On a Mixed Problem for a Constant Coefficient Second-Order System
© Rita Cavazzoni. 2010
Received: 2 July 2010
Accepted: 1 December 2010
Published: 15 December 2010
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© Rita Cavazzoni. 2010
Received: 2 July 2010
Accepted: 1 December 2010
Published: 15 December 2010
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 , where 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 and the time-variable belongs to the bounded interval , with sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution , 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.
The coefficients of the operator satisfy the following assumptions:
(i) is a positive real number;
(ii) for all , are symmetric matrices with real entries;
(iii) for every , where is a positive constant;
(iv) for all ; in addition, there exist two positive constants and such that for every , and ;
(v) for every , for all , with positive constant.
We will denote by a point of , by the first coordinates of , and by the time variable.
where is the unknown vector-valued function, whereas and are given functions, which take values in and are defined in and , respectively.
Under suitable assumptions on the functions and and on the coefficients of the operator , we will prove that, in the case where the positive real number is sufficiently small, there exists a function , 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  to prove that the initial value problem for the system , admits a solution : the main result of  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 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 .
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).
We will assume that the vector-valued function belongs to the space , while, as for the initial data, we suppose that and . Let us notice that in the problem (1.3) an initial value for the unknown vector field and a Dirichlet boundary condition only are prescribed. Due to the a priori estimates that we will derive for the operator , 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 satisfy a prescribed condition at the boundary of the domain . 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 , while the latter will be studied like the boundary value problem considered in Section 2. Subsequently, thanks to the linearity of the operator , a solution to (1.3) belonging to the space will be determined by superposition of the solutions to the preliminary mixed problems.
By adopting the same strategy of  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.
For all , let be the norms of the matrices , respectively.
Consider the operator defined in (1.1) and the corresponding adjoint . Let the conditions (i)–(iv), listed in the Introduction, be fulfilled. In addition, assume the sums , , , and for every , to be positive real numbers. Moreover, denote by the sum and suppose that, as long as the positive number is sufficiently small, .
where are suitable positive constants.
Consider a vector function , with compact support in the subset . Applying the Fourier transform with respect to both the tangential variable and the time variable , we can obtain a priori estimates, which, by substituting for and carrying out similar calculations, turn out to be like the estimates obtained in  for the initial value problem. Subsequently, assuming the function belongs to , we deduce the a priori estimates (2.3) and (2.4) for the adjoint operator .
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).
Consider the boundary value problem (1.2), and let the assumptions of Proposition 2.1 be satisfied. Furthermore, suppose that and . Then, the boundary value problem (1.2) has a solution .
The functional turns out to be well defined, since is injective on . Moreover, is continuous with respect to the norm of the space , because of the energy estimates proved in Proposition 2.1. Due to the Hahn-Banach Theorem, the functional can be extended to the space . Let us denote by this functional.
the function 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 , we have to study the regularity of the solution . For this purpose, let us extend the functions and by zero outside the interval . By means of an approximation argument, we construct a sequence of smooth functions , so that turns out to be convergent to the function , with respect to the norm of the space . We define the approximating sequence in such a way for all vanishes outside a compact neighbourhood of , for example, .
As a result, the sequence of functions is weakly convergent to the function in the space . Hence, the sequence turns out to be bounded in .
Let us consider again the system , for every .
Since there exists a suitable constant such that, for all ., the sequence turns out to be bounded in . Thus, the sequence of functions also turns out to be bounded in .
The sequence is convergent to in .
Due to the convergence of to in , the sequence turns out to converge to in .
Furthermore, the sequence is weakly convergent in . Therefore, it is bounded in the space . Similarly, the sequence also turns out to be bounded in . Hence, is bounded in . Since the sequences of functions and are bounded in , the sequence turns out to satisfy the assumptions of the Riesz-Fréchet-Kolmogorov theorem.
Thus the function admits a first-order weak derivative with respect to in . Therefore the function belongs to the space .
By setting and , the system (1.2) becomes .
Because of the regularity properties of the function , and turn out to belong to .
Multiplying both members of the system by any function of the space , we prove that the vector function has a weak partial derivative with respect to the variable . Thus and , a.e. in .
Since belongs to , the traces of and are well-defined on the hyperplane , and belong to and , respectively.
Hence, we obtain , a.e. in .
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 ; 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 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.
and quote from  the following result.
Consider the operator defined in (1.1) and the corresponding adjoint . Assume the conditions (i)–(iv) listed in the Introduction to be fulfilled. In addition, let the sums , , and for every , be positive real numbers. Moreover, we denote by the sum and suppose is positive, provided that is small enough.
Then, the operator satisfies the following estimates:
for every ,
with being positive constants that are independent of .
By taking into account the energy estimates of Proposition 3.1, we establish the following existence result.
Consider the initial boundary value problem (3.1), and let the assumptions of Proposition 3.1 be satisfied. If the function , then the problem (3.1) has a solution .
Through the procedure followed in , we can prove there exists a function , such that for every .
In addition, due to the results proved in , , a.e. , and , a.e. .
Consider a vector function , which, as long as is nonnegative and sufficiently small, has the form , with . Hence, we deduce by means of a standard argument that , a.e. in .
We assume that , whereas .
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.
Consider the initial boundary value problem (3.9), and let the assumptions of Proposition 2.1 be satisfied. If and , then there exists a function , which provides a solution to (3.9).
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 , and there exists a function , so that for every , . After studying the regularity properties of the function as in the previous section and in , we can prove that , , a.e. in , and satisfies the boundary condition.
The function 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 , for all , the trace of on turns out to belong to the space . Thus, . Consider a function , such that . Hence, .
By means of a suitable choice of the function , we prove that , a.e. in . Therefore, the function turns out to be a solution to (3.9).
Finally, both the solution to the auxiliary problem (3.1) and the solution to (3.9) belong to the space . Denote by the sum . The function , and, due to the previous results, has second-order partial derivatives with respect to and , which belong to . Hence, 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 and satisfy compatibility conditions: if and are smooth functions up to the boundary, we assume that, for every , .
Let us state now the main result.
Consider the initial boundary value problem (1.3). Suppose that the hypotheses of Propositions 3.2 and 3.3 are satisfied. If , , and , then there exists a function , which provides a solution to (1.3).
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