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,
.

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,

Hence, we obtain
, a.e. in
.