In this section, we would like to present the main results of the study which is based on our previous results (cf. [5, 6]) and the results of elliptic equations in cusp domains (cf. [7]). For the start of this section, we denote by
the operator corresponding to the parameter-depending boundary value problem

For each
we have the operator pencil
to be Fredholm, and its spectrum consists of a countable number of isolated eigenvalues. Similarly to Theorem
in [7], we have the following lemma.

Lemma 4.1.

Assume that

, where

are real numbers. Additionally, the authors suppose that no eigenvalues of

line in strips

and

, where

and

are eigenvalues of

, and

. Then the generalized solution

of the Dirichlet problem for elliptic equation (

3.13), such that

if

, belongs to the

and satisfies the inequality

where the constant
is independent of
.

Proof.

by the Friederichs inequality, we have

for all

. Hence,

Let
be the function which arises from
via the coordinate change
. We set
; then from the properties of the mapping (2.2) and from inequality (4.6), it follows that
. Since
is the solution of an elliptic equation in
with coefficients which stabilize for
, that is,

where

, we obtain

(cf. [

7, Lemma

]). This implies

. Using the fact that

as
if
, we conclude that
From Corollary
in [7] it follows that
. Furthermore, (4.2) holds.

Lemma 4.2.

Suppose that

, and the strip

does not contain eigenvalues of

. Then the generalized solution

of problem (3.6)–(3.8), such that

if

, belongs to the

and satisfies the inequality

where the constant
is independent of
.

Proof.

Using the smoothness of the generalized solution of problem (3.6)–(3.8) with respect to

in Theorem 3.3 and Proposition 3.4, we can see that for a.e.

is the generalized solution of Dirichlet problem for (3.13) with

. From Lemma 4.1, it implies that

for a.e.

and satisfies the inequality

By integrating the inequality above with respect to
from
to
, and using the estimates for derivatives of
with respect to
again, we obtain
, which satisfies inequality (4.9).

Theorem 4.3.

Let the assumptions of Lemma 4.2 be satisfied, and

,

,

for

. Then the generalized solution

of problem (3.6)–(3.8), such that

if

, belongs to the

and satisfies the inequality

where the constant
is independent of
.

Proof.

Let us first prove that

belong to the

for

and satisfy

The proof is an induction on

. According to Lemma 4.2, it is valid for

. Now let this assertion be true for

; we will prove that this also holds for

. Due to Lemma 4.2,

satisfies (3.6). By differentiating both sides of (3.6) with respect to

times, we obtain

By the supposition of the theorem and the inductive assumption, the right-hand side of (4.13) belongs to

. By the arguments analogous to the proof of Lemma 4.2, we get

and

where
is a constant independent of
, and
.

By using (4.15) and estimates for derivatives of
with respect to
in Theorem 3.3, we have

Remark 4.4.

Let
be a sufficiently small positive number. Suppose that
and the strip
contains no eigenvalues of
,
; then the generalized solution
of problem (3.6)–(3.8), such that
if
, belongs to the
. In fact, setting
, we obtain the first initial boundary value problem which differs little from (3.6)–(3.8). Therefore,
, and then
. Using the remark above and Lemma 4.1, we obtain the following theorem.

Theorem 4.5.

Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that

,

and

for

. Then the generalized solution

of problem (3.6)–(3.8), such that

if

, belongs to the

and satisfies the inequality

where the constant
is independent of
.

This theorem is proved by arguments analogous to those proofs of Lemma 4.2 and Theorem 4.3. Next, we will prove the well regularity of the generalized solution of problem (3.6)–(3.8).

Theorem 4.6.

Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that

,

and

for

. Then the generalized solution

of problem (3.6)–(3.8), such that

if

, belongs to the

and satisfies the inequality

where the constant
is independent of
and
.

Proof.

The theorem is proved by induction on

. Thanks to Theorem 4.5, this theorem is obviously valid for

. Assume that the theorem is true for

, we will prove that it also holds for

. It is only needed to show that

Differentiating both sides of (3.6) again with respect to

times, we obtain

By the supposition of the theorem and the inductive assumption, the right-hand side of (4.20) belongs to

for a.e.

. Using Lemma 4.1, we conclude that

. It implies that (4.19) holds for

. Suppose that (4.19) is true for

and set

, we obtain

where

. By the inductive assumption with respect to

,

belongs to

for a.e.

. Thus, the right-hand side of (4.21) belongs to

. Applying Lemma 4.1 again for

, we get that

for a.e.

. It means that

belongs to

. Furthermore, we have

It implies that (4.19) holds for
. The proof is complete.

Now we will prove the global regularity of the solution.

Theorem 4.7.

Let the hypotheses of Lemma 4.1 be satisfied. Furthermore, suppose

,

and

,for

. Then the generalized solution

of problem (3.6)–(3.8) belongs to the

and satisfies the inequality

where the constant
is independent of
and
.

Proof.

We denote by

the unit ball, and suppose that

and

in the neighborhood of the origin

. It is easy to get that

where

is a differential operator, whose coefficients have compact support in a neighborhood of the origin. By arguments analogous to the proof of Theorem 4.6, we obtain

Set

, then

in a neighborhood of the origin and

, and using the smoothness of the solution of this problem in domain with smooth boundary, we get

The proof is complete.