Consider the second-order linear differential operator

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.

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

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 [1] to prove that the initial value problem for the system , admits a solution : 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 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 , as the time variable belongs to the bounded interval , where is a suitable positive real number:

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 [1], 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 [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 the adjoint operator of :

For all , let be the norms of the matrices , respectively.

Proposition 2.1.

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, .

Define the linear space

Then, for all functions , the following estimates hold:

where are suitable positive constants.

Proof.

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 [1] 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).

Proposition 2.2.

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 .

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 is one-to-one on the linear space . Let , and define the following linear functional on the space :

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.

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

Hence, since for every

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 [1], 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, .

Moreover, let us denote by the differential operator . Therefore, . Set for all , . Let . Thus,

Since the sequence is convergent to the function with respect to the norm of , integrating by parts, we obtain

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 .

By means of integration on the interval , with , we obtain

Let us estimate the -norm of the r.h.s. of (2.10). We deduce that

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 .

Let . Differentiating with respect to or to both members of (2.10), we have

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 .

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

Thus, the system (1.2) can be rewritten as

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.

Let us consider a function , which, in a neighbourhood of has the form , with . Thus,

Integrating by parts,

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 [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 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 .

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 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 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 ,

for all ,

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.

Proposition 3.2.

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 .

Proof.

Let us define on the linear functional

where .

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

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

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

Thus,

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 .

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 , 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.

Proposition 3.3.

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).

Proof.

We consider the boundary value problem

Since the operator satisfies the assumptions of Proposition 2.1, energy estimates can be proved for the adjoint operator . Let us define the linear space

For every , 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 , and there exists a function , so that for every , . After studying the regularity properties of the function as in the previous section and in [1], 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 integrating by parts, we have

whence

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.

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 , , and , then there exists a function , which provides a solution to (1.3).