a sufficiently large positive http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq481_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ92_HTML.gif
        (5.4)

        (b)the system of root functions of the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq482_HTML.gif is complete in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq483_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq484_HTML.gif denote the embedding operator from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq485_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq486_HTML.gif . From Result 2, there exists a resolvent operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq487_HTML.gif which is bounded from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq488_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq489_HTML.gif . Moreover, from Theorem A4 and Remark 3.1, we get that the embedding operator
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ93_HTML.gif
        (5.5)
        is compact and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ94_HTML.gif
        (5.6)
        It is clear that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ95_HTML.gif
        (5.7)
        Hence, from relations (5.6) and (5.7), we obtain (5.4). Now, Result 1 implies that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq490_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq491_HTML.gif is positive in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq492_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ96_HTML.gif
        (5.8)

        Then, from (4.52) and (5.6), we obtain assertion (b).

        Consider now the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq493_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq494_HTML.gif generated by the nondegenerate BVP obtained from (5.1) under the mapping (3.7); that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ97_HTML.gif
        (5.9)

        From Theorem 5.2 and Remark 3.1, we get the following.

        Result 3.

        Let all the conditions of Theorem 5.1 hold. Then, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq495_HTML.gif is Fredholm from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq496_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq497_HTML.gif .

        Result 4.

        Then,

        (a)for a sufficiently large positive http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq498_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ98_HTML.gif
        (5.10)

        (b)the system of root functions of the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq499_HTML.gif is complete in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq500_HTML.gif .

        6. BVPs for Degenerate Quasielliptic PDE

        In this section, maximal regularity properties of degenerate anisotropic differential equations are studied. Maximal regularity properties for PDEs have been studied, for example, in [3] for smooth domains and in [28] for nonsmooth domains.

        Consider the BVP
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ99_HTML.gif
        (6.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq501_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq502_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq503_HTML.gif are complex number, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq504_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ100_HTML.gif
        (6.2)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq505_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq506_HTML.gif . Now, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq507_HTML.gif will denote the space of all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq508_HTML.gif -summable scalar-valued functions with mixed norm (see, e.g., [29, Section 1, page 6]), that is, the space of all measurable functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq509_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq510_HTML.gif , for which
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ101_HTML.gif
        (6.3)

        Analogously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq511_HTML.gif denotes the Sobolev space with corresponding mixed norm.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq512_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq513_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq514_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq515_HTML.gif denote the roots of the equations
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ102_HTML.gif
        (6.4)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq516_HTML.gif denote the operator generated by BVP (6.1). Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ103_HTML.gif
        (6.5)

        Theorem 6.1.

        Let the following conditions be satisfied:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq517_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq518_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq519_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq520_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq521_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq522_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq523_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq524_HTML.gif ,

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq525_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq526_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq527_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq528_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq529_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq530_HTML.gif ,

        (3)for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq531_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq532_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq533_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq534_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq535_HTML.gif let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ104_HTML.gif
        (6.6)
        (4)for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq536_HTML.gif , the local BVPs in local coordinates corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq537_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ105_HTML.gif
        (6.7)

        has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq538_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq539_HTML.gif and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq540_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq541_HTML.gif ,

        (5) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq542_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq543_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ106_HTML.gif
        (6.8)

        Then,

        (a)the following coercive estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ107_HTML.gif
        (6.9)

        holds for the solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq544_HTML.gif of problem (6.1),

        (b)for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq545_HTML.gif and for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq546_HTML.gif , there exists a resolvent http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq547_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ108_HTML.gif
        (6.10)

        (c)the problem (6.1) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq548_HTML.gif is Fredholm in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq549_HTML.gif ,

        (d)the relation with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq550_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ109_HTML.gif
        (6.11)

        holds,

        (e)for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq551_HTML.gif the system of root functions of the BVP (6.1) is complete in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq552_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq553_HTML.gif . Then, from [3, Theorem 3.6], part (1.1) of Condition 1 is satisfied. Consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq554_HTML.gif which is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ110_HTML.gif
        (6.12)
        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq555_HTML.gif , we also consider operators
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ111_HTML.gif
        (6.13)
        The problem (6.1) can be rewritten as the form of (3.1)-(3.2), where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq556_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq557_HTML.gif are functions with values in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq558_HTML.gif . From [3, Theorem 8.2] problem
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ112_HTML.gif
        (6.14)
        has a unique solution for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq559_HTML.gif and arg http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq560_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq561_HTML.gif . Moreover, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq562_HTML.gif , generated by (5.8) is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq563_HTML.gif -positive in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq564_HTML.gif ; that is, part (2.2) of Condition 1 holds. From (2.2), (3.7), and by [29, Section 18], we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ113_HTML.gif
        (6.15)
        that is, all the conditions of Theorem 5.2 and Result 4 are fulfilled. As a result, we obtain assertion (a) and (b) of the theorem. Also, it is known (e.g., [27, Theorem 3.2.5, Section 4.10]) that the embedding http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq565_HTML.gif is compact and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ114_HTML.gif
        (6.16)

        Then, Results 3 and 4 imply assertions (c), (d), (e).

        7. Boundary Value Problems for Infinite Systems of Degenerate PDE

        Consider the infinity systems of BVP for the degenerate anisotropic PDE
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ115_HTML.gif
        (7.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq566_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq567_HTML.gif are complex-valued functions, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq568_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq569_HTML.gif are complex numbers. Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ116_HTML.gif
        (7.2)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq570_HTML.gif denote the operator in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq571_HTML.gif generated by problem (7.1). Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ117_HTML.gif
        (7.3)

        Theorem 7.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq572_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq573_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq574_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq575_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq576_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq577_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq578_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq579_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq580_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq581_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq582_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq583_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq584_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq585_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ118_HTML.gif
        (7.4)

        Then,

        (a)for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq586_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq587_HTML.gif and sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq588_HTML.gif , the problem (7.1) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq589_HTML.gif that belongs to the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq590_HTML.gif and the following coercive estimate holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ119_HTML.gif
        (7.5)
        (b)there exists a resolvent http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq591_HTML.gif of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq592_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ120_HTML.gif
        (7.6)

        (c)for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq593_HTML.gif , the system of root functions of the BVP (7.1) is complete in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq594_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq595_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq596_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq597_HTML.gif be infinite matrices such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ121_HTML.gif
        (7.7)
        It is clear that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq598_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq599_HTML.gif -positive in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq600_HTML.gif . The problem (7.1) can be rewritten in the form (1.1). From Theorem 4.4, we obtain that problem (7.1) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq601_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq602_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq603_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ122_HTML.gif
        (7.8)

        From the above estimate, we obtain assertions (a) and (b). The assertion (c) is obtained from Result 4.

        Authors’ Affiliations

        (1)
        Department of Mathematical Sciences, Florida Institute of Technology
        (2)
        Department of Mathematics, National University of Ireland
        (3)
        Department of Electronics Engineering and Communication, Okan University, Akfirat, Tuzla

        References

        1. Agranovich MS:Spectral problems in Lipschitz domains for strongly elliptic systems in the Banach spaces http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq604_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq605_HTML.gif . Functional Analysis and Its Applications 2008, 42(4):249-267. 10.1007/s10688-008-0039-xMathSciNetView Article
        2. Ashyralyev A: On well-posedness of the nonlocal boundary value problems for elliptic equations. Numerical Functional Analysis and Optimization 2003, 24(1-2):1-15. 10.1081/NFA-120020240MathSciNetView Article
        3. Denk R, Hieber M, Prüss J: R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc.. American Mathematical Society; 2003.
        4. Favini A, Shakhmurov V, Yakubov Y: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup Forum 2009, 79(1):22-54. 10.1007/s00233-009-9138-0MathSciNetView Article
        5. Gorbachuk VI, Gorbachuk ML: Boundary Value Problems for Differential-Operator Equations. "Naukova Dumka", Kiev, Ukraine; 1984:284.
        6. Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245.
        7. Lions J-L, Magenes E: Nonhomogenous Boundary Value Problems. Mir, Moscow, Russia; 1971.
        8. Shklyar AYa: Complete Second Order Linear Differential Equations in Hilbert Spaces, Operator Theory: Advances and Applications. Volume 92. Birkhäuser, Basel, Switzerland; 1997:xii+219.
        9. Shakhmurov VB: Nonlinear abstract boundary-value problems in vector-valued function spaces and applications. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3):745-762. 10.1016/j.na.2006.06.027MathSciNetView Article
        10. Shakhmurov VB: Theorems on the embedding of abstract function spaces and their applications. Matematicheskiĭ Sbornik. Novaya Seriya 1987, 134(176)(2):260-273.
        11. Shakhmurov VB: Embedding theorems and their applications to degenerate equations. Differentsial'nye Uravneniya 1988, 24(4):672-682.MathSciNet
        12. Yakubov S: Completeness of Root Functions of Regular Differential Operators, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 71. Longman Scientific & Technical, Harlow, UK; 1994:x+245.
        13. Yakubov S: A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integral Equations and Operator Theory 1999, 35(4):485-506. 10.1007/BF01228044MathSciNetView Article
        14. Yakubov S, Yakubov Y: Differential-Operator Equations: Ordinary and Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Volume 103. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xxvi+541.
        15. Agarwal RP, Bohner R, Shakhmurov VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Boundary Value Problems 2005, 2005(1):9-42. 10.1155/BVP.2005.9MathSciNetView Article
        16. Shakhmurov VB: Separable anisotropic differential operators and applications. Journal of Mathematical Analysis and Applications 2007, 327(2):1182-1201. 10.1016/j.jmaa.2006.05.007MathSciNetView Article
        17. Amann H: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Mathematische Nachrichten 1997, 186: 5-56.MathSciNetView Article
        18. Amann H: Linear and Quasi-Linear Equations. Birkhauser; 1995.
        19. Krein SG: Linear Differential Equations in Banach Space. American Mathematical Society, Providence, RI, USA; 1971.
        20. Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser, Basel, Switzerland; 1995:xviii+424.
        21. Orlov VP: Regular degenerate differential operators of arbitrary order with unbounded operators coefficients. Proceedings of Voronej State University 1974, 2: 33-41.
        22. Sobolevskiĭ PE: Inequalities coerciveness for abstract parabolic equations. Doklady Akademii Nauk SSSR 1964, 57: 27-40.
        23. Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. Journal of Mathematical Analysis and Applications 2004, 292(2):605-620. 10.1016/j.jmaa.2003.12.032MathSciNetView Article
        24. Shakhmurov VB: Degenerate differential operators with parameters. Abstract and Applied Analysis 2007, 2007:-27.
        25. Weis Lutz:Operator-valued Fourier multiplier theorems and maximal http://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq606_HTML.gif -regularity. Mathematische Annalen 2001, 319(4):735-758. 10.1007/PL00004457MathSciNetView Article
        26. Burkholder DL: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II, 1981, Chicago, Ill, USA, Wadsworth Math. Ser. 270-286.
        27. Triebel H: Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library. Volume 18. North-Holland, Amsterdam, The Netherlands; 1978:528.
        28. Grisvard P: Elliptic Problems in Non Smooth Domains. Pitman; 1985.
        29. Besov OV, Ilin VP, Nikolskii SM: Integral Representations of Functions and Embedding Theorems. Wiley, New York, NY, USA; 1978.

        Copyright

        © Ravi Agarwal 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.