On the Harmonic Problem with Nonlinear Boundary Integral Conditions

Abstract

In the present work, we deal with the harmonic problems in a bounded domain of with the nonlinear boundary integral conditions. After applying the Boundary integral method, a nonlinear boundary integral equation is obtained; the existence and uniqueness of the solution will be a consequence of applying theory of monotone operators.

1. Introduction

For the harmonic problem, the simplest boundary condition we can impose specifies at all points on the boundary 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 , is specified at all points on the boundary , that is, the Neumann boundary condition, with , then given the value of at one point on 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 is an open bounded region in with a smooth boundary , and are given real value functions.

By the Green representation formula, we formulate a nonlinear integral equation on the boundary of the domain . Under some assumptions on the Kernel of the nonlinear integral equation of Urysohn we prove the existence and uniqueness of the solution.

1.1. Definitions and Notations

Definition 1 (see [1, 9]). Let ; one denotes by the Sobolev space:

Definition 2 (see [1, 9]). Let ; one denotes by the Sobolev space: and the associated norm: with the Fourier transform.

Definition 3 (see [1, 9]). Let a bounded domain and ; one defined

Definition 4 (see [1, 9]). The Fichera trace spaces , for , is defined to be the completion of with respect to the norm

2. The Boundary Integral Method

2.1. 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 , the single layer potential is and the double layer potential is

Using Green’s identity for harmonic functions, we get for , which can be written as Sending in (15) . The continuity of the simple layer potential and the jump relation of the double layer potential . we can write the integral equation on the boundary as follows: where Clearly, if is the solution of (1), then the Cauchy data and satisfies the integral equation (16).

Then the boundary conditions yield Equation (19) can be written as Conversely, if solves (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.(H1)We assume a .(H2)The Kernel of the Urysohn operator is a Caratheodory function [3].(H3)We assume that is measurable satisfying for some constants and .

Remark 5. The operator may have eigenfunctions [1]; then ensures that the integral operator is an isomorphism for every and for all with some positive constant [1]. By we denote the scalar product.
The Kernel is a Caratheodory function ; that is, is measurable for all and is continuous for almost all .
The assumption implies that the Nemytskii operator is Lipschitz continuous and strongly monotonous such that for all .

Theorem 6. Let assumptions (H1), (H2), and (H3) hold. Then, for every the nonlinear boundary integral equation (20) has a unique solution in .

Proof. The proof follows from the well-known theorem by Browder and Minty on monotone operators [6, 10].
Since the simple layer potential operator on 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.
The function defined by for is the normal derivative of the harmonic function for ; this means that satisfies the problem
Then Green’s theorem yields Hence, for all , where denotes the harmonic function corresponding to the Cauchy data and .
On the other hand, we note that there exists , such that on [1]. Hence for all , we have The simple layer potential is continuous, for all [1]. Hence for , we find for some positive constants , , and .
Hence we have 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.

Now we prove the regularity of the solution of the nonlinear boundary integral equation (20).

Theorem 7. For all , , the unique solution of the nonlinear boundary integral equation (20) belongs to the space .

In the proof of this theorem we will need the following lemma.

Lemma 8. For every , , one has and the mapping is bounded.

Proof. For , has already been proved.
For , , 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 ; 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.

Proof of Theorem 7. Let , , be given. By Theorem 6, there exists a unique solution of the nonlinear boundary integral equation: Lemma 8 ensure that therefore
This implies together with the Fredholm property of the double layer potential operator that , .
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 , , and such that
The Kernel of the nonlinear boundary integral equation of Urysohn type is a Caratheodory function. And is measurable satisfying implying that the Nemytskii operator is Lipschitz continuous and strongly monotonous such that for all .

Conflict of Interests

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

Acknowledgment

The author would like to thank the referee for his very careful reading of the paper and his detailed comments and valuable suggestions which improved both the content and the presentation of this paper.