Let us first consider the BVP for the anisotropic type DOE with constant coefficients

are boundary conditions defined by (3.2),

are complex numbers,

is a complex parameter, and

is a linear operator in a Banach space

. Let

be the roots of the characteristic equations

By applying the trace theorem [27, Section 1.8.2], we have the following.

Theorem A5.

Let

and

be integer numbers,

,

,

. Then, for any

, the transformations

are bounded linear from

onto

, and the following inequality holds:

Proof.

Then, by applying the trace theorem [27, Section 1.8.2] to the space
, we obtain the assertion.

Condition 1.

Assume that the following conditions are satisfied:

(1)
is a Banach space satisfying the multiplier condition with respect to
and the weight function
,
;

(2)
is an
-positive operator in
for
;

(3)

, and

for
,
.

Let
denote the operator in
generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.

Theorem A6.

Let Condition 1 be satisfied. Then,

- (a)
the problem (4.1) for

and

with sufficiently large

has a unique solution

that belongs to

and the following coercive uniform estimate holds:

(b)the operator
is
-positive in
.

From Theorems A5 and A6 we have.

Theorem A7.

Suppose that Condition 1 is satisfied. Then, for sufficiently large

with

the problem (4.1) has a unique solution

for all

and

. Moreover, the following uniform coercive estimate holds:

Consider BVP (3.11). Let

be roots of the characteristic equations

Condition 2.

Suppose the following conditions are satisfied:

(1)

and

(2)
is a Banach space satisfying the multiplier condition with respect to
and the weighted function
,
.

Remark 4.1.

Let
and
, where
are real-valued positive functions. Then, Condition 2 is satisfied for
.

Consider the inhomogenous BVP (3.1)-(3.2); that is,

Lemma 4.2.

Assume that Condition 2 is satisfied and the following hold:

(1)
is a uniformly
-positive operator in
for
, and
are continuous functions on
,
,

(2)
and
for
.

Then, for all

and for sufficiently large

the following coercive uniform estimate holds:

for the solution of problem (4.13).

Proof.

Let

be regions covering

and let

be a corresponding partition of unity; that is,

,

and

. Now, for

and

, we get

here,

and

are boundary operators which orders less than

. Freezing the coefficients of (4.15), we have

It is clear that

on neighborhoods of

and

on neighborhoods of

and

on other parts of the domains

, where

are positive constants. Hence, the problems (4.17) are generated locally only on parts of the boundary. Then, by Theorem A7 problem (4.17) has a unique solution

and for

the following coercive estimate holds:

From the representation of

,

and in view of the boundedness of the coefficients, we get

Now, applying Theorem A1 and by using the smoothness of the coefficients of (4.16), (4.18) and choosing the diameters of

so small, we see there is an

and

such that

Then, using Theorem A5 and using the smoothness of the coefficients of (4.16), (4.18), we get

Now, using Theorem A1, we get that there is an

and

such that

Using the above estimates, we get

Consequently, from (4.22)–(4.26), we have

Choosing

from the above inequality, we obtain

Then, by using the equality
and the above estimates, we get (4.14).

Condition 3.

Suppose that part (1.1) of Condition 1 is satisfied and that
is a Banach space satisfying the multiplier condition with respect to
and the weighted function
,
,
.

Consider the problem (3.11). Reasoning as in the proof of Lemma 4.2, we obtain.

Proposition 4.3.

Assume Condition 3 hold and suppose that

(1)
is a uniformly
-positive operator in
for
, and that
are continuous functions on
,
,

(2)
and
for
.

Then, for all

and for sufficiently large

, the following coercive uniform estimate holds

for the solution of problem (3.11).

Let

denote the operator generated by problem (3.11) for

; that is,

Theorem 4.4.

Assume that Condition 3 is satisfied and that the following hold:

(1)
is a uniformly
-positive operator in
, and
are continuous functions on
,

(2)
, and
for
.

Then, problem (3.11) has a unique solution

for

and

with large enough

. Moreover, the following coercive uniform estimate holds:

Proof.

By Proposition 4.3 for

, we have

Hence, by using the definition of

and applying Theorem A1, we obtain

From the above estimate, we have

The estimate (4.35) implies that problem (3.11) has a unique solution and that the operator

has a bounded inverse in its rank space. We need to show that this rank space coincides with the space

; that is, we have to show that for all

, there is a unique solution of the problem (3.11). We consider the smooth functions

with respect to a partition of unity

on the region

that equals one on

, where supp

and

. Let us construct for all

the functions

that are defined on the regions

and satisfying problem (3.11). The problem (3.11) can be expressed as

Consider operators

in

that are generated by the BVPs (4.17); that is,

By virtue of Theorem A6, the operators

have inverses

for

and for sufficiently large

. Moreover, the operators

are bounded from

to

, and for all

, we have

Extending

to zero outside of

in the equalities (4.36), and using the substitutions

, we obtain the operator equations

where

are bounded linear operators in

defined by

In fact, because of the smoothness of the coefficients of the expression

and from the estimate (4.38), for

with sufficiently large

, there is a sufficiently small

such that

Moreover, from assumption (2.2) of Theorem 4.4 and Theorem A1 for

, there is a constant

such that

Hence, for

with sufficiently large

, there is a

such that

. Consequently, (4.39) for all

have a unique solution

. Moreover,

Thus,

are bounded linear operators from

to

. Thus, the functions

are solutions of (4.38). Consider the following linear operator

in

defined by

It is clear from the constructions

and from the estimate (4.39) that the operators

are bounded linear from

to

, and for

with sufficiently large

, we have

Therefore,

is a bounded linear operator in

. Since the operators

coincide with the inverse of the operator

in

, then acting on

to

gives

where

are bounded linear operators defined by

Indeed, from Theorem A1 and estimate (4.46) and from the expression

, we obtain that the operators

are bounded linear from

to

, and for

with sufficiently large

, there is an

such that

. Therefore, there exists a bounded linear invertible operator

; that is, we infer for all

that the BVP (3.11) has a unique solution

Result 1.

Theorem 4.4 implies that the resolvent

satisfies the following anisotropic type sharp estimate:

for
,
.

Let
denote the operator generated by BVP (3.1)-(3.2). From Theorem 4.4 and Remark 3.1, we get the following.

Result 2.

Assume all the conditions of Theorem 4.4 hold. Then,

(a)the problem (3.1)-(3.2) for

,

and for sufficiently large

has a unique solution

, and the following coercive uniform estimate holds

(b)if
, then the operator
is Fredholm from
into
.

Example 4.5.

Now, let us consider a special case of (3.1)-(3.2). Let

,

and

,

,

and

; that is, consider the problem

Theorem 4.4 implies that for each

, problem (4.52) has a unique solution

satisfying the following coercive estimate:

Example 4.6.

Let
and
, where
are positive continuous function on
,
and
is a diagonal matrix-function with continuous components
.

Then, we obtain the separability of the following BVPs for the system of anisotropic PDEs with varying coefficients:

in the vector-valued space
.