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Boundary Value Problems

, 2010:526917 | Cite as

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

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Research Article
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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 Open image in new window , where Open image in new window 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 Open image in new window and the time-variable belongs to the bounded interval Open image in new window , with Open image in new window sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution Open image in new window , 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.

Keywords

Cauchy Problem Differential Operator Linear Space Existence Result Positive Real Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Consider the second-order linear differential operator

The coefficients of the operator Open image in new window satisfy the following assumptions:

(i) Open image in new window is a positive real number;

(ii) for all Open image in new window , Open image in new window are Open image in new window symmetric matrices with real entries;

(iii) for every Open image in new window , where Open image in new window is a positive constant;

(iv) for all Open image in new window ; in addition, there exist two positive constants Open image in new window and Open image in new window such that for every Open image in new window , and Open image in new window ;

(v) for every Open image in new window , for all Open image in new window , with Open image in new window positive constant.

We will denote by Open image in new window a point of Open image in new window , by Open image in new window the first Open image in new window coordinates of Open image in new window , and by Open image in new window the time variable.

Let Open image in new window be a positive real number, and denote by Open image in new window the subset of Open image in new window , Open image in new window . In the first section of the paper we will be concerned with the following boundary value problem

where Open image in new window is the unknown vector-valued function, whereas Open image in new window and Open image in new window are given functions, which take values in Open image in new window and are defined in Open image in new window and Open image in new window , respectively.

Under suitable assumptions on the functions Open image in new window and Open image in new window and on the coefficients of the operator Open image in new window , we will prove that, in the case where the positive real number Open image in new window is sufficiently small, there exists a function Open image in new window , 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 Open image in new window , admits a solution Open image in new window : 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 Open image in new window 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 Open image in new window , as the time variable Open image in new window belongs to the bounded interval Open image in new window , where Open image in new window is a suitable positive real number:

We will assume that the vector-valued function Open image in new window belongs to the space Open image in new window , while, as for the initial data, we suppose that Open image in new window and Open image in new window . Let us notice that in the problem (1.3) an initial value for the unknown vector field Open image in new window and a Dirichlet boundary condition only are prescribed. Due to the a priori estimates that we will derive for the operator Open image in new window , 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 Open image in new window satisfy a prescribed condition at the boundary of the domain Open image in new window . 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 Open image in new window , a solution to (1.3) belonging to the space Open image in new window 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 Open image in new window the adjoint operator of Open image in new window :

For all Open image in new window , let Open image in new window be the norms of the matrices Open image in new window , respectively.

Proposition 2.1.

Consider the operator Open image in new window defined in (1.1) and the corresponding adjoint Open image in new window . Let the conditions (i)–(iv), listed in the Introduction, be fulfilled. In addition, assume the sums Open image in new window , Open image in new window , Open image in new window , and for every Open image in new window , Open image in new window to be positive real numbers. Moreover, denote by Open image in new window the sum Open image in new window and suppose that, as long as the positive number Open image in new window is sufficiently small, Open image in new window .

Define the linear space
Then, for all functions Open image in new window , the following estimates hold:

where Open image in new window are suitable positive constants.

Proof.

Consider a vector function Open image in new window , with compact support in the subset Open image in new window . Applying the Fourier transform with respect to both the tangential variable Open image in new window and the time variable Open image in new window , we can obtain a priori estimates, which, by substituting Open image in new window for Open image in new window and carrying out similar calculations, turn out to be like the estimates obtained in [1] for the initial value problem. Subsequently, assuming the function Open image in new window belongs to Open image in new window , we deduce the a priori estimates (2.3) and (2.4) for the adjoint operator Open image in new window .

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 Open image in new window and Open image in new window . Then, the boundary value problem (1.2) has a solution Open image in new window Open image in new window .

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 Open image in new window is one-to-one on the linear space Open image in new window . Let Open image in new window , and define the following linear functional on the space Open image in new window :

The functional Open image in new window turns out to be well defined, since Open image in new window is injective on Open image in new window . Moreover, Open image in new window is continuous with respect to the norm of the space Open image in new window , because of the energy estimates proved in Proposition 2.1. Due to the Hahn-Banach Theorem, the functional Open image in new window can be extended to the space Open image in new window . Let us denote by Open image in new window this functional.

By the Riesz Theorem, there exists a function Open image in new window , which belongs to the dual space Open image in new window , so that for every Open image in new window , Open image in new window . In particular, in the case where Open image in new window ,

the function Open image in new window 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 Open image in new window . For this purpose, let us extend the functions Open image in new window and Open image in new window by zero outside the interval Open image in new window . By means of an approximation argument, we construct a sequence of smooth functions Open image in new window , so that Open image in new window turns out to be convergent to the function Open image in new window , with respect to the norm of the space Open image in new window . We define the approximating sequence in such a way for all Open image in new window vanishes outside a compact neighbourhood of Open image in new window , for example, Open image in new window .

Since the sequence Open image in new window is convergent to the function Open image in new window with respect to the norm of Open image in new window , integrating by parts, we obtain

As a result, the sequence of functions Open image in new window is weakly convergent to the function Open image in new window in the space Open image in new window . Hence, the sequence Open image in new window turns out to be bounded in Open image in new window .

Let us consider again the system Open image in new window , for every Open image in new window .

By means of integration on the interval Open image in new window , with Open image in new window , we obtain
Let us estimate the Open image in new window -norm of the r.h.s. of (2.10). We deduce that

Since there exists a suitable constant such that, for all Open image in new window ., the sequence Open image in new window turns out to be bounded in Open image in new window . Thus, the sequence of functions Open image in new window also turns out to be bounded in Open image in new window .

Let Open image in new window . Differentiating with respect to Open image in new window or to Open image in new window both members of (2.10), we have

The sequence Open image in new window is convergent to Open image in new window in Open image in new window .

Due to the convergence of Open image in new window to Open image in new window in Open image in new window , the sequence Open image in new window turns out to converge to Open image in new window in Open image in new window .

Furthermore, the sequence Open image in new window is weakly convergent in Open image in new window . Therefore, it is bounded in the space Open image in new window . Similarly, the sequence Open image in new window also turns out to be bounded in Open image in new window . Hence, Open image in new window is bounded in Open image in new window . Since the sequences of functions Open image in new window and Open image in new window are bounded in Open image in new window , the sequence Open image in new window turns out to satisfy the assumptions of the Riesz-Fréchet-Kolmogorov theorem.

Thus the function Open image in new window admits a first-order weak derivative with respect to Open image in new window in Open image in new window . Therefore the function Open image in new window belongs to the space Open image in new window .

If we introduce a new variable, the system (1.2) may be reduced to a first-order system with respect to the variable Open image in new window . Let us denote by Open image in new window the vector function Open image in new window and by Open image in new window and Open image in new window the following differential operators
Thus, the system (1.2) can be rewritten as

By setting Open image in new window and Open image in new window , the system (1.2) becomes Open image in new window .

Because of the regularity properties of the function Open image in new window , Open image in new window and Open image in new window turn out to belong to Open image in new window .

Multiplying both members of the system by any function of the space Open image in new window , we prove that the vector function Open image in new window has a weak partial derivative with respect to the variable Open image in new window . Thus Open image in new window and Open image in new window , a.e. in Open image in new window .

Since Open image in new window belongs to Open image in new window , the traces of Open image in new window and Open image in new window are well-defined on the hyperplane Open image in new window , and belong to Open image in new window and Open image in new window , respectively.

Let us consider a function Open image in new window , which, in a neighbourhood of Open image in new window has the form Open image in new window , with Open image in new window . Thus,
Integrating by parts,

Hence, we obtain Open image in new window , a.e. in Open image in new window .

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 Open image in new window 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:

where Open image in new window .

We consider the Cauchy problem
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 Open image in new window in order for energy estimates to be satisfied. Furthermore, let us define the linear space

and quote from [1] the following result.

Proposition 3.1.

Consider the operator Open image in new window defined in (1.1) and the corresponding adjoint Open image in new window . Assume the conditions (i)–(iv) listed in the Introduction to be fulfilled. In addition, let the sums Open image in new window , Open image in new window , and for every Open image in new window , be positive real numbers. Moreover, we denote by Open image in new window the sum Open image in new window and suppose Open image in new window is positive, provided that Open image in new window is small enough.

Then, the operator Open image in new window satisfies the following estimates:

for every Open image in new window ,

with Open image in new window being positive constants that are independent of Open image in new window .

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 Open image in new window , then the problem (3.1) has a solution Open image in new window .

Proof.

Let us define on Open image in new window the linear functional

where Open image in new window .

Through the procedure followed in [1], we can prove there exists a function Open image in new window , such that for every Open image in new window .

In addition, due to the results proved in [1], Open image in new window , a.e. Open image in new window , and Open image in new window , a.e. Open image in new window .

Moreover, since for all Open image in new window , the function Open image in new window , the trace of Open image in new window on the boundary of Open image in new window belongs to Open image in new window . Let us determine the trace of Open image in new window on Open image in new window . Let Open image in new window be a function of the space Open image in new window , such that supp Open image in new window is a compact subset of Open image in new window . Therefore, Open image in new window and Open image in new window . By integrating by parts,

Consider a vector function Open image in new window , which, as long as Open image in new window is nonnegative and sufficiently small, has the form Open image in new window , with Open image in new window . Hence, we deduce by means of a standard argument that Open image in new window , a.e.  in Open image in new window .

Let us define now the second auxiliary initial boundary value problem in order to obtain by superposition a solution to  (1.3):

We assume that Open image in new window , whereas Open image in new window .

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 Open image in new window and Open image in new window , then there exists a function Open image in new window , which provides a solution to (3.9).

Proof.

We consider the boundary value problem
Since the operator Open image in new window satisfies the assumptions of Proposition 2.1, energy estimates can be proved for the adjoint operator Open image in new window . Let us define the linear space
For every Open image in new window , consider the following functional:

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 Open image in new window , and there exists a function Open image in new window , so that for every Open image in new window , Open image in new window . After studying the regularity properties of the function Open image in new window as in the previous section and in [1], we can prove that Open image in new window , Open image in new window , a.e. in Open image in new window , and Open image in new window satisfies the boundary condition.

The function Open image in new window 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 Open image in new window , for all Open image in new window , the trace of Open image in new window on Open image in new window turns out to belong to the space Open image in new window . Thus, Open image in new window . Consider a function Open image in new window , such that Open image in new window . Hence, Open image in new window .

By integrating by parts, we have

By means of a suitable choice of the function Open image in new window , we prove that Open image in new window , a.e. in Open image in new window . Therefore, the function Open image in new window turns out to be a solution to (3.9).

Finally, both the solution Open image in new window to the auxiliary problem (3.1) and the solution Open image in new window to (3.9) belong to the space Open image in new window . Denote by Open image in new window the sum Open image in new window . The function Open image in new window , and, due to the previous results, Open image in new window has second-order partial derivatives with respect to Open image in new window and Open image in new window , which belong to Open image in new window . Hence, Open image in new window 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 Open image in new window and Open image in new window satisfy compatibility conditions: if Open image in new window and Open image in new window are smooth functions up to the boundary, we assume that, for every Open image in new window , Open image in new window .

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 Open image in new window , Open image in new window , and Open image in new window , then there exists a function Open image in new window , which provides a solution to (1.3).

References

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

Copyright information

© 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.

Authors and Affiliations

  1. 1.TurinItaly

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