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On solvability of a nonlocal problem for the Laplace equation with the fractional-order boundary operator

Moldir A Muratbekova*, Kanat M Shinaliyev and Batirkhan K Turmetov

Author Affiliations

Department of Mathematics, Akhmet Yasawi International Kazakh-Turkish University, B. Sattarkhanov Street 29, Turkistan, 161200, Kazakhstan

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Boundary Value Problems 2014, 2014:29  doi:10.1186/1687-2770-2014-29


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/29


Received:15 August 2013
Accepted:9 January 2014
Published:30 January 2014

© 2014 Muratbekova et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present work, we study properties of some integro-differential operators of the Hadamard-Marchaud type in the class of harmonic functions. As an application of these properties, we consider the question of the solvability of a nonlocal boundary value problem for the Laplace equation in the unit ball.

MSC: 35J05, 35J25, 26A33.

Keywords:
Hadamard-Marchaud operator; fractional derivative; nonlocal problem

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M1">View MathML</a> be the unit ball, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M2">View MathML</a>. A paradigm in the theory of elliptic partial differential equations and harmonic functions is the Laplace equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M3">View MathML</a>

(1)

If we prescribe the values of the solution at the boundary Ω of Ω, then we can solve equation (1) uniquely. Of course, one can consider many other boundary conditions such as Neumann’s boundary conditions.

In some applied problems of hydrodynamics [1], it is necessary to prescribe the value of a fractional derivative of the solution on the boundary. Fractional differential equations and boundary value problems involving fractional derivatives appear in many applied problems ranging from the spring-pot model [2] to geology [3] or from nonlinear circuits [4] to alternative models to differential equations [5].

Hence, in this paper we study the Laplace equation concentrating on some conditions on the boundary involving derivatives of fractional order.

Note that numerous works of authors [6-13] were dedicated to the research questions of the solvability of boundary value problems for partial differential equations with boundary operators of high (whole and fractional) order. In the paper of A.N. Tikhonov [14] boundary value problems with boundary conditions containing derivatives of higher order have been investigated for the heat equation. Research questions as regards the solvability of similar problems for higher-order equations with boundary operators of whole and fractional order were carried out in [15,16]. Later in [17], these results were generalized for partial differential equations of fractional order. In [18-20] questions about the solvability of boundary value problems with boundary operators of high order were studied for the Laplace equation. In the studies of these authors the exact conditions for the solvability have been established and the integral representations of solutions of the studied problems have been found. The cycle of studies by the authors [21-26] is devoted to the study of the existence and smoothness of solutions of boundary value problems for the second-order elliptic equations with boundary operators of fractional order. In the paper mentioned above local boundary value problems with boundary operators of fractional order in the Riemann-Liouville or Caputo sense are studied. In this paper we study nonlocal problems with boundary operators of fractional-order derivatives of Hadamard type. Definitions of Hadamard operators, a statement of the main problems, and the history of the questions on this topic are in Section 3.

The organization of this paper is as follows. In Section 2, we present the operators of integration and differentiation in the Hadamard sense and some modifications. In the third section we provide a formulation of the basic problem of this paper and some historical information as regards nonlocal boundary value problems. In the fourth section we study the properties of integral and differential Hadamard-Marchaud operators in the class of harmonic functions in the ball. In Section 5 we provide some auxiliary propositions. Finally, Section 6 is devoted to the study of the fundamental problem, where we formulate and prove the main statement of the paper.

2 Definition of Hadamard operators of integration and differentiation and some modifications

In this section, we give a statement on the operators of fractional differentiation in the sense of Hadamard, Hadamard-Marchaud, and their modifications.

For any positive α, fractional integrals and derivatives of the order α in the sense of Hadamard are defined by the following formulas [27]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M4">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M5">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M6">View MathML</a> is the Dirac operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M7">View MathML</a> is the integral part of α.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M8">View MathML</a>, then, in the class of sufficiently ‘good’ functions, operator (3) can be reduced to the following form [27]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M9">View MathML</a>

(4)

This operator is said to be the differentiation operator of order α in the sense of Hadamard-Marchaud.

In [28], the following modification of the Hadamard-Marchaud operator was considered:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M10">View MathML</a>

(5)

In [18], in the class of harmonic functions in a ball, the properties and applications of the operators in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M11">View MathML</a>

(6)

are considered. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M14">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M15">View MathML</a> is a differential operator in the form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M16">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a> be a harmonic function in the domain Ω, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M19">View MathML</a> be arbitrary real numbers. Let us consider a modification of the Bavrin operator (6).

Introduce the operators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M20">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, then we obtain the Bavrin operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M23">View MathML</a>.

3 Statement of the problem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M25">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M26">View MathML</a>, be continuous mappings, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27">View MathML</a> be continuous functions satisfying the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M28">View MathML</a>

(7)

We assume that the series (7) converges uniformly on Ω.

Further, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31">View MathML</a>, i.e.α and β are not equal to zero simultaneously.

Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M32">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M33">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M34">View MathML</a>.

A harmonic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M35">View MathML</a> from the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M36">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M37">View MathML</a> and condition (9) is realized in the classical sense, will be called a solution of problem (8)-(9).

The above-mentioned problem is a simple generalization of Bitsadze-Samarskii’s nonlocal problem [29]. For convenience of the reader, we formulate the Bitsadze-Samarskii problem.

Let D be a finite simply-connected domain of the plane of complex variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M38">View MathML</a> with the smooth boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M39">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M40">View MathML</a> be a closed simple smooth curve lying in D.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M43">View MathML</a> a diffeomorphism between S and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M44">View MathML</a>.

Formulation of the problem: We are to find a harmonic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M45">View MathML</a> in D, which is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M46">View MathML</a> and satisfies the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M47">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M48">View MathML</a> is a given function.

Similar problems with operators of integer order were considered in [30-32], and for operators of fractional order with fractional-order derivatives in the sense of Riemann-Liouville and Caputo in [33-41]. It should also be noted that some questions of solvability of nonlocal problems for fractional-order equations in the one-dimensional case were studied in [42-44].

4 Properties of operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50">View MathML</a>

In this section, we study some properties of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50">View MathML</a> in the class of harmonic functions. Further, for convenience, we shall take everywhere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M53">View MathML</a>.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M56">View MathML</a>be a homogeneous harmonic polynomial of the power<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M57">View MathML</a>. Then the following equalities are correct:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M58">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M59">View MathML</a>

(11)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>. Then, using homogeneity of the polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M62">View MathML</a>

The value of the last integral can easily be calculated with the help of the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M63">View MathML</a>. In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M64">View MathML</a>

The equality (10) is proved.

Further, note that the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M65">View MathML</a>

(12)

holds for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M66">View MathML</a>.

Now, let us study actions of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67">View MathML</a> to the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61">View MathML</a>. Using the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67">View MathML</a> and the homogeneity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M61">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M71">View MathML</a>

Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M72">View MathML</a> and integrating by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M73">View MathML</a>

After the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M74">View MathML</a>, as in the proof of equality (10), we easily obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M75">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M76">View MathML</a>.

Further, taking into account fulfilling of equality (12), we obtain in the general case for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M77">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M78">View MathML</a>

The lemma is proved. □

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M81">View MathML</a>be a harmonic function in the ball Ω. Then the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M83">View MathML</a>are also harmonic in Ω.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a> be a harmonic function in the ball Ω. Then it is known [45] that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a> is represented in the form of the series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M86">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M87">View MathML</a> is a complete system of homogeneous harmonic polynomials of power k, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M88">View MathML</a> are coefficients of the expansion (13). Applying formally the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50">View MathML</a> to the series (13) and taking into account equality (11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M90">View MathML</a>

(14)

Now let us check convergence of the series (13) and (14). The following asymptotical estimate is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M91">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M92">View MathML</a>

Moreover, the series (13) converges absolutely and uniformly by x at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M93">View MathML</a>, hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M94">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M95">View MathML</a>, the equalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M96">View MathML</a> hold. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M97">View MathML</a>, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M99">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M100">View MathML</a>

Therefore, the series (14) converges absolutely and uniformly by x at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M98">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M103">View MathML</a>, and its sum is a harmonic function. By virtue of the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M103">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82">View MathML</a> is defined in the whole ball Ω.

Let us study the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49">View MathML</a>. Applying formally the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49">View MathML</a> to the series (13), taking into account equality (10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M109">View MathML</a>

Convergence of this series can be checked as in the case of series (14), and that is why <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M110">View MathML</a> is a harmonic function in the ball Ω. The lemma is proved. □

Now we show that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a> can be represented in terms of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M112">View MathML</a>.

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M115">View MathML</a>be a harmonic function in the domain Ω. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116">View MathML</a>the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M117">View MathML</a>

is valid.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>. Represent a harmonic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a> in the form of the series (13) and transform it to the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M120">View MathML</a>

(15)

Further, taking into account equalities (10)-(11), and the absolute and uniform convergence of the series (15) by x at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M93">View MathML</a>, it can be reduced to the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M122">View MathML</a>

The lemma is proved. □

One can similarly prove the following lemma.

Lemma 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a>be a harmonic function in the domain Ω. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116">View MathML</a>the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M127">View MathML</a>

(16)

is valid.

Lemma 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M115">View MathML</a>be a harmonic function in the domain Ω. Then the following equalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M131">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>. Applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M49">View MathML</a> to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M134">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M135">View MathML</a>

By virtue of Lemma 3, the value of the last integral is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M136">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M137">View MathML</a>.

To prove the second equality, apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M67">View MathML</a> to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M139">View MathML</a>. We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M140">View MathML</a>

Then, in the general case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M141">View MathML</a>

The lemma is proved. □

5 Some auxiliary propositions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27">View MathML</a> satisfy the conditions from Section 2.

Consider the following problem in the domain Ω:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M144">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M145">View MathML</a>

(18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31">View MathML</a>, i.e.α and β are not equal to zero simultaneously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M149">View MathML</a>.

A harmonic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M150">View MathML</a> from the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M151">View MathML</a>, satisfying condition (18) in the classical case, will be called a solution of problem (17)-(18).

It should be noted that problem (17)-(18) was investigated for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M152">View MathML</a> in [30].

Let us investigate uniqueness for the solution of problem (17)-(18). The following statement holds.

Lemma 6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M154">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M156">View MathML</a> , be continuous functions satisfying the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M157">View MathML</a>

(19)

and let a solution of problem (17)-(18) exist.

Then:

(1) If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M158">View MathML</a>

(20)

then the solution of problem (17)-(18) is unique.

(2) If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M159">View MathML</a>

(21)

then the solution of problem (17)-(18) is unique up to a constant summand.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M150">View MathML</a> be the solution of problem (17)-(18) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161">View MathML</a>.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M163">View MathML</a>.

Then if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M164">View MathML</a>, then, by virtue of the maximum principle for harmonic functions [46], the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M165">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M116">View MathML</a>.

The boundary condition (18) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M168">View MathML</a>

Further, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M25">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M26">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M172">View MathML</a>, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M174">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M175">View MathML</a>.

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M176">View MathML</a>

If now condition (19) is realized , then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M177">View MathML</a>, and we obtain from this the contradiction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M178">View MathML</a>.

Hence, if condition (19) holds, it is necessary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M179">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M180">View MathML</a>, substituting the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M181">View MathML</a> into the boundary condition (18), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M182">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M183">View MathML</a>

The last equality is equivalent to the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M184">View MathML</a>

We obtain from this the result that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M185">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M186">View MathML</a>.

Thus, if conditions (19) and (20) are fulfilled, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M185">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M188">View MathML</a>.

If the conditions (21) are fulfilled, then any constant is a solution of the homogeneous problem (17)-(18). In fact, substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M189">View MathML</a> into equation (18), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M190">View MathML</a>

The lemma is proved. □

Now investigate existence of a solution of problem (17)-(18). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M191">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M192">View MathML</a> be the Poisson kernel of the Dirichlet problem, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M193">View MathML</a> the area of the unit sphere.

Introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M194">View MathML</a>

(22)

and consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M195">View MathML</a>

(23)

The following statement holds.

Lemma 7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M198">View MathML</a> , be continuous functions satisfying the condition (19). Then:

(1) If the condition (20) is realized, then problem (17)-(18) is uniquely solvable at any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M199">View MathML</a>.

(2) If the condition (21) is realized, then problem (17)-(18) is solvable if the following condition is realized:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M200">View MathML</a>

(24)

where the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M201">View MathML</a>is a solution of equation (23), moreover the number of independent solutions of this equation under these conditions is equal to 1.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> is a harmonic function, a solution of problem (17)-(18) can be found in the form of the Poisson integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M203">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M204">View MathML</a> is an unknown function. Substituting this function into the boundary condition (18), we obtain the integral equation with respect to the unknown function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M205">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M206">View MathML</a>

(25)

Designate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M207">View MathML</a>

Then equation (25) can be rewritten in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M208">View MathML</a>

(26)

To investigate the solvability of the integral equation (26), we study the properties of the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209">View MathML</a>. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209">View MathML</a> is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211">View MathML</a>.

In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M212">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M213">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M214">View MathML</a>, and therefore the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M215">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211">View MathML</a>. Further, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M217">View MathML</a> has an integrable singularity, and that is why the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M218">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M211">View MathML</a>. Then by virtue of the uniform convergence of the series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M220">View MathML</a>, the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M209">View MathML</a> is also a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M222">View MathML</a>.

Hence, one can apply Fredholm theory to equation (26). Since in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161">View MathML</a> and fulfillment of the condition (20), the solution of problem (17)-(18) can only be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M224">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M161">View MathML</a> the integral equation (26) has only a trivial solution.

Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M199">View MathML</a> the solution of equation (26) exists, is unique, and belongs to the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M227">View MathML</a>. Using this solution, we construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> which will satisfy all the conditions of problem (17)-(18).

If the condition (21) is valid, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M229">View MathML</a> satisfies the condition (18) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M230">View MathML</a>, i.e. the corresponding homogeneous equation (26) has the nonzero solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M231">View MathML</a>. Then the adjoint homogeneous equation has also a nonzero solution, and that is why in this case fulfillment of the condition (24) is necessary and sufficient for solvability of problem (17)-(18). The lemma is proved. □

6 Study of the basic problem

We now formulate the basic statement.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M198">View MathML</a> , be continuous functions satisfying the condition (19). Then:

(1) If the condition (20) is fulfilled, then problem (8)-(9) is uniquely solvable at any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M238">View MathML</a>.

(2) If the condition (21) is fulfilled, then the condition (24) is necessary and sufficient for solvability of problem (8)-(9) where the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M201">View MathML</a>is a solution of equation (23). If a solution of the problem exists, then it is unique up to the constant summand.

(3) If a solution of problem (8)-(9) exists, then it is represented in the form of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a>is a solution of problem (17)-(18).

Proof (1) Let a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242">View MathML</a> of problem (8)-(9) exist. Apply to this function the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M50">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244">View MathML</a>. Take the problem which the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> satisfies. Since by Lemma 2, in the case of harmonicity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M17">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M82">View MathML</a> is also harmonic in Ω, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> is harmonic.

Further, since according to Lemma 5 the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M249">View MathML</a> holds, the boundary condition of problem (8)-(9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M250">View MathML</a>

with respect to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> will be rewritten in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M252">View MathML</a>

In addition, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M253">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M254">View MathML</a>. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242">View MathML</a> is a solution of problem (8)-(9), then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244">View MathML</a> will be a solution of problem (17)-(18).

Now, let the conditions (19) and (20) be realized. Then by Lemmas 6 and 7, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M257">View MathML</a> the solution of problem (17)-(18) exists, is unique, and designate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M259">View MathML</a>. Then we have by Lemma 5 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M260">View MathML</a> in Ω, and therefore we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M253">View MathML</a>. Harmonicity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242">View MathML</a> follows from Lemma 2, and fulfillment of the conditions (9) can be checked immediately:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M263">View MathML</a>

The first statement of the theorem is proved.

(2) Let now the condition (21) be fulfilled, and let the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M242">View MathML</a> of problem (8)-(9) exist. Consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M244">View MathML</a>. As in the first case, we show that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M202">View MathML</a> satisfies the conditions of problem (17)-(18). Then according to Lemma 7, fulfillment of the condition (24) is necessary. Thus, we prove the necessity of the condition (24) at fulfillment of the equality (21).

We show that if the equality (21) is fulfilled, then the condition (24) is also sufficient for the existence of the solution of problem (8)-(9).

In fact, if the conditions (21) and (24) are realized, a solution of problem (17)-(18) exists, is unique up to constant summand, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M267">View MathML</a>. Then, similarly to the proof of the first statement of the theorem, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M240">View MathML</a> satisfies all the conditions of problem (8)-(9). The theorem is proved. □

Remark 1 One can show that in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M269">View MathML</a>, the corresponding homogeneous problem (8)-(9) has nontrivial solutions.

Example 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M272">View MathML</a> , and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M274">View MathML</a>. Further, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M275">View MathML</a> be a homogeneous harmonic polynomial of the power k. By virtue of the equality (11), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M276">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M277">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M278">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M279">View MathML</a> the harmonic polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M280">View MathML</a> will be the solution of the homogeneous problem (8)-(9). If δ is a number close to zero, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M281">View MathML</a>.

If the dimension of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M282">View MathML</a>, then the number of these polynomials is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/29/mathml/M283">View MathML</a>[47].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

This work has been supported by the MON Republic of Kazakhstan under Research Grant No. 0713/GF.

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