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# Schwartz-Type Boundary Value Problems for Monogenic Functions in a Biharmonic Algebra

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## Abstract

We consider Schwartz-type boundary value problems for monogenic functions in a commutative algebra \(\mathbb {B}\) over the field of complex numbers with the bases {*e*_{1}, *e*_{2}} satisfying the conditions \((e_1^2+e_2^2)^2=0\), \(e_1^2+e_2^2\ne 0\). The algebra \(\mathbb {B}\) is associated with the biharmonic equation, and considered problems have relations to the plane elasticity. We develop methods of its solving which are based on expressions of solutions by hypercomplex integrals analogous to the classic Schwartz and Cauchy integrals.

## Keywords

Biharmonic equation Biharmonic algebra Biharmonic plane Monogenic function Schwartz-type boundary value problem## Mathematics Subject Classification (2010)

Primary 30G35; Secondary 31A30## 1 Monogenic Functions in a Biharmonic Algebra

### Definition 1.1

*biharmonic*(see [1, 2]) if in \(\mathbb B\) there exists a basis {

*e*

_{1},

*e*

_{2}} satisfying the conditions

Such a basis {*e*_{1}, *e*_{2}} is also called *biharmonic*.

In the paper [2] I.P. Mel’nichenko proved that there exists the unique biharmonic algebra \(\mathbb {B}\), and he constructed all biharmonic bases in \(\mathbb {B}\). Note that the algebra \(\mathbb {B}\) is isomorphic to four-dimensional over the field of real numbers \(\mathbb R\) algebras considered by A. Douglis [3] and L. Sobrero [4].

*e*

_{1},

*e*

_{2}} with the following multiplication table (see [1]):

*i*is the imaginary complex unit. We consider also a basis {1,

*ρ*} (see [2]), where a nilpotent element

*ρ*

^{2}= 0 .

We use the Euclidean norm \(\|a\|:=\sqrt {|z_1|{ }^2+|z_2|{ }^2}\) in the algebra \(\mathbb {B}\), where *a* = *z*_{1}*e*_{1} + *z*_{2}*e*_{2} and \(z_1, z_2\in \mathbb {C}\).

Consider a *biharmonic plane* \(\mu _{e_1,e_2}:=\{\zeta =x\,e_1+y\,e_2 : x,y\in \mathbb R\}\) which is a linear span of the elements *e*_{1}, *e*_{2} of the biharmonic basis (1.1) over the field \(\mathbb R\).

With a domain *D* of the Cartesian plane *xOy* we associate the congruent domain \(D_{\zeta }:= \{\zeta =xe_1+ye_2 \in \mu _{e_1, e_2} : (x,y)\in D\}\) in the biharmonic plane \(\mu _{e_1, e_2}\) and the congruent domain *D*_{z} := {*z* = *x* + *iy* : (*x*, *y*) ∈ *D*} in the complex plane \(\mathbb {C}\). Its boundaries are denoted by *∂D*, *∂D*_{ζ} and *∂D*_{z}, respectively. Let \(\overline {D_{\zeta }}\) (or \(\overline {D_{z}}\), \(\overline {D}\)) be the closure of domain *D*_{ζ} (or *D*_{z}, *D*, respectively).

In what follows, *ζ* = *x* *e*_{1} + *y* *e*_{2}, *z* = *x* + *iy*, where (*x*, *y*) ∈ *D*, and *ζ*_{0} = *x*_{0} *e*_{1} + *y*_{0} *e*_{2}, *z*_{0} = *x*_{0} + *iy*_{0}, where (*x*_{0}, *y*_{0}) ∈ *∂D*.

*l*= 1, 2, 3, 4, are real-valued component-functions. We shall use the following notation: \(\mathrm {U}_{l}\left [\Phi \right ]:=U_{l}\),

*l*= 1, 2, 3, 4.

### Definition 1.2

*monogenic*in a domain

*D*

_{ζ}if it has the classical derivative Φ

*′*(

*ζ*) at every point

*ζ*∈

*D*

_{ζ}:

*D*

_{ζ}if and only if its each real-valued component-function in (1.3) is real differentiable in

*D*and the following analog of the Cauchy–Riemann condition is fulfilled:

*U*

_{k}, \(k=\overline {1,4}\), in (1.3):

*U*

_{l},

*l*= 1, 2, 3, 4, in the expansion (1.3) of any monogenic function \(\Phi \colon D_{\zeta }\longrightarrow \mathbb {B}\) are biharmonic functions (cf., e.g., [5, 6]), i.e., satisfy the biharmonic equation in

*D*:

At the same time, every biharmonic in a simply-connected domain *D* function *U*(*x*, *y*) is the first component *U*_{1} ≡ *U* in the expression (1.3) of a certain function \(\Phi \colon D_{\zeta } \longrightarrow \mathbb {B}\) monogenic in *D*_{ζ} and, moreover, all such functions Φ are found in [5, 6] in an explicit form.

*z*in the form (cf., e.g., [5, 6]):

*D*

_{ζ}and pairs of complex-valued analytic functions

*F*,

*F*

_{0}in the domain

*D*

_{z}.

## 2 Schwartz-Type BVP’s for Monogenic Functions

*D*

_{ζ}when limiting values of two component-functions in (1.3) are given on the boundary

*∂D*

_{ζ}, i.e., the following boundary conditions are satisfied:

*k*<

*m*≤ 4, where

*u*

_{k},

*u*

_{m}are given continuous functions.

*D*

_{ζ}is unbounded as well as the assumption that every given function

*u*

_{l},

*l*∈{

*k*,

*m*}, has a finite limit

*∂D*

_{ζ}is unbounded.

We shall call such a problem by the (*k*-*m*)-problem.

*k*-

*m*)-problems with additional assumptions that the sought-for function \(\Phi : \overline {D_{\zeta }} \longrightarrow \mathbb {B}\) is continuous in \(\overline {D_{\zeta }}\) and has the limit

*D*

_{ζ}is unbounded. He named such problems as

*biharmonic Schwartz problems*owing to their analogy with the classic Schwartz problem on finding an analytic function of a complex variable when values of its real part are given on the boundary of domain. We shall call problems of such a type as (

*k*-

*m*)-

*problems in the sense of Kovalev*.

Note, that in previous papers [5, 6, 8, 9, 10, 11, 12, 13] we interpret the (*k*-*m*)-problem as the (*k*-*m*)-problem in the sense of Kovalev.

It was established in [7] that all (*k*-*m*)-problems are reduced to the main three problems: with *k* = 1 and *m* ∈{2, 3, 4}, respectively.

It is shown (see [7, 8, 9]) that the main biharmonic problem is reduced to the (1–3)-problem. In [8], we investigated the (1–3)-problem for cases where *D*_{ζ} is either a half-plane or a unit disk in the biharmonic plane. Its solutions were found in explicit forms with using of some integrals analogous to the classic Schwartz integral.

In [9, 10], using a hypercomplex analog of the Cauchy type integral, we reduced the (1–3)-problem to a system of integral equations and established sufficient conditions under which this system has the Fredholm property. It was made for the case where the boundary of domain belongs to a class being wider than the class of Lyapunov curves that was usually required in the plane elasticity theory (cf., e.g., [14, 15, 16, 17, 18]). The similar is done for the (1–4)-problem in [12].

In [12, 13], there is considered a relation between (1–4)-problem and boundary value problems of the plane elasticity theory. Namely, there is considered a problem on finding an elastic equilibrium for isotropic body occupying *D* with given limiting values of partial derivatives \(\frac {\partial u}{\partial x}\) , \(\frac {\partial v}{\partial y}\) for displacements *u* = *u*(*x*, *y*) , *v* = *v*(*x*, *y*) on the boundary *∂D*. In particular, it is shown in [13] that such a problem is reduced to (1–4)-problem.

## 3 (1–3)-Problem and a Biharmonic Problem

*biharmonic problem*(cf., e.g., [14, p. 13]) is a boundary value problem on finding a biharmonic function \(V \colon D \longrightarrow \mathbb {R}\) with the following boundary conditions:

It is well-known a great importance of the biharmonic problem in the plane elasticity theory (see, e.g., [14, 19]).

_{1}be monogenic in

*D*

_{ζ}function having the sought-for function

*V*(

*x*,

*y*) of the problem (3.1) as the first component:

*x*and using a condition of the type (1.5) for the monogenic function Φ

_{1}, we obtain

In what follows, let us agree to use the same denomination *u* for functions \(u \colon \partial D \longrightarrow \mathbb {R}\), \(u \colon \partial D_z \longrightarrow \mathbb {R}\), \(u \colon \partial D_{\zeta } \longrightarrow \mathbb {R}\) taking the same values at corresponding points of boundaries *∂D*, *∂D*_{z}, *∂D*_{ζ}, respectively, i.e., *u*(*x*_{0}, *y*_{0}) = *u*(*z*_{0}) = *u*(*ζ*_{0}) for all (*x*_{0}, *y*_{0}) ∈ *∂D* .

## 4 Boundary Value Problems Associated with a (1–4)-Problem

*D*is a bounded simply connected domain in the Cartesian plane

*xOy*. For a function \(u \colon D\longrightarrow \mathbb {R}\) we denote a limiting value at a point (

*x*

_{0},

*y*

_{0}) ∈

*∂D*by

*D*partial derivatives \(\mathcal {V}_{1}:=\frac {\partial u}{\partial x}\), \(\mathcal {V}_{2}:=\frac {\partial v}{\partial y}\) for displacements

*u*=

*u*(

*x*,

*y*),

*v*=

*v*(

*x*,

*y*) of an elastic isotropic body occupying

*D*, when their limiting values are given on the boundary

*∂D*:

*k*= 1, 2, are given functions.

We shall call this problem as the (*u*_{x}, *v*_{y})*-problem*. This problem has been posed in [13].

*μ*and

*λ*are Lamé constants (cf., e.g., [19, p. 2]).

*D*(cf., e.g., [19, pp. 8–9],[14, p. 5]):

*u*

_{x},

*v*

_{y})-problem is reduced to finding the functions C

_{k}[

*W*],

*k*= 1, 2, in

*D*with an unknown biharmonic function \(W \colon D \longrightarrow \mathbb {R}\), when their limiting values satisfy the system

Consider some auxiliary statements.

### Lemma 4.1

*[*13 ])

*Let W be a biharmonic function in a domain D and Φ*

_{∗}

*be a monogenic in D*

_{ζ}

*function such that*\(\mathrm {U}_{1}\left [{\Phi }_{\ast }\right ]=W\)

*. Then the following equalities are true*:

*for every* (*x*, *y*) ∈* D, where* \(\Phi :={\Phi }_{\ast }^{\prime \prime }\).

### Lemma 4.2

*The*(

*u*

_{x},

*v*

_{y})

*-problem is equivalent to a boundary value problem on finding in D the second derivatives*\(\frac {\partial ^{2} W(x,y)}{\partial x^2}\) , \(\frac {\partial ^{2} W(x,y)}{\partial y^2}\)

*of a biharmonic function W, which have limiting values at all*(

*x*

_{0},

*y*

_{0}) ∈

*∂D and satisfy the boundary data*:

*Then the general solution of*(

*u*

_{x},

*v*

_{y})

*-problem is expressed by the formula:*

The following theorem establishes relations between solutions of (*u*_{x}, *v*_{y})-problem and corresponding (1–4)-problem.

### Theorem 4.3

*Let W be a biharmonic function satisfying the boundary conditions*(4.2)

*. Then W rebuilds the general solution of*(

*u*

_{x},

*v*

_{y})

*-problem with boundary data*(4.1)

*by the formula*(4.4)

*. The general solution Φ of (1–4)-problem with boundary data*

*generates the second order derivatives*\(\frac {\partial ^2 W}{\partial x^2}\) , \(\frac {\partial ^2 W}{\partial y^2}\)

*in D by the formulas*(4.3)

*. The general solution of*(

*u*

_{x},

*v*

_{y})

*-problem is expressed for every*(

*x*,

*y*) ∈

*D by the equalities*

A theorem analogous to Theorem 4.3 is proved in [13, Theorem 4] in assumption that the (1–4)-problem is understood in the sense of Kovalev. But it is still valid with the same proof for the (1–4)-problem formulated in this paper. It happens due to Lemmas 4.1, 4.2 and the fact that the left-hand sides of (4.3) have limiting values on *∂D* if and only if \(\mathrm {U}_{1}\left [\Phi \right ]\), \(\mathrm {U}_{4}\left [\Phi \right ]\) have limiting values on *∂D*_{ζ}.

The elastic equilibrium in terms of displacements and stresses can be found by use of the generalized Hooke’s law and solutions \(\mathcal {V}_{1}\), \(\mathcal {V}_{2}\) of the (*u*_{x}, *v*_{y})-problem (see [13, sect. 5]).

## 5 Solving Process of (1–4)-Problem via Analytic Functions of a Complex Variable

A method for solving the (1–4)-problem by means of its reduction to classic Schwartz boundary value problems for analytic functions of a complex variable is delivered in [11]. Let us formulate some results of such a kind.

In what follows, we assume that the domain *D*_{z} is simply connected (bounded or unbounded), and in this case we shall say that the domains *D* and *D*_{ζ} are also simply connected.

For a function \(F\colon D_{z} \longrightarrow \mathbb {C}\) we denote its limiting value at a point *z*_{0} ∈ *∂D*_{z} by *F*^{+}(*z*_{0}) if it exists.

*classic Schwartz problem*is a problem on finding an analytic function \(F \colon D_{z} \longrightarrow \mathbb {C}\) of a complex variable when values of its real part are given on the boundary of domain, i.e.,

### Theorem 5.1

*Let*\( u_l \colon \partial D_{\zeta } \longrightarrow \mathbb {R}\)

*, l*∈{1, 4}

*, be continuous functions and F be a solution of the classic Schwartz problem with boundary condition*:

*and, furthermore, the function*

*have continuous limiting values on ∂D*

_{z}

*. Then a solution of the (1–4)-problem is expressed by the formula*(1.6)

*or, the same, by the formula*(1.7)

*, where the function F*

_{0}

*is a solution of the classic Schwartz problem with boundary condition*:

### Proof

*D*

_{z}functions

*F*,

*F*

_{0}satisfying the following boundary conditions:

In the case where the function \(\widetilde {F}_{\ast }\) has continuous limiting values on *∂D*_{z}, the conditions (5.4) are equivalent to the boundary conditions (5.2), (5.3) of classic Schwartz problems. □

### Theorem 5.2

*The general solution of the homogeneous*(1–4)

*-problem for an arbitrary simply connected domain D*

_{ζ}

*is expressed by the formula*

*where a* _{1} *, a* _{2} *are any real constants*

### Proof

By Theorem 5.1, a solving process of the homogeneous (1–4)-problem consists of consecutive finding of solutions of two homogeneous classic Schwartz problems, viz.:

- a)
to find an analytic in

*D*_{z}function*F*satisfying the boundary condition (Re*F*)^{+}(*t*) = 0 for all*t*∈*∂D*_{z}. As a result, we have*F*(*z*) =*ai*, where*a*is an arbitrary real constant; - b)
to find similarly an analytic in

*D*_{z}function*F*_{0}satisfying the boundary condition (Re*F*_{0})^{+}(*t*) = 0 for all*t*∈*∂D*_{z}.

Consequently, getting a general solution of the homogeneous (1–4)-problem in the form (1.7), we can rewrite it in the form (5.5). □

### Remark 5.3

A statement similar to Theorem 5.2 is proved for homogeneous (1–4)-problem in the sense of Kovalev in [11], where the formula of solutions is the same as (5.5).

### Remark 5.4

*∂D*

_{z}if and only if \(\mathrm {U}_{1}\left [\widetilde {\Phi }\right ]\) is continuously extended on

*∂D*.

### Remark 5.5

Theorem 5.2 shows an example of the (1–4)-problem (with *u*_{1} = *u*_{4} ≡ 0 and with no extra assumptions on a domain *D*_{ζ}) when the condition on existence of continuous limiting values \(\bigl (\widetilde {F}_{\ast }\bigr )^{+}\) is satisfied. Evidently, we have another similar trivial case, where *u*_{1} and *u*_{4} are real constants. In the next section we consider else a nontrivial case of the (1–4)-problem when the continuous limiting values \(\bigl (\widetilde {F}_{\ast }\bigr )^{+}\) exist.

## 6 (1–4)-Problem for a Half-Plane

Consider the (1–4)-problem in the case where the domain *D*_{ζ} is the half-plane Π^{+} := {*ζ* = *xe*_{1} + *ye*_{2} : *y* > 0}.

*biharmonic Schwartz integral*for the half-plane Π

^{+}:

^{+}of the complex Schwartz integral

*z*=

*x*+

*iy*:

*y*> 0} of the complex plane \(\mathbb {C}\) with the given boundary values

*u*(

*t*) of real part on the real line \(\mathbb {R}\). Furthermore, the equality

*u*.

### Theorem 6.1

*Let every function*\(u_l \colon \mathbb {R}\longrightarrow \mathbb {R}\)

*, l*∈{1, 4}

*, have a finite limit of the type*(2.1)

*. Then the general solution of the*(1–4)

*-problem for the half-plane Π*

^{+}

*is expressed by the formula*

*where a* _{1} *, a* _{2} *are any real constants.*

### Proof

^{+}.

The general solution of the (1–4)-problem in the form (6.5) is obtained by summarizing the particular solution (6.6) of the inhomogeneous (1–4)-problem and the general solution (5.5) of the homogeneous (1–4)-problem. □

### Remark 6.2

In Theorem 3 [11] we obtain the general solution of (1–4)-problem in the sense of Kovalev in the form (6.5) but under complementary assumptions that for every given function \(u_l \colon \mathbb {R}\longrightarrow \mathbb {R}\), *l* ∈{1, 4}, its modulus of continuity and the local centered (with respect to the infinitely remote point) modulus of continuity satisfy Dini conditions.

## 7 Solving Process of (1–4)-Problem for Bounded Simply Connected Domain with Use of the Complex Green Function

Now, for solving the (1–4)-problem we shall use solutions of the classic Schwartz problem for analytic functions of a complex variable in the form of an appropriate Schwartz operator involving the complex Green function.

Here we assume that *D*_{z} is a bounded simply connected domain with a smooth boundary *∂D*_{z}. Let *g*(*z*, *z*_{0}) be the Green function of *D*_{z} for the Laplace operator (cf, e.g., [21, p. 22]).

*a*

_{0}and the Schwartz operator (

*Su*)(

*z*) having the form

*M*(cf, e.g., [21, p. 32]) is of the form

*M*(

*w*,

*z*) =

*g*(

*w*,

*z*) +

*ih*(

*w*,

*z*),

*h*is a conjugate harmonic function to the Green function

*g*with respect to

*w*∈

*D*

_{z}:

*w*≠

*z*,

**n**

_{t}is the outward normal unit vector at the point

*t*∈

*∂D*

_{z},

*s*

_{t}is an arc coordinate of the point

*t*.

*u*

_{1}and

*u*

_{4}are given functions of the (1–4) problem.

Therefore, using expressions of solutions of the classic Schwartz problems with the boundary conditions (5.2), (5.3) in the form (7.1) via appropriate Schwartz operators of the type (7.2), by Theorem 5.1 we obtain the following statement.

### Theorem 7.1

*Let the function*(7.3)

*have the continuous limiting values*\(\bigl (\widetilde {F}_{\ast }\bigr )^{+}\)

*on ∂D*

_{z}

*. Then the general solution of (1–4)-problem is expressed in the form*

*where*

*and a* _{1} *, a* _{2} *are any real constants.*

In the next section we develop a method for solving the inhomogeneous (1–4)-problem without an essential in Theorem 5.1 assumption that the function \(\widetilde {F}_{\ast }\) have continuous limiting values on the boundary *∂D*_{z}.

## 8 Solving BVP’s by Means Hypercomplex Cauchy-Type Integrals

Let the boundary *∂D*_{ζ} of the bounded domain *D*_{ζ} be a closed smooth Jordan curve.

Below, we show a method for reducing (1–3)-problem and (1–4)-problem to systems of the Fredholm integral equations. Such a method was developed in [9, 12]. Obtained results are appreciably similar for the mentioned problems, however, in contrast to (1–3)-problem, which is solvable in a general case if and only if a certain natural condition is satisfied, the (1–4)-problem is solvable unconditionally.

*φ*given on

*∂D*

_{ζ}:

*ω*(

*φ*,

*ε*) satisfies the Dini condition:

*biharmonic*Cauchy type integral:

*ζ*

_{0}∈

*∂D*

_{ζ}that are represented by the Sokhotski–Plemelj formulas:

We assume that boundary functions *u*_{k}, *k* ∈{1, 3} or *k* ∈{1, 4}, of the (1–3) problem or the (1–4) problem, respectively, satisfy Dini conditions of the type (8.1).

*k*∈{1, 3, 4}, satisfies a Dini condition of the type (8.1).

We use a conformal mapping *z* = *τ*(*t*) of the upper half-plane \(\{t\in \mathbb {C}: \mathrm {Im}\,t>0\}\) onto the domain *D*_{z}. Denote *τ*_{1}(*t*) := Re *τ*(*t*), *τ*_{2}(*t*) := Im *τ*(*t*).

*∂D*

_{ζ}.

*k*(

*t*,

*s*) =

*k*

_{1}(

*t*,

*s*)

*e*

_{1}+

*iρ*

*k*

_{2}(

*t*,

*s*) ,

Evidently, *g*(*s*) = *g*_{1}(*s*)*e*_{1} + *g*_{3}(*s*)*e*_{2} for the (1–3) problem and *g*(*s*) = *g*_{1}(*s*)*e*_{1} + *g*_{4}(*s*)*ie*_{2} for the (1–4) problem, where \(g_{l}(s):=\varphi _{l}\left ( \widetilde {\tau }(s)\right )\) for all \(s\in \overline {\mathbb {R}}\), *l* ∈{1, 3, 4}.

*l*∈{1, 3}, and after the substitution them into the boundary conditions of the (1–3)-problem, we shall obtain the following system of integral equations for finding the functions

*g*

_{1}and

*g*

_{3}:

*l*∈{1, 3}.

*l*∈{1, 4}, and after the substitution them into the boundary conditions of the (1–4)-problem, we shall obtain the following system of integral equations for finding the functions

*g*

_{1}and

*g*

_{4}:

*l*∈{1, 4}.

Let \(C(\overline {\mathbb {R}})\) denote the Banach space of functions \(g_{\ast } \colon \overline {\mathbb {R}}\longrightarrow \mathbb {C}\) that are continuous on the extended real axis \(\overline {\mathbb {R}}\) with the norm \(\|g_{\ast }\|{ }_{C(\overline {\mathbb {R}})}:=\sup \limits _{t\in \mathbb {R}}|g_{\ast }(t)|\).

In Theorem 6.13 [9] there are conditions which are sufficient for compactness of integral operators on the left-hand sides of equations of the systems (8.6), (8.7) in the space \(C(\overline {\mathbb {R}})\).

To formulate such conditions, consider the conformal mapping *σ*(*T*) of the unit disk \(\{T\in \mathbb {C}: |T|<1\}\) onto the domain *D*_{z} such that \(\tau (t)=\sigma \left (\frac {t-i}{t+i}\right )\) for all \(t\in \{t\in \mathbb {C}: \mathrm {Im}\,t>0\}\).

*σ*(

*T*) have the nonvanishing continuous contour derivative

*σ′*(

*T*) on the unit circle \(\Gamma :=\{T\in \mathbb {C}: |T|=1\}\), and its modulus of continuity

Since the sought-for function *φ* in (8.2) has to satisfy the condition (8.1), it is necessary to require that the corresponding functions *g*_{1}, *g*_{3} in (8.4) or *g*_{1}, *g*_{4} in (8.5) should belong to the class \({\mathcal D}(\overline {\mathbb {R}})\). In the next theorems we state a condition on the conformal mapping *σ*(*T*), under which all solutions of the system (8.6), (8.7) satisfy the mentioned requirement.

### Theorem 8.1

*Assume that the functions*\( u_l \colon \partial D_{\zeta } \longrightarrow \mathbb {R}\)

*, l*∈{1, 3}

*, satisfy conditions of the type*(8.1)

*. Also, assume that the conformal mapping σ*(

*T*)

*has the nonvanishing continuous contour derivative σ′*(

*T*)

*on the circle Γ, and its modulus of continuity ω*

_{Γ}(

*σ′*,

*ε*)

*satisfies the condition*

*Then all functions* \(g_1, g_3\in C(\overline {\mathrm {R}})\) *satisfying the system of Fredholm integral equations* (8.6) *belong to the class* \({\mathcal D}(\overline {\mathrm {R}})\)*, and the corresponding function φ in* (8.4) *satisfies the Dini condition* (8.1).

*Assume additionally:*

- 1)
*all solutions*\((g_1,g_3)\in C(\overline {\mathbb {R}})\times C(\overline {\mathbb {R}})\)*of the homogeneous system of equations*(8.6) (*with*\(\widetilde {u}_{k} \equiv 0\)*for k*∈{1, 3})*are differentiable on*\(\mathbb {R}\) ; - 2)
*for every mentioned solution*(*g*_{1},*g*_{3})*of the homogeneous system of equations*(8.6)*, the integral*\(\mathcal {B}\left [ \varphi ' \right ]\)*is finite in D*_{ζ}*and*\(\mu _{e_1,e_2}\setminus \overline {D_{\zeta }}\)*, and the functions*$$\displaystyle \begin{aligned} \mathrm{U}_{1}\left[\mathcal{B}\left[\varphi'\right](\zeta) \right]- \mathrm{U}_{4}\left[\mathcal{B}\left[\varphi'\right](\zeta) \right] \quad \forall\, \zeta\in D_{\zeta},\end{aligned}$$$$\displaystyle \begin{aligned} \mathrm{U}_{2}\left[\mathcal{B}\left[\varphi'\right](\zeta) \right]+ \mathrm{U}_{3}\left[\mathcal{B}\left[\varphi'\right](\zeta) \right] \quad \forall\, \zeta\in \mu_{e_1,e_2}\setminus\overline{D_{\zeta}}\end{aligned}$$*are bounded, where φ′ is the contour derivative of the corresponding function φ in*(8.4)*, i.e.,*\(\varphi \left (\zeta \right ) \equiv \varphi (\widetilde {\tau }(s)):= g_{1}(s) e_1 + g_{3}(s) e_2\)*for all*\(s\in \mathbb {R}\).

### Theorem 8.2

*Assume that the functions*\( u_l \colon \partial D_{\zeta } \longrightarrow \mathbb {R}\)

*, l*∈{1, 4}

*, satisfy conditions of the type*(8.1)

*. Also, assume that the conformal mapping σ*(

*T*)

*has the nonvanishing continuous contour derivative σ′*(

*T*)

*on the circle Γ, and its modulus of continuity satisfy the condition*(8.8)

*. Then the following assertions are true*:

- (i)
*the system of Fredholm integral equations*(8.7)*has the unique solution in*\(C(\overline {\mathbb {R}})\) ; - (ii)
*all functions*\(g_1, g_4\in C(\overline {\mathbb {R}})\)*satisfying the system*(8.7)*belong to the class*\({\mathcal D}(\overline {\mathbb {R}})\)*, and the corresponding function φ in*(8.5)*satisfies the Dini condition*(8.1).

## Notes

### Acknowledgements

This research is partially supported by the State Program of Ukraine (Project No. 0117U004077) and Grant of Ministry of Education and Science of Ukraine (Project No. 0116U001528).

## References

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