Questions on solvability of exterior boundary value problems with fractional boundary conditions

In this paper we study questions on solvability of some boundary value problems for the Laplace equation with boundary integro-differential operators in the exterior of a unit ball. We study properties of the given integral differential operators of fractional order in a class of functions which are harmonic outside a ball. We prove theorems about existence and uniqueness of a solution of the problem. We construct explicit form of the solution of the problem in integral form, by solving the Dirichlet problem.


Introduction
Let D be a bounded domain in the space R n , n ≥ 3, with sufficiently smooth boundary S.
It is known (see e.g.[4]) that any function u(x), which belongs to the class C 2 (D) and satisfies the Laplace equation ∆u(x) = 0, x ∈ D, is called harmonic function in the domain D.
Laplace's equation is the most simple example of elliptic partial differential equations.The general theory of solutions to Laplace's equation is known as potential theory.The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, robotic technique, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials (see [5,10,12,21,24]).In the study of heat conduction, the Laplace equation is the steady-state heat equation.The Laplace operator has a great importance in quantum physics, in particular in the study of the Schrödinger equation.
The present work is devoted to the study of a exterior boundary value problem for the Laplace equation with a boundary operator of fractional order.
Boundary value problem with boundary operators of fractional order appear in the problem of diffraction of waves and in the processes of electromagnetic waves.Details about this can be seen in [1,31,32].
In the study of boundary value problems for the Laplace equation on infinite domains additionally regular solutions are required.Namely, a function u(x), harmonic in the domain D 1 = R n \D, is called regular harmonic (at infinity) if the following condition holds as |x| → ∞: where C = const.Note that if for the function u(x) the estimation (1.1) holds, then for any multi-index [33, p. 373]: Let Ω = {x ∈ R n : |x| < 1}, n ≥ 3, be a unit ball, ∂Ω = {x ∈ R n : |x| = 1} be a unit sphere.We denote by Ω + = R n \Ω the exterior of the unit ball.
Assume that u(x) is the regular harmonic function in the domain For the formulation of the problem, we need to define the fractional differentiation operator.In the class of functions, harmonic in the domain Ω + , we define integral-differentiation operators of the fractional order.For a positive number α a fractional integration operator in Riemann-Liouville sense of α order is the expression [15]: and the expression: is called a fractional differentiation operator in Riemann-Liouville sense of α order, where d dr denotes differentiation operator of the form Furthermore, we will suppose, that therefore, when α = 1 the operator (1.3) coincides with derivative by direction of the vector r = |x|.
Introduce the additional notation It is easy to show that the operators B α − and B β − commute for α, β ∈ (0, 1].Similarly, we can show that the operators B −α − and B −β − also commute.Hence, we see that in the general case: Let now α=(α 1 , . . ., α m ) and 0 < α j ≤ 1, j = 1, . . ., m.Consider more general operators: Note that properties and applications of similar operators in the class of functions, harmonic in the ball Ω, were studied in [14,28].Moreover, note that in [6] in the class of functions, harmonic in the ball, properties and applications of operators of the following form were studied:

Formulation and solution of boundary value problems
Now let us consider formulation and solution of some exterior boundary value problems, including the operators B α − and B − α − on the boundary.
Problem 2.1.Find a function u(x), harmonic in the domain Ω + , for which the function and the condition (1.1).
Problem 2.2.Find a function u(x), harmonic in the domain Ω + , for which the function and the condition (1.1).
Let v(x) be a regular solution of the Dirichlet problem in the domain Ω + , i.e.
It is well known (see [8, p. 73]) that if f (x) ∈ C(∂Ω), then the solution of the problem (2.1) exists, unique and can be represented as: where is the area of the unit sphere in R n .Since when α = 1 we have the equality: where ν is a vector of normal to the sphere ∂Ω, then the Problem 2.1 coincides with the exterior Neumann problem for the Laplace equation.It is known that (see e.g.[23]), for any f (x) ∈ C(∂Ω) the solution of the exterior Neumann problem exists, unique and can be represented as where v(x) is a solution of the Dirichlet problem (2.1).Analogously, the Problem 2.2 when α j = 1, j = 1, . . ., m coincides with external problem with the boundary operator of the form: Let us formulate the main propositions concerning to the Problems 2.1 and 2.2.
Then for any f (x) ∈ C(∂Ω) a solution of the Problem 2.1 exists, unique and can be represented as: where v(x) is a solution of the Dirichlet problem (2.1).
Hence, statement of Theorem 2.3 implies that the Problem 2.1 for any 0 < α ≤ 1 behaves as a solution of the exterior Neumann problem.

Properties of the operators B α
− and B − α

−
In this section we investigate some properties of the operator B α − and B − α − .Lemma 3.1.Let 0 < α < 1 and u(x) is a regular harmonic function in the domain Ω + .Then the following inequalities hold: where C is a some constant.
Proof.Let u(x) be the regular harmonic function in the domain Ω + .Then Here B(α, β) is the Euler beta function.Furthermore, we represent the function in the form: Consequently Lemma 3.1 is proved.
Remark 3.2.From now on we will consider regular harmonic functions in Ω + .Therefore all further investigated integrals converge.
Proof.Let u(x) be a regular harmonic function in the domain Ω + .We represent B α − [u](x) in the form: Formally applying the operator Laplace ∆ to the function B α − [u](x), we get

Now we show harmony of the function
. By direct calculation we find, that in the domain Ω + the following equality holds: Consequently, functions B α − [u](x) and B −α − [u](x) are harmonic in Ω + .Further, since the function u(x) is regular at infinity, then the condition (1.1) holds for this function.Then as in the Lemma 3.1 for the functions B α − [u](x) and B −α − [u](x) are regulars at infinity.Lemma 3.3 is proved.Lemma 3.4.Let u(x) be a regular harmonic function in the domain Ω + , then the functions B α − [u](x) and B − α − [u](x) are also harmonic in Ω + .
Proof.Let a function u(x) be regular harmonic in the domain Ω + .Then as in Lemma 3.3, the function B α − [u](x) can be represented as: where (s − 1) Applying Laplace operator to the function B α − [u](x), we have Further, now we show that the function Regularity of the functions B α − [u](x) and B − α − [u](x) at infinity can be checked as in the case of Lemma 3.3.Lemma 3.4 is proved.Lemma 3.5.Let u(x) be a regular harmonic function in the domain Ω + and 0 < α ≤ 1.Then, for any x ∈ Ω + the following equality holds: Consider the function: Since u(x) is the regular harmonic function, then it satisfies the estimate (1.1).Then, by the assertion of Lemma 3.1 for the function B α − [u](x) satisfies the estimate (3.1).Therefore for all t ≥ 1 the integral t [u](x) exists.
We represent t [u](x) as: Further, using definition of the operator B α − , we get Then If now we put t = 1, then we get the equality (3.2).Lemma 3.5 is proved.
Lemma 3.6.Let α = (α 1 , . . . ,α m ), 0 < α j ≤ 1, j = 1, . . ., m and u(x) be harmonic function in the domain Ω.Then for any x ∈ Ω the following equality holds: Consider the function: Denote . By using results of Lemma 3.5, we obtain: Further, repeating this process by all t j , j = 1, . . ., m − 1, we have , If now we put t 1 = 1, t 2 = 1, . . ., t m = 1, then we get the equality (3.3).Lemma 3.6 is proved.Lemma 3.7.If function u(x) is harmonic in the domain Ω, then the following equalities hold: Proof.Let us prove the first equality of Lemma 3.7.We apply operator B −α − to the function . By definition of the operator B −α − [u](x) and according to Lemma 3.5, we have To prove the second equality of Lemma 3.7 we apply the operator Since u(x) is the regular harmonic function, then it satisfies the estimate (1.1).Therefore, each of the considered integrals exist and by Fubini's theorem, we can change the order of integration.Then Further, it is easily seen correctness of the following equalities: Consequently, the second equality of Lemma 3.7 is proved.
Lemma 3.8.Let a function u(x) be harmonic in the domain Ω.Then the following equalities hold: Proof.Let us prove the first equality.To the function B α − [u](x) we apply the operator B − α − .Then by definition of the operator B − α − and according to Lemma 3.6, we get To prove the second equality of Lemma 3.8 we apply the operator It is easy to show implementation of the following equality: Consequently, taking into account definition of the operator we can write that The second equality of Lemma 3.8 is proved.
Therefore, Lemma 3.8 yields that B α − and B − α − are inverse on functions, which are harmonic in Ω + .