Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq1_HTML.gif , where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq4_HTML.gif , with https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq7_HTML.gif satisfy the following assumptions:

(i) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq10_HTML.gif are https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq12_HTML.gif , where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq16_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq17_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq18_HTML.gif ;

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

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

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq29_HTML.gif the subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq30_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ2_HTML.gif
(1.2)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq33_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq35_HTML.gif and are defined in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq36_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq38_HTML.gif and https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq43_HTML.gif , admits a solution https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq46_HTML.gif , as the time variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq47_HTML.gif belongs to the bounded interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq49_HTML.gif is a suitable positive real number:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq50_HTML.gif belongs to the space https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq52_HTML.gif and https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq60_HTML.gif the adjoint operator of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq61_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ4_HTML.gif
(2.1)

For all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq62_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq63_HTML.gif be the norms of the matrices https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq67_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq69_HTML.gif , and for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq70_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq72_HTML.gif the sum https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq74_HTML.gif is sufficiently small, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq75_HTML.gif .

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq78_HTML.gif , with compact support in the subset https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq80_HTML.gif and the time variable https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq82_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq84_HTML.gif belongs to https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq87_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq89_HTML.gif https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq92_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq94_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ8_HTML.gif
(2.5)

The functional https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq96_HTML.gif is injective on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq97_HTML.gif . Moreover, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq100_HTML.gif can be extended to the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq101_HTML.gif . Let us denote by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq103_HTML.gif , which belongs to the dual space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq104_HTML.gif , so that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq106_HTML.gif . In particular, in the case where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq107_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ9_HTML.gif
(2.6)
Hence, since for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq108_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ10_HTML.gif
(2.7)

the function https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq112_HTML.gif by zero outside the interval https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq114_HTML.gif , so that https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq118_HTML.gif vanishes outside a compact neighbourhood of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq119_HTML.gif , for example, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq120_HTML.gif .

Moreover, let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq121_HTML.gif the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq122_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq123_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq124_HTML.gif . Set for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq126_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq127_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ11_HTML.gif
(2.8)
Since the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq128_HTML.gif is convergent to the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq129_HTML.gif with respect to the norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq130_HTML.gif , integrating by parts, we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq131_HTML.gif is weakly convergent to the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq132_HTML.gif in the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq133_HTML.gif . Hence, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq134_HTML.gif turns out to be bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq135_HTML.gif .

Let us consider again the system https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq136_HTML.gif , for every https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq138_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq139_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ13_HTML.gif
(2.10)
Let us estimate the https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq141_HTML.gif ., the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq142_HTML.gif turns out to be bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq143_HTML.gif . Thus, the sequence of functions https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq145_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq146_HTML.gif . Differentiating with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq147_HTML.gif or to https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ15_HTML.gif
(2.12)

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

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

Furthermore, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq158_HTML.gif is weakly convergent in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq160_HTML.gif . Similarly, the sequence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq162_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq163_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq164_HTML.gif . Since the sequences of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq166_HTML.gif are bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq167_HTML.gif , the sequence https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq170_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq171_HTML.gif . Therefore the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq172_HTML.gif belongs to the space https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq174_HTML.gif . Let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq175_HTML.gif the vector function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq176_HTML.gif and by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq178_HTML.gif the following differential operators
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ17_HTML.gif
(2.14)

By setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq180_HTML.gif , the system (1.2) becomes https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq184_HTML.gif turn out to belong to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq186_HTML.gif , we prove that the vector function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq188_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq189_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq190_HTML.gif , a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq191_HTML.gif .

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

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

Hence, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq203_HTML.gif , a.e. in https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ20_HTML.gif
(3.1)

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

We consider the Cauchy problem
https://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 https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq211_HTML.gif , and for every https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq213_HTML.gif the sum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq214_HTML.gif and suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq215_HTML.gif is positive, provided that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq216_HTML.gif is small enough.

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

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

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

with https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq225_HTML.gif .

Proof.

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq228_HTML.gif , such that for every https://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], https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq230_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq231_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq232_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq233_HTML.gif .

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

Consider a vector function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq247_HTML.gif , which, as long as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq249_HTML.gif , with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq251_HTML.gif , a.e.  in https://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):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ28_HTML.gif
(3.9)

We assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq253_HTML.gif , whereas https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq255_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq256_HTML.gif , then there exists a function https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ29_HTML.gif
(3.10)
Since the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq259_HTML.gif . Let us define the linear space
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ30_HTML.gif
(3.11)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq260_HTML.gif , consider the following functional:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq261_HTML.gif , and there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq262_HTML.gif , so that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq263_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq266_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq267_HTML.gif , a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq268_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq269_HTML.gif satisfies the boundary condition.

The function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq271_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq272_HTML.gif , the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq273_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq275_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq276_HTML.gif . Consider a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq277_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq278_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq279_HTML.gif .

By integrating by parts, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_Equ32_HTML.gif
(3.13)
whence
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq280_HTML.gif , we prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq281_HTML.gif , a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq282_HTML.gif . Therefore, the function https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq286_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq287_HTML.gif the sum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq288_HTML.gif . The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq289_HTML.gif , and, due to the previous results, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq292_HTML.gif , which belong to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq293_HTML.gif . Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq295_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq296_HTML.gif satisfy compatibility conditions: if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq297_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq299_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq301_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq302_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F526917/MediaObjects/13661_2010_Article_936_IEq303_HTML.gif , then there exists a function https://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.