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On solvability of the Neumann boundary value problem for a non-homogeneous polyharmonic equation in a ball

Batirkhan K Turmetov1* and Ravshan R Ashurov2

Author Affiliations

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

2 Institute of Mathematics National University of Uzbekistan, Hodjaeva st., 29, Tashkent, 100125, Uzbekistan

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Boundary Value Problems 2013, 2013:162  doi:10.1186/1687-2770-2013-162


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/162


Received:23 April 2013
Accepted:19 June 2013
Published:5 July 2013

© 2013 Turmetov and Ashurov; 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 this work the Neumann boundary value problem for a non-homogeneous polyharmonic equation is studied in a unit ball. Necessary and sufficient conditions for solvability of this problem are found. To do this we first reduce the Neumann problem to the Dirichlet problem for a different non-homogeneous polyharmonic equation and then use the Green function of the Dirichlet problem.

MSC: 35J40, 35J30, 35A01.

Keywords:
non-homogeneous polyharmonic equation; the Neumann problem; the necessary and sufficient conditions for solvability

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M1">View MathML</a> be a unit ball, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M2">View MathML</a> be a unit sphere and m be a positive integer.

Consider on the domain Ω the following Neumann boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M3">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M4">View MathML</a>

(2)

where ν is the unit outer normal vector to sphere Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a> are given functions; we always suppose that these functions are sufficiently smooth, and from here on we do not pay any attention to their smoothness.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M7">View MathML</a> is called a solution of problem (1), (2) if it satisfies (1), (2) in a classical sense.

It is well known (see, for example, [1]) that even in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M8">View MathML</a> the considered problem (1), (2) does not have solutions for arbitrary (even, as we supposed, smooth) functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a>; in the case of the Poisson equation, the necessary and sufficient solvability condition for the Neumann problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M11">View MathML</a>

In the paper [2] by Kanguzhin and Koshanov, in particular, it is shown that in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M12">View MathML</a> the necessary and sufficient condition for solvability of problem (1), (2) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M14">View MathML</a> is the scalar product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M16">View MathML</a>.

The authors of the paper [3] presented this condition in a different form, which could be easily verified,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M17">View MathML</a>

In the above paper [2] the authors found a solvability condition for Neumann problem (1), (2) for arbitrary m as well (see [2], Theorem 4.2). This condition follows from equality to zero of the determinant of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M18">View MathML</a> matrix, one column of which consists of integrals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M21">View MathML</a> is a constant. Note that the equation which one has as a result is very difficult to verify.

The main goal of the present paper is to find a solvability condition for problem (1), (2) in a more simple form. It should be noted that in our study of problem (1), (2) the Green function of the Dirichlet problem for equation (1) is essentially used. In the paper [4] a similar method was used in the solution of the boundary value problem for the Poisson equation with the boundary operator of fractional-order.

The paper is organized as follows. In the next section we study the properties of some integro-differential operators, which we then use throughout the paper. In Section 3 we investigate the Dirichlet problem for a polyharmonic equation, making use of the explicit form of the Green function found in [5-7]. Then, in the following section, reducing Neumann problem (1), (2) to the considered Dirichlet problem, we give the necessary and sufficient solvability condition for problem (1), (2) with homogeneous boundary conditions. In the same way we consider in Section 5 the Neumann boundary value problem for the homogeneous equation with non-homogeneous boundary conditions. Finally, in Section 6 we study problem (1), (2) in the general case. To present the necessary and sufficient conditions for solvability, here we apply the Almansi formula for constructing solutions to the Dirichlet problem.

2 Properties of some integro-differential operators

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> be a sufficiently smooth function in Ω. Consider the following operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M23">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25">View MathML</a> is a constant. Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M26">View MathML</a> is not defined for functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M28">View MathML</a>.

Note that in the class of harmonic functions in a ball the properties of operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M30">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31">View MathML</a> have previously been studied in the paper [8].

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a>be a smooth function. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33">View MathML</a>one has

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M35">View MathML</a>

(4)

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M37">View MathML</a>

(5)

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M40">View MathML</a>

(6)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M42">View MathML</a>

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M44">View MathML</a>

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M46">View MathML</a>

(7)

Hence, equality (5) and the second equality of (4) are proved.

As we noted above, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39">View MathML</a>, then the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M48">View MathML</a> is defined. Now apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49">View MathML</a> to this expression. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M50">View MathML</a>

But due to equality (7) and the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39">View MathML</a>, the last expression is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a>. Hence, equality (6) is proved. The first equality in (4) can be proved in the same way. □

The following statement can be proved by a direct calculation.

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a>be a smooth function. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33">View MathML</a>one has

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M56">View MathML</a>

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M58">View MathML</a>

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M61">View MathML</a>

Corollary 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a>be a smooth function. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33">View MathML</a>one has

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M65">View MathML</a>

(8)

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M31">View MathML</a>, or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M39">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M69">View MathML</a>

(9)

3 Some properties of the solutions of the Dirichlet problem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> be a solution of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M71">View MathML</a>

(10)

It is known (see, for example, [5-7]) that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M72">View MathML</a> is a sufficiently smooth function, then the solution of problem (10) exists, it is unique and has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M73">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M74">View MathML</a> is the Green function of Dirichlet problem (10).

We make use of the following explicit form of the Green function [5]:

if n is odd, or even and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M75">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M76">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M77">View MathML</a>

if n is even and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M78">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M79">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M80">View MathML</a>

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81">View MathML</a>in Dirichlet problem (10), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a>be the unique solution of this problem. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M84">View MathML</a>

(12)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> be the solution of problem (10). Then it has the form (11). To use the explicit form of the Green function, we shall deal only with the case n is odd or even and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M75">View MathML</a>, the other cases being exactly similar. So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a>, then from (11) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M88">View MathML</a>

If we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M90">View MathML</a>, then the last integral can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M91">View MathML</a>

Now we consider the inner integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M92">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M93">View MathML</a>, we introduce the following two integrals:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M94">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M95">View MathML</a>.

Integrating by part in the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M96">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M97">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M98">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M99">View MathML</a>

It is not hard to prove by induction that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M100">View MathML</a>

(13)

Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M101">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M102">View MathML</a>

Now let us suppose that (13) holds true for some j and prove it for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M103">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M104">View MathML</a>

Thus equality (13) holds true for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M105">View MathML</a>. In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M106">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M107">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M108">View MathML</a>

and going back to the Cartesian coordinate system, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M109">View MathML</a>

 □

4 The Neumann problem with homogeneous boundary conditions

In this section we study problem (1), (2) with homogeneous boundary conditions.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5">View MathML</a>be sufficiently smooth. Then the necessary and sufficient solvability condition for Neumann problem (1), (2) has the form (12).

If a solution exists, then it is unique up to a constant and can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M111">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a>is the solution of Dirichlet problem (10) with the right-hand side<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81">View MathML</a>, which satisfies the additional condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a>.

Proof Let a solution of problem (1), (2) exist and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> be this solution. We apply an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49">View MathML</a> to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M118">View MathML</a>. Now we obtain the conditions for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a>. If we apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M121">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a>, then by virtue of (8) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M123">View MathML</a>

Further, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M124">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M125">View MathML</a>

It is not hard to verify that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M126">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127">View MathML</a> one has [9]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M128">View MathML</a>

Therefore from homogeneous conditions (2) we finally have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M129">View MathML</a>

Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> is a solution of problem (1), (2) with homogeneous boundary conditions, then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M118">View MathML</a> is the solution of Dirichlet problem (10) with the right-hand side

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M132">View MathML</a>

Moreover, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> satisfies the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a> and according to Lemma 3, the necessary condition for this is (12). Hence, if a solution of problem (1), (2) exists, then it is necessary for condition (12) to be satisfied.

Now we prove that if condition (12) is satisfied, then the solution of problem (1), (2) with homogeneous boundary conditions exists.

Indeed, if (12) is satisfied, then according to Lemma 3 the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> of Dirichlet problem (10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81">View MathML</a> exists and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M83">View MathML</a>.

Therefore, we may apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M26">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> and consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M140">View MathML</a>. It is not hard to show that this function is the solution of problem (1), (2).

Indeed, by virtue of (9) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M141">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M142">View MathML</a>, then one can show as above that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> satisfies all homogeneous boundary conditions. □

5 The Neumann problem for the homogeneous equation

In the present section we consider Neumann problem (1), (2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M144">View MathML</a>.

Let A be the following matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M145">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M147">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M148">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M149">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M151">View MathML</a>, the determinant of the matrix obtained from A by deleting the elements of the first column and the jth row. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M152">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a> be a solution of the following Dirichlet problem with sufficiently smooth boundary functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M154">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M155">View MathML</a>

(15)

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M144">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158">View MathML</a>, be sufficiently smooth functions. Then the necessary and sufficient solvability condition for Neumann problem (1), (2) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M159">View MathML</a>

(16)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M160">View MathML</a>.

If a solution exists, then it is unique up to a constant and can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M161">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a>is the solution of Dirichlet problem (15) with boundary functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M165">View MathML</a>, which satisfies the additional condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a>.

Proof Let a solution of problem (1), (2) exist and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> be this solution. We apply an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49">View MathML</a> to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M170">View MathML</a>. Now we prove that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a> is the solution of Dirichlet problem (15) with the additional condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a>.

From the properties of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M33">View MathML</a>. By virtue of the following formula [9]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M176">View MathML</a>

(17)

one has for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M178">View MathML</a>

We rewrite these conditions in a more convenient form. To do this we first consider the last two of them:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M179">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M180">View MathML</a>

(19)

We multiply expression (19) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M181">View MathML</a> and sum to (18). Then, making use of (17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M182">View MathML</a>

Further, by repeating this argument for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M183">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M184">View MathML</a>

Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> is the solution of Neumann problem (1), (2), then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M170">View MathML</a> will be the solution of Dirichlet problem (15) with the additional condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a>. Note that, under the conditions of Theorem 2, the solution of problem (15) exists and it is unique (see, for example, [10]).

Next we find the conditions to the boundary functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a>, which guarantee the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a>. Making use of the Almansi formula (see, for example, [11], p.188) we write the solution of problem (15) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M190">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M191">View MathML</a> are harmonic functions in the ball Ω. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M193">View MathML</a>.

Substituting function (20) into the boundary condition of (15) and integrating over the sphere, taking into account the equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M194">View MathML</a>

we get the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M195">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M196">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M197">View MathML</a>. The matrix of this system is matrix A, defined by (14). As we noted above, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M152">View MathML</a>. By reducing to the Vandermonde determinant, it is not hard to find the value of this determinant; one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M199">View MathML</a>.

Making use of Cramer’s rule, we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M200">View MathML</a> from the above system of equations: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M201">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M202">View MathML</a> is the determinant of the following matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M203">View MathML</a>

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M204">View MathML</a>

where determinants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158">View MathML</a>, are defined above. Therefore, the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M193">View MathML</a> holds if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M208">View MathML</a>. But by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M209">View MathML</a>, this condition is equivalent to (16).

Thus, if the solution of the considered Neumann problem exists, then necessarily condition (16) holds.

We now prove the converse, i.e., if (16) holds, then the solution of the Neumann problem exists.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a> be the solution of Dirichlet problem (15). If condition (16) holds, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M166">View MathML</a> and we may consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M212">View MathML</a>

and prove that this function is in fact the solution of the Neumann problem.

Indeed, after changing of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M213">View MathML</a>, the last integral can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M214">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M215">View MathML</a>

(21)

In the subsequent discussions, we use formulas (17) and (21) and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M127">View MathML</a>. So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M217">View MathML</a>

Further, for the second derivative one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M218">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M219">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M220">View MathML</a>

Using the same argument, we have for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M221">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M222">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M223">View MathML</a>

Therefore, finally we have by induction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M224">View MathML</a>

 □

6 The Neumann problem in the general case

In this final section we consider Neumann problem (1), (2) in the case when both the equation and the conditions are non-homogeneous.

Let the solution of problem (1), (2) exist and denote this solution by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a>. Apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M49">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M22">View MathML</a> and put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M228">View MathML</a>. Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229">View MathML</a> is a solution of the following Dirichlet problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M230">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M197">View MathML</a>. Moreover, by definition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229">View MathML</a> satisfies the additional condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234">View MathML</a>. Since functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M236">View MathML</a> are sufficiently smooth, then the solution of problem (22) exists and it is unique.

Next we find the conditions to functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a>, which guarantee the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234">View MathML</a>. To do this we present <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M241">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a> are the considered above solutions of the corresponding Dirichlet problems (10) and (15). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M245">View MathML</a>.

We represent the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M153">View MathML</a>, according to (11) and (20), in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M248">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M249">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M250">View MathML</a>

It is not hard to see that using the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M251">View MathML</a> we can simplify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M252">View MathML</a> and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M253">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M254">View MathML</a>, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M245">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M256">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M160">View MathML</a>.

Thus, we proved the following statement on the necessary and sufficient solvability condition for the general Neumann boundary value problem.

Theorem 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M5">View MathML</a>be sufficiently smooth. Then the necessary and sufficient solvability condition for Neumann boundary value problem (1), (2) has the form (23).

If a solution exists, then it is unique up to a constant and can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M261">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M229">View MathML</a>is the solution of Dirichlet problem (22) with the right-hand side<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M81">View MathML</a>, which satisfies the additional condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M234">View MathML</a>.

Example 1 Let us consider the biharmonic equation, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M12">View MathML</a>.

In this case

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M266">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M267">View MathML</a>

Therefore the solvability condition has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M268">View MathML</a>

i.e., this condition coincides with the result of the paper [3].

Example 2 Let us consider the so-called three-harmonic equation, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M269">View MathML</a>.

In this case

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M270">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M271">View MathML</a>

Therefore the solvability condition has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M272">View MathML</a>

Remark 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M273">View MathML</a>. Obviously, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M274">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M275">View MathML</a>. Therefore, if we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/162/mathml/M158">View MathML</a>, then all the considered boundary value problems have solutions and these solutions are unique (see, for example, [2]).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This work has been supported by the MON Republic of Kazakhstan under Research Grant 0830/GF2 and the Ministry of Higher and Secondary Special Education of Uzbekistan under Research Grant F4-FA-F010.

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