On a Mixed Problem for a Constant Coefficient Second-Order System

Boundary Value Problems20102010:526917

DOI: 10.1155/2010/526917

Received: 2 July 2010

Accepted: 1 December 2010

Published: 15 December 2010

Abstract

The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq1_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq2_HTML.gif is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq3_HTML.gif and the time-variable belongs to the bounded interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq4_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq5_HTML.gif sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq6_HTML.gif , by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.

1. Introduction

Consider the second-order linear differential operator
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ1_HTML.gif
(1.1)

The coefficients of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq7_HTML.gif satisfy the following assumptions:

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq8_HTML.gif is a positive real number;

(ii) for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq10_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq11_HTML.gif symmetric matrices with real entries;

(iii) for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq12_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq13_HTML.gif is a positive constant;

(iv) for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq14_HTML.gif ; in addition, there exist two positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq16_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq17_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq18_HTML.gif ;

(v) for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq19_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq20_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq21_HTML.gif positive constant.

We will denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq22_HTML.gif a point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq23_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq24_HTML.gif the first http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq25_HTML.gif coordinates of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq26_HTML.gif , and by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq27_HTML.gif the time variable.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq28_HTML.gif be a positive real number, and denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq29_HTML.gif the subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq31_HTML.gif . In the first section of the paper we will be concerned with the following boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq32_HTML.gif is the unknown vector-valued function, whereas http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq34_HTML.gif are given functions, which take values in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq35_HTML.gif and are defined in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq37_HTML.gif , respectively.

Under suitable assumptions on the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq39_HTML.gif and on the coefficients of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq40_HTML.gif , we will prove that, in the case where the positive real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq41_HTML.gif is sufficiently small, there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq42_HTML.gif , which provides a solution to the boundary value problem (1.2). The existence of the solution is established by means of the techniques applied in [1] to prove that the initial value problem for the system http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq43_HTML.gif , admits a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq44_HTML.gif : the main result of [1] states that if the assumptions (i)–(iv) listed above along with other suitable conditions are fulfilled (see Proposition 3.1), then the adjoint operator of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq45_HTML.gif satisfies a priori estimates, which allow proving the existence of the solution to the Cauchy problem, through the definition of a suitable functional and a duality argument. As we will explain below, the boundary value problem (1.2) can be regarded exactly as an initial value problem. For this reason, the result of Section 2 does not represent any significant advance with respect to the results proved in [1].

The main novelty of the paper is represented by the study of a mixed problem in the third section. The interest in this kind of problems relies on the fact that they appear frequently as physical models: mixed problems for second-order hyperbolic equations and systems of equations occur in the theory of sound to describe for instance the evolution of the air pressure inside a room where noise is produced, as well as in the electromagnetism to describe the evolution of the electromagnetic field in some region of space (the system of Maxwell equations accounts for this kind of phenomenon).

The existence results, stated both for the initial value problem in [1] and for the boundary value problem in Section 2 of this paper, turn out to be the backbone in proving the existence of the solution to the following mixed problem in the strip http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq46_HTML.gif , as the time variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq47_HTML.gif belongs to the bounded interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq48_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq49_HTML.gif is a suitable positive real number:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ3_HTML.gif
(1.3)

We will assume that the vector-valued function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq50_HTML.gif belongs to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq51_HTML.gif , while, as for the initial data, we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq53_HTML.gif . Let us notice that in the problem (1.3) an initial value for the unknown vector field http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq54_HTML.gif and a Dirichlet boundary condition only are prescribed. Due to the a priori estimates that we will derive for the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq55_HTML.gif , it is not required in problem (1.3), in contrast with classical mixed problems for hyperbolic second-order systems, that the first-order derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq56_HTML.gif satisfy a prescribed condition at the boundary of the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq57_HTML.gif . This lack of information about the initial value of the first-order derivatives results in a possible nonuniqueness of the solution to (1.3). The existence result of Section 3 will be achieved by means of the definition of two mixed problems related to (1.3): the existence of the solution to the former will be established similarly to the result obtained in [1], while the latter will be studied like the boundary value problem considered in Section 2. Subsequently, thanks to the linearity of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq58_HTML.gif , a solution to (1.3) belonging to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq59_HTML.gif will be determined by superposition of the solutions to the preliminary mixed problems.

2. Boundary Value Problem

By adopting the same strategy of [1] to prove the existence of the solution to the initial value problem, let us determine the existence of a solution to (1.2) through a duality argument, by proving energy estimates.

Let us denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq60_HTML.gif the adjoint operator of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq61_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ4_HTML.gif
(2.1)

For all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq62_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq63_HTML.gif be the norms of the matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq64_HTML.gif , respectively.

Proposition 2.1.

Consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq65_HTML.gif defined in (1.1) and the corresponding adjoint http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq66_HTML.gif . Let the conditions (i)–(iv), listed in the Introduction, be fulfilled. In addition, assume the sums http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq69_HTML.gif , and for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq71_HTML.gif to be positive real numbers. Moreover, denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq72_HTML.gif the sum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq73_HTML.gif and suppose that, as long as the positive number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq74_HTML.gif is sufficiently small, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq75_HTML.gif .

Define the linear space
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ5_HTML.gif
(2.2)
Then, for all functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq76_HTML.gif , the following estimates hold:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ6_HTML.gif
(2.3)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ7_HTML.gif
(2.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq77_HTML.gif are suitable positive constants.

Proof.

Consider a vector function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq78_HTML.gif , with compact support in the subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq79_HTML.gif . Applying the Fourier transform with respect to both the tangential variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq80_HTML.gif and the time variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq81_HTML.gif , we can obtain a priori estimates, which, by substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq82_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq83_HTML.gif and carrying out similar calculations, turn out to be like the estimates obtained in [1] for the initial value problem. Subsequently, assuming the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq84_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq85_HTML.gif , we deduce the a priori estimates (2.3) and (2.4) for the adjoint operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq86_HTML.gif .

Due to (2.3) and (2.4), by means of a duality argument, we can prove the existence result for the solution to the boundary value problem (1.2).

Proposition 2.2.

Consider the boundary value problem (1.2), and let the assumptions of Proposition 2.1 be satisfied. Furthermore, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq88_HTML.gif . Then, the boundary value problem (1.2) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq89_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq90_HTML.gif .

Proof.

The result can be proved by means of the same tools used to establish the existence result for the initial value problem in [1]. For the sake of completeness, let us sketch the proof. Because of the a priori estimate (2.3), the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq91_HTML.gif is one-to-one on the linear space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq92_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq93_HTML.gif , and define the following linear functional on the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq94_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ8_HTML.gif
(2.5)

The functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq95_HTML.gif turns out to be well defined, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq96_HTML.gif is injective on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq97_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq98_HTML.gif is continuous with respect to the norm of the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq99_HTML.gif , because of the energy estimates proved in Proposition 2.1. Due to the Hahn-Banach Theorem, the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq100_HTML.gif can be extended to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq101_HTML.gif . Let us denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq102_HTML.gif this functional.

By the Riesz Theorem, there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq103_HTML.gif , which belongs to the dual space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq104_HTML.gif , so that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq106_HTML.gif . In particular, in the case where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq107_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ9_HTML.gif
(2.6)
Hence, since for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq108_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ10_HTML.gif
(2.7)

the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq109_HTML.gif turns out to be a solution of the system (1.2) in the sense of distributions.

In order to prove that the boundary condition is satisfied, by adopting the same strategy followed in [1], we have to study the regularity of the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq110_HTML.gif . For this purpose, let us extend the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq111_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq112_HTML.gif by zero outside the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq113_HTML.gif . By means of an approximation argument, we construct a sequence of smooth functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq114_HTML.gif , so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq115_HTML.gif turns out to be convergent to the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq116_HTML.gif , with respect to the norm of the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq117_HTML.gif . We define the approximating sequence in such a way for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq118_HTML.gif vanishes outside a compact neighbourhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq119_HTML.gif , for example, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq120_HTML.gif .

Moreover, let us denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq121_HTML.gif the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq122_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq123_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq124_HTML.gif . Set for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq126_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq127_HTML.gif . Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ11_HTML.gif
(2.8)
Since the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq128_HTML.gif is convergent to the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq129_HTML.gif with respect to the norm of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq130_HTML.gif , integrating by parts, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ12_HTML.gif
(2.9)

As a result, the sequence of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq131_HTML.gif is weakly convergent to the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq132_HTML.gif in the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq133_HTML.gif . Hence, the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq134_HTML.gif turns out to be bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq135_HTML.gif .

Let us consider again the system http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq136_HTML.gif , for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq137_HTML.gif .

By means of integration on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq138_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq139_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ13_HTML.gif
(2.10)
Let us estimate the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq140_HTML.gif -norm of the r.h.s. of (2.10). We deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ14_HTML.gif
(2.11)

Since there exists a suitable constant such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq141_HTML.gif ., the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq142_HTML.gif turns out to be bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq143_HTML.gif . Thus, the sequence of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq144_HTML.gif also turns out to be bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq145_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq146_HTML.gif . Differentiating with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq147_HTML.gif or to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq148_HTML.gif both members of (2.10), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ15_HTML.gif
(2.12)

The sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq149_HTML.gif is convergent to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq150_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq151_HTML.gif .

Due to the convergence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq152_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq153_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq154_HTML.gif , the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq155_HTML.gif turns out to converge to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq156_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq157_HTML.gif .

Furthermore, the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq158_HTML.gif is weakly convergent in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq159_HTML.gif . Therefore, it is bounded in the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq160_HTML.gif . Similarly, the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq161_HTML.gif also turns out to be bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq162_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq163_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq164_HTML.gif . Since the sequences of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq166_HTML.gif are bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq167_HTML.gif , the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq168_HTML.gif turns out to satisfy the assumptions of the Riesz-Fréchet-Kolmogorov theorem.

Thus the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq169_HTML.gif admits a first-order weak derivative with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq170_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq171_HTML.gif . Therefore the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq172_HTML.gif belongs to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq173_HTML.gif .

If we introduce a new variable, the system (1.2) may be reduced to a first-order system with respect to the variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq174_HTML.gif . Let us denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq175_HTML.gif the vector function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq176_HTML.gif and by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq178_HTML.gif the following differential operators
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ16_HTML.gif
(2.13)
Thus, the system (1.2) can be rewritten as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ17_HTML.gif
(2.14)

By setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq180_HTML.gif , the system (1.2) becomes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq181_HTML.gif .

Because of the regularity properties of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq184_HTML.gif turn out to belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq185_HTML.gif .

Multiplying both members of the system by any function of the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq186_HTML.gif , we prove that the vector function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq187_HTML.gif has a weak partial derivative with respect to the variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq188_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq189_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq190_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq191_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq192_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq193_HTML.gif , the traces of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq195_HTML.gif are well-defined on the hyperplane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq196_HTML.gif , and belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq198_HTML.gif , respectively.

Let us consider a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq199_HTML.gif , which, in a neighbourhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq200_HTML.gif has the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq201_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq202_HTML.gif . Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ18_HTML.gif
(2.15)
Integrating by parts,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ19_HTML.gif
(2.16)

Hence, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq203_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq204_HTML.gif .

3. Mixed Problem

This section deals with the study of the initial boundary value problem (1.3). We will prove the existence of the solution after solving two auxiliary problems: first we will determine the solution of an initial value problem, by means of the techniques developed in [1]; next, we will find the solution of a suitable boundary value problem, in accordance with the results stated in the previous section. Since the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq205_HTML.gif is linear, the solution to the mixed problem (1.3) will be determined by superposition. As a matter of fact, both auxiliary problems are mixed problems, but, as we will explain below, the solution of the former will be found as in the case of initial value problems, whereas the latter may be studied in the framework of boundary value problems.

Let us define the first problem as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ20_HTML.gif
(3.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq206_HTML.gif .

We consider the Cauchy problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ21_HTML.gif
(3.2)
and determine the solution by means of a duality argument through the procedure followed in [1]. For this purpose, we have to assume conditions on the coefficients of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq207_HTML.gif in order for energy estimates to be satisfied. Furthermore, let us define the linear space
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ22_HTML.gif
(3.3)

and quote from [1] the following result.

Proposition 3.1.

Consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq208_HTML.gif defined in (1.1) and the corresponding adjoint http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq209_HTML.gif . Assume the conditions (i)–(iv) listed in the Introduction to be fulfilled. In addition, let the sums http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq210_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq211_HTML.gif , and for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq212_HTML.gif , be positive real numbers. Moreover, we denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq213_HTML.gif the sum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq214_HTML.gif and suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq215_HTML.gif is positive, provided that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq216_HTML.gif is small enough.

Then, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq217_HTML.gif satisfies the following estimates:

for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq219_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ23_HTML.gif
(3.4)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq221_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ24_HTML.gif
(3.5)

with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq222_HTML.gif being positive constants that are independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq223_HTML.gif .

By taking into account the energy estimates of Proposition 3.1, we establish the following existence result.

Proposition 3.2.

Consider the initial boundary value problem (3.1), and let the assumptions of Proposition 3.1 be satisfied. If the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq224_HTML.gif , then the problem (3.1) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq225_HTML.gif .

Proof.

Let us define on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq226_HTML.gif the linear functional
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ25_HTML.gif
(3.6)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq227_HTML.gif .

Through the procedure followed in [1], we can prove there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq228_HTML.gif , such that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq229_HTML.gif .

In addition, due to the results proved in [1], http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq230_HTML.gif , a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq231_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq232_HTML.gif , a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq233_HTML.gif .

Moreover, since for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq234_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq235_HTML.gif , the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq236_HTML.gif on the boundary of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq237_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq238_HTML.gif . Let us determine the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq239_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq240_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq241_HTML.gif be a function of the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq242_HTML.gif , such that supp http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq243_HTML.gif is a compact subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq244_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq245_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq246_HTML.gif . By integrating by parts,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ26_HTML.gif
(3.7)
Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ27_HTML.gif
(3.8)

Consider a vector function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq247_HTML.gif , which, as long as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq248_HTML.gif is nonnegative and sufficiently small, has the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq249_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq250_HTML.gif . Hence, we deduce by means of a standard argument that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq251_HTML.gif , a.e.  in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq252_HTML.gif .

Let us define now the second auxiliary initial boundary value problem in order to obtain by superposition a solution to  (1.3):
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ28_HTML.gif
(3.9)

We assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq253_HTML.gif , whereas http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq254_HTML.gif .

The existence of the solution to (3.9) can be proved by means of the duality argument and a procedure similar to the previous problem.

Proposition 3.3.

Consider the initial boundary value problem (3.9), and let the assumptions of Proposition 2.1 be satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq255_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq256_HTML.gif , then there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq257_HTML.gif , which provides a solution to (3.9).

Proof.

We consider the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ29_HTML.gif
(3.10)
Since the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq258_HTML.gif satisfies the assumptions of Proposition 2.1, energy estimates can be proved for the adjoint operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq259_HTML.gif . Let us define the linear space
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ30_HTML.gif
(3.11)
For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq260_HTML.gif , consider the following functional:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ31_HTML.gif
(3.12)

As proved in the previous section, the functional turns out to be well defined and continuous as a consequence of the energy estimates. Furthermore, the functional can be extended to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq261_HTML.gif , and there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq262_HTML.gif , so that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq263_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq264_HTML.gif . After studying the regularity properties of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq265_HTML.gif as in the previous section and in [1], we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq266_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq267_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq268_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq269_HTML.gif satisfies the boundary condition.

The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq270_HTML.gif will be a solution to the mixed problem (3.9) after proving that the initial condition is satisfied. First of all, we have to remark that, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq271_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq272_HTML.gif , the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq273_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq274_HTML.gif turns out to belong to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq275_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq276_HTML.gif . Consider a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq277_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq278_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq279_HTML.gif .

By integrating by parts, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ32_HTML.gif
(3.13)
whence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ33_HTML.gif
(3.14)

By means of a suitable choice of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq280_HTML.gif , we prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq281_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq282_HTML.gif . Therefore, the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq283_HTML.gif turns out to be a solution to (3.9).

Finally, both the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq284_HTML.gif to the auxiliary problem (3.1) and the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq285_HTML.gif to (3.9) belong to the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq286_HTML.gif . Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq287_HTML.gif the sum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq288_HTML.gif . The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq289_HTML.gif , and, due to the previous results, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq290_HTML.gif has second-order partial derivatives with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq291_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq292_HTML.gif , which belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq293_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq294_HTML.gif turns out to be a solution to the initial boundary value problem (2.3). To avoid inconsistencies in the auxiliary mixed problems (3.1) and (3.9) as well as in (1.3), we have to require that the data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq295_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq296_HTML.gif satisfy compatibility conditions: if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq297_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq298_HTML.gif are smooth functions up to the boundary, we assume that, for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq299_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq300_HTML.gif .

Let us state now the main result.

Theorem 3.4.

Consider the initial boundary value problem (1.3). Suppose that the hypotheses of Propositions 3.2 and 3.3 are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq301_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq302_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq303_HTML.gif , then there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq304_HTML.gif , which provides a solution to (1.3).

Authors’ Affiliations

(1)

References

  1. Cavazzoni R: Initial value problem for a constant coefficient second order system. submitted

Copyright

© Rita Cavazzoni. 2010

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.