International Journal of Analysis

Volume 2014 (2014), Article ID 976520, 5 pages

http://dx.doi.org/10.1155/2014/976520

Research Article

On the Harmonic Problem with Nonlinear Boundary Integral Conditions

Saker Hacene

L.M.A. Department of Mathematics, Faculty of Sciences, University of Badji Mokhtar, P.O. Box 12, 23000 Annaba, Algeria

Received 18 November 2013; Revised 7 January 2014; Accepted 8 January 2014; Published 23 February 2014

Academic Editor: Shamsul Qamar

Copyright © 2014 Saker Hacene. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

G. C. Hsiao and W. Wendland, Boundary Integral Equations, Applied Mathematical Sciences, Springer, Berlin, Germany, 2008.

A. jafarian, Z. Esmailzadeh, and L. Khoshbakhti, “A numerical method for solving nonlinear integral equation in the Urysohn form,” Applied Mathematical Sciences, vol. 7, no. 28, pp. 1375–1385, 2013.

M. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, NY, USA, 1964.

K. Atkinson and G. Chandler, “Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions,” Mathematics of Computation, vol. 55, no. 192, pp. 451–472, 1990.

K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997.

K. Ruotsalainen and W. Wendland, “On the boundary element method for some nonlinear boundary value problems,” Numerische Mathematik, vol. 53, no. 3, pp. 299–314, 1988.

R. Bialecki and A. J. Nowak, “Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions,” Applied Mathematical Modelling, vol. 5, no. 6, pp. 417–421, 1981.

C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary Element Techniques, Springer, Berlin, Germany, 1984.

R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.

D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980.