Abstract
In this work the Neumann boundary value problem for a nonhomogeneous 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 nonhomogeneous polyharmonic equation and then use the Green function of the Dirichlet problem.
MSC: 35J40, 35J30, 35A01.
Keywords:
nonhomogeneous polyharmonic equation; the Neumann problem; the necessary and sufficient conditions for solvability1 Introduction
Let
Consider on the domain Ω the following Neumann boundary value problem:
where ν is the unit outer normal vector to sphere ∂Ω,
A function
It is well known (see, for example, [1]) that even in case
In the paper [2] by Kanguzhin and Koshanov, in particular, it is shown that in case
where
The authors of the paper [3] presented this condition in a different form, which could be easily verified,
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
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 fractionalorder.
The paper is organized as follows. In the next section we study the properties of some integrodifferential 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 [57]. 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 nonhomogeneous 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 integrodifferential operators
Let
where
Note that in the class of harmonic functions in a ball the properties of operators
Lemma 1Let
(1) if
(2) if
(3) if
Proof Let
Therefore, if
and if
Hence, equality (5) and the second equality of (4) are proved.
As we noted above, if
But due to equality (7) and the condition
The following statement can be proved by a direct calculation.
Lemma 2Let
(1) if
(2) if
(3) if
Corollary 1Let
(1) if
(2) if
3 Some properties of the solutions of the Dirichlet problem
Let
It is known (see, for example, [57]) that if
where
We make use of the following explicit form of the Green function [5]:
if n is odd, or even and
where
if n is even and
where
Lemma 3Let
Proof Let
If we denote
Now we consider the inner integral
Obviously,
Integrating by part in the integral
Therefore
where
It is not hard to prove by induction that
Indeed, if
Now let us suppose that (13) holds true for some j and prove it for
Thus equality (13) holds true for any
where
Therefore
and going back to the Cartesian coordinate system, we have
□
4 The Neumann problem with homogeneous boundary conditions
In this section we study problem (1), (2) with homogeneous boundary conditions.
Theorem 1Let
If a solution exists, then it is unique up to a constant and can be represented as
where
Proof Let a solution of problem (1), (2) exist and let
Obviously,
Further, since
then
It is not hard to verify that for any
Therefore from homogeneous conditions (2) we finally have
Thus, if
Moreover, the function
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
Therefore, we may apply the operator
Indeed, by virtue of (9) one has
Since
5 The Neumann problem for the homogeneous equation
In the present section we consider Neumann problem (1), (2) with
Let A be the following matrix
where
Let
Theorem 2Let
where
If a solution exists, then it is unique up to a constant and can be represented as
where
Proof Let a solution of problem (1), (2) exist and let
From the properties of the operator
one has for
We rewrite these conditions in a more convenient form. To do this we first consider the last two of them:
We multiply expression (19) by
Further, by repeating this argument for all
Thus, if
Next we find the conditions to the boundary functions
where
Substituting function (20) into the boundary condition of (15) and integrating over the sphere, taking into account the equalities
we get the system of equations
where
and
Making use of Cramer’s rule, we find
Obviously,
where determinants
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
and prove that this function is in fact the solution of the Neumann problem.
Indeed, after changing of variable
Therefore,
In the subsequent discussions, we use formulas (17) and (21) and assume that
Further, for the second derivative one has
Then
Hence
Using the same argument, we have for any
Consequently,
Therefore, finally we have by induction
□
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 nonhomogeneous.
Let the solution of problem (1), (2) exist and denote this solution by
where
Next we find the conditions to functions
where
We represent the functions
Then
where
It is not hard to see that using the formula
Since
where
Thus, we proved the following statement on the necessary and sufficient solvability condition for the general Neumann boundary value problem.
Theorem 3Let
If a solution exists, then it is unique up to a constant and can be represented as
where
Example 1 Let us consider the biharmonic equation, i.e.,
In this case
Then
Therefore the solvability condition has the form
i.e., this condition coincides with the result of the paper [3].
Example 2 Let us consider the socalled threeharmonic equation, i.e.,
In this case
Then
Therefore the solvability condition has the form
Remark 1 Let
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 F4FAF010.
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