On the Harmonic Problem with Nonlinear Boundary Integral Conditions

For the harmonic problem, the simplest boundary condition we can impose specifies u at all points on the boundary Γ and is known as the Dirichlet boundary condition. The Dirichlet problem for the Laplace equation can easily be solved using the boundary integral equation [1]. If the normal derivative of u, that is, ∂u/∂n, where n is the outward normal to the boundary Γ, is specified at all points on the boundary Γ, that is, the Neumann boundary condition, with ∫ Γ (∂u/∂n)ds = 0, then given the value of u at one point on Γ enables a unique solution to be obtained [1]. In this work, we impose more general boundary conditions, namely, the nonlinear integral equation of Urysohn type [2, 3]. Much attention has been paid to the resolution of boundary value problems for partial differential operators with nonlinear boundary conditions by the method of integral equations in many directions (see, e.g., Atkinson and Chandler [4, 5] and Ruotsalainen and Wendland [6]). Problems involving nonlinearities form a basis of mathematical models of various steady-state phenomena and processes in mechanics, physics, and many other areas of science. Among these is the steady-state heat transfer. Also some electromagnetic problems contain nonlinearities in the boundary conditions, for instance, problems where the electrical conductivity of the boundary is variable [7]. Further applications arise in heat radiation and heat transfer [7, 8]. In the present paper, we look for the solution of the Laplacian equation with nonlinear data of the form


Introduction
For the harmonic problem, the simplest boundary condition we can impose specifies at all points on the boundary Γ and is known as the Dirichlet boundary condition. The Dirichlet problem for the Laplace equation can easily be solved using the boundary integral equation [1]. If the normal derivative of , that is, / , where is the outward normal to the boundary Γ, is specified at all points on the boundary Γ, that is, the Neumann boundary condition, with ∫ Γ ( / ) = 0, then given the value of at one point on Γ enables a unique solution to be obtained [1].
In this work, we impose more general boundary conditions, namely, the nonlinear integral equation of Urysohn type [2,3].
Much attention has been paid to the resolution of boundary value problems for partial differential operators with nonlinear boundary conditions by the method of integral equations in many directions (see, e.g., Atkinson and Chandler [4,5] and Ruotsalainen and Wendland [6]).
Problems involving nonlinearities form a basis of mathematical models of various steady-state phenomena and processes in mechanics, physics, and many other areas of science. Among these is the steady-state heat transfer. Also some electromagnetic problems contain nonlinearities in the boundary conditions, for instance, problems where the electrical conductivity of the boundary is variable [7]. Further applications arise in heat radiation and heat transfer [7,8].
In the present paper, we look for the solution of the Laplacian equation with nonlinear data of the form We recall that the nonlinear boundary integral operator defined by is the nonlinear integral operator of Urysohn type. In (1), we assume Ω is an open bounded region in R 2 with a smooth boundary Γ = Ω, and

Representative Formula and Boundary Operator.
We introduce the fundamental solution of the Laplacian operator in the plane defined by: We first consider some standard boundary integral operators. For ∈ Ω, the single layer potential is and the double layer potential is Using Green's identity for harmonic functions, we get for ∈ Ω, which can be written as Sending in (15) → Γ. The continuity of the simple layer potential Ω and the jump relation of the double layer potential Ω . we can write the integral equation on the boundary as follows: where Clearly, if ∈ 1 (Ω) is the solution of (1), then the Cauchy data | Γ and / | Γ satisfies the integral equation (16). Then the boundary conditions Equation (19) can be written as (20), then the solution of (1) can be given by the representation formula (15) and will satisfy due to (20). For studying the solvability of the nonlinear equation (20), we give some assumptions to be made here.
Since the simple layer potential operator on Γ is an isomorphism, it is sufficient to consider the unique solvability of the following equation: We will prove that the operator is continuous and strongly monotonous.
(i) In the first we show that is continuous.
It is clear from the continuity of the mapping properties of the simple and double layer operators that is continuous. And from (H3), is continuous. Hence the boundary integral operator is continuous.
(ii) In the second we show that is strongly monotone operator.
On the other hand, we note that there exists (] 1 − ] 2 ) ∈ −1/2 (Γ), such that on Γ [1]. Hence for all ∈ Ω, we have The simple layer potential is continuous, for all ∈ R [1]. Hence for = −3/2, we find for some positive constants 1 , 2 , and 3 . Hence we have International Journal of Analysis Then with (28) and (37) we get and with (26) we get the inequality hence with (42) we have by the trace theorem [1,9], which completes the proof. In the proof of this theorem we will need the following lemma. Proof. For = 0, ∈ 0 (Γ) := 2 (Γ) has already been proved. For = 1, ∈ 1 (Γ), is an absolutely continuous function. By assumption (H3) the function ( ) is Lipschitz continuous. Hence ( ) is also absolutely continuous function.
It remains to prove the case 0 < < 1; by the assumption (H3) and due to the definition of the Sobolev space in Definition 4, we have which completes the proof of Lemma 8.
Example 9. Here we give an example to illustrate the theoretical results. We consider the harmonic problems: where the nonlinear boundary integral equation of Urysohn type is defined by and the domain is Clearly, the nonlinearity satisfies our assumptions ( 1 ), ( 2 ), and ( 3 ) such that diam (Ω) = 2 < 1.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.