Abstract

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

1. Introduction

The 2D Dirichlet boundary value problem for the Helmholtz equation in an exterior multiply connected domain bounded by closed curves is considered in monographs on mathematical physics, for instance, in [1–3]. The review on studies of the Dirichlet problem for this equation in the exterior of cracks is given in [4]. The present paper is an attempt to combine these problems and to consider exterior domains containing cracks. From a practical stand point such domains have great significance, because cracks model both cracks in solids and wings or double-sided screens in fluids.

So, we study Dirichlet problem in an exterior domain bounded by closed curves and cracks. The theorems on existence and uniqueness of a classical solution are proved. The integral representation for a solution to a problem in the form of potentials is obtained. The problem is reduced to the uniquely solvable Fredholm integral equation of the second kind and index zero. To derive uniquely solvable integral equation on the whole boundary we use the modified integral equation approach [5, 6] on the closed curves. Since the derived integral equation of the 2nd kind is uniquely solvable, we may obtain its numerical solution in a very simple way, just by discretization and inversion of the matrix. Substituting numerical solution of the integral equation into potentials, we obtain numerical solution to the exterior Dirichlet problem in a very simple way as well. The integral representation for a solution presented in the present paper enables us to derive asymptotic formulae for singularities of a gradient of the solution at the tips of the cracks.

The Dirichlet problem for the Helmholtz equation in the exterior of several closed curves in a plane and the Dirichlet problem for the Helmholtz equation in the exterior of several curvilinear cracks in a plane are particular cases of our problem.

It is important to stress that the boundary data on the closed curves in the present paper is assumed to be only continuous. This means that weak solution may not exist in our problem, though classical solution exists. In other words, the problem in this paper is studied under conditions, which are not sufficient for existence of a weak solution in and weak solution may not exist, but these conditions are sufficient for existence of a classical solution. This curious fact follows from the Hadamard example of a nonexistence of a weak solution to the Dirichlet problem for Laplacian in the unit disc with continuous boundary data (classical solution exists in this example). Roughly speaking, continuity of a Dirichlet data on smooth closed curves does not ensure existence of a weak solution in the Dirichlet problem. The Hadamard example of existence of a classical solution and nonexistence of a weak solution is presented and is discussed in the book [7, section 12.5] by Sobolev himself, who invented Sobolev's spaces for analysis of weak solvability of boundary value problems.

Numerical methods for the Dirichlet and Neumann problems for the Laplace and Helmholtz equations in the exterior of cracks in a plane have been developed in [8, 9] on the basis of boundary integral equations. Numerical simulation for engineering problems with cracks is presented in [10–12] using boundary element method. Problems with a crack in electromagnetoelasticity have been reduced to integral equations in [13], and numerical solutions for some model problems have been obtained. The Dirichlet problem for elasticity equations in an exterior of several arbitrary curvilinear cracks in a plane has been reduced to the uniquely solvable integral equations in [14].

2. Formulation of the Problem

By an open curve we mean a simple smooth nonclosed arc of finite length without self-intersections [15]. In the plane we consider an exterior multiply connected domain bounded by simple open curves and simple closed curves , so that the curves do not have common points; in particular, they do not have common endpoints. Suppose that . We set , , and . The exterior connected domain bounded by will be denoted by . We assume that each curve is parametrized by the arc length : so that and the domain is to the right when the parameter increases on . Therefore points and values of the parameter are in one-to-one correspondence except for and , which correspond to the same point for . Below the sets of the intervals on the axis will be denoted by , and also.

We set , and

The tangent vector to at the point is denoted by , where , and . Let be the normal vector to at . The direction of is chosen such that it will coincide with the direction of if is rotated anticlockwise through an angle of .

We consider as a set of cracks. The side of which is on the left, when the parameter increases, will be denoted by and the opposite side will be denoted by .

We say that the function belongs to the smoothness class   if(1) , (2) , where is a point-set, consisting of the endpoints of ,(3)in the neighbourhood of any endpoint for some constants the inequality holds, where and or for .

Remark 2.1. By we denote the class of continuous in functions, which are continuously extensible to the sides of the cracks from the left and from the right, but their limiting values on can be different from the left and from the right, so that these functions may have a jump on . The functions of class belong to the class if they are continuously extensible to from and if they are continuously extensible to the tips of the cracks .

Let us formulate the Dirichlet problem for the Helmholtz equation in the exterior domain (see Figure 1).


Figure 1 
An example of an exterior domain.

Problem . Find a function of the class   which satisfies the Helmholtz equation: the boundary conditions: and the radiating conditions at infinity:

All conditions of the Problem must be satisfied in the classical sense. Problem includes two particular cases. In the first particular case, there are no cracks (i.e., ), then we get the Dirichlet problem for the Helmholtz equation in the exterior of several closed curves in a plane (see [1–3]). In another particular case, there are no closed curves (i.e., ), and we obtain the Dirichlet problem for the Helmholtz equation in the exterior of several curvilinear cracks in a plane [4].

By we mean . On the basis of the Rellich lemma [1], energy equalities [2], and the regularity of the solution to the homogeneous Dirichlet problem near the boundary (see [16, lemma 6.18]), we can easily prove the following assertion.

Theorem 2.2. If , then the Problem has at most one solution.

Proof. It is sufficiently to prove that the homogeneous Problem admits the trivial solution only. Let be a solution to the homogeneous Problem with and . Let be an open disc with a center in the origin and with sufficiently large radius . Assume that for some and assume that .
Since , (remind that ), and since , and owing to the lemma on regularity of solutions of elliptic equations near the boundary [16, lemma 6.18], we obtain . Since , we observe that . We envelope each crack by a closed contour and write first Green's formula for in a domain, bounded by these contours, by and by . Then we allow to shrink closed contours onto cracks and use smoothness of the function . In this way we arrive at the identity for in the domain which is true for any , where is some constant. The curviliniear integral of the 1st kind is taken over . By the complex conjugate function to is denoted. Clearly, belongs to class   and satisfies homogeneous boundary conditions.
By the superscripts + and − we denote the limiting values of functions on and on , respectively. Since satisfies the homogeneous boundary condition (2.4b) on , we rewrite identity in the following form: Taking the imaginary part, we obtain the identity which is true for any . Tending in this identity and taking into account conditions (2.4c) at infinity, we obtain Since , we have whence in on the basis of the Rellich lemma [1, section 229]. Thus, is a trivial solution to the homogeneous Problem . Consequently, the homogeneous Problem has only the trivial solution, and the theorem is proved owing to the linearity of the Problem .

3. Integral Equations at the Boundary

To prove existence of a solution to the Problem ,  we assume that The conditions (3.1b) are compatibility conditions for functions and at the tips of the cracks. To solve Problem we discuss some preliminary matter.

If and are Banach spaces of functions given on and , then for functions given on we introduce the Banach space with the norm .

We consider the angular potential from [4] for (2.4a) on : The kernel is defined on each curve , by where is the Hankel function of the first kind [3]:

Here in after we suppose that belongs to and satisfies the following additional conditions:

As shown in [4], for such the angular potential belongs to the class . In particular, the condition (2.3) is satisfied for any . Moreover, integrating by parts and using (3.5) we express the angular potential in terms of a double-layer potential: with the density Consequently, satisfies both equation (2.4a) outside and the conditions at infinity (2.4c).

Let us construct a solution to the Problem . This solution can be obtained with the help of potential theory for the Helmholtz equation (2.4a). We look for a solution to the problem in the following form: where is given by (3.2), (3.6), and

The density must belong to and must satisfy conditions (3.5).

We will look for in the Banach space with the norm . We say that belongs to the Banach space with some and if where is a Hölder space with the exponent . The norm in the Banach space is defined by

It can be checked directly with the help of [4] that for such the function satisfies (2.4a) and belongs to the class . In particular, the inequality (2.3) holds with if . The potential satisfies (2.4a) and belongs to . It is clear that the function (3.8) satisfies conditions at infinity (2.4c). So, the function (3.8) with densities and subject to requirements described before satisfies all conditions of the Problem except for the boundary conditions (2.4b).

To satisfy the boundary conditions we substitute (3.8) in (2.4b) and arrive at the system of integral equations for the densities and : where is defined in terms of in (3.7).

To derive limit formulas for the angular potential, we used its expression in the form of a double-layer potential (3.6).

Equation (3.12a) is obtained as and comprises two integral equations. The upper sign denotes the integral equation on , and the lower sign denotes the integral equation on .

In addition to the integral equations written before we have the conditions (3.5).

Subtracting the integral equations (3.12a) and using (3.7), we find

We note that is found completely and satisfies all required conditions, in particular, conditions (3.5). Hence, the angular potential of (3.2) and (3.6) is found completely as well.

We introduce the function on by where is specified on in (2.4b) and if . As shown in [4], if , then where if and for any if . Consequently, .

We set Adding the integral equations (3.12a) and taking into account (3.12b) we obtain the integral equation for on : where is given in (3.14).

Thus, if is a solution of (3.16) in the space , then the potential (3.8) with from (3.13) satisfies all conditions of the Problem . We arrive at the following statement.

Theorem 3.1. If   , if conditions (3.1a) and (3.1b) hold, and if equation (3.16) has a solution from the Banach space ,  then a solution to the Problem exists and is given by formula (3.8), where is defined in (3.13).

If , then (3.16) is an equation of the second kind. If , then (3.16) is an equation of the first kind and its kernel has a logarithmic singularity, because where is a smooth function. Indeed, as ,

Our further treatment will be aimed to the proof of the solvability of equation (3.16) in the Banach space . Moreover, we reduce equation (3.16) to a Fredholm equation of the second kind and of index zero, which can be easily computed by classical methods.

By differentiating equation (3.16) on we reduce it to the following singular integral equation on : where the function is defined by (3.17), and is the angle between the vector and the direction of the normal . The angle is taken to be positive if it is measured anticlockwise from and negative if it is measured clockwise from . Besides, is continuous in if .

Equation (3.16) on we rewrite in the following form: where Here (see [3, page 339]), We note that (3.19a) is equivalent to (3.16) on if and only if (3.19a) is accompanied by the following additional conditions:

The system of (3.19a), (3.19b), and (3.22) is equivalent to (3.16).

It can be easily proved (see [4] for details) that Therefore we can rewrite (3.19a) in the following form: where (see [4])

4. The Fredholm Integral Equation and the Solution to the Problem

Inverting the singular integral operator in (3.24), we arrive at the following integral equation of the second kind [4, 15]: where and are arbitrary constants. We set as ; then the function belongs to in variable for .

It can be shown using the properties of singular integrals [15] that and are Hölder continuous functions if and . Consequently, any solution of (4.1) belongs to with some , and here in after we look for on in this space.

We set

Instead of we introduce the new unknown function and rewrite (4.1) and (3.19b) in the form of one equation: where

To derive equations for we substitute from (4.1) and (3.19b) in the conditions (3.22); then in terms of we obtain where By we denote the variable of integration in the potential in (3.9).

Thus, the system of (3.19a), (3.19b), and (3.22) for has been reduced to the system of (4.4) and (4.6) for the function and constants . It is clear from our consideration that any solution of system of (4.4) and (4.6) generates a solution to the system of (3.19a), (3.19b), and (3.22).

As noted before, and are Hölder continuous functions if , and . More precisely (see [4]), , and belongs to in uniformly with respect to . Using these properties we can prove the following.

Lemma 4.1. If , and if from satisfies equation (4.4), then .

The condition holds if conditions (3.1a) and (3.1b) hold.

Hence here in after we will look for from .

Since , and due to the special representation for from (3.19b), the integral operator from (4.4) is a compact operator mapping into itself. Indeed, one can check using Arzela theorem [17] that the integral operator with the kernel is a compact operator mapping into , while the integral operator with the kernel is a compact operator mapping into . Moreover, it can be verified directly with the help of the Arzela theorem that the integral operator is a compact operator mapping into . To verify equicontinuity in the Arzela theorem, we may use the property of uniform continuity in on for the functions and . In doing so, we may use the Cauchy-Bunyakovski inequality for estimates.

We rewrite (4.4) in the following operator form: where is the operator of multiplication of the row by the column . The operator is finite-dimensional from into and therefore compact [18].

Now we rewrite (4.6) in the following form: where is a column of elements, is the identity operator in , is a   matrix consisting of the elements from (4.7). The operator acts from into , so that , where The operators and are finite-dimensional and therefore compact [19].

We consider the columns in the Banach space with the norm .

We write the system of (4.9) and (4.10) in the form of one equation: where is the identity operator in the space . It is clear that is a compact operator mapping into itself, since it consists of compact operators. Therefore, (4.13) is a Fredholm equation of the second kind and of index zero in the space (see [17, 18, 20]).

Let us show that homogeneous equation (4.13) has only a trivial solution. Then, according to Fredholm's alternative [17, 18, 20], the inhomogeneous equation (4.13) has a unique solution for any right-hand side. Let be an arbitrary solution of the homogeneous equation (4.13). According to Lemma 4.1, Therefore the function and the column convert the homogeneous equations (4.1), (3.19b) and (4.6) into identities. For instance, (3.19b) takes the following form: Using the homogeneous identities (4.1) and (3.19b), we check that the homogeneous identities (4.6) are equivalent to Besides, acting on the homogeneous identity (4.1) with a singular operator with the kernel we observe that satisfies the homogeneous equation (3.24): It follows from (4.16a), (4.16b) and (4.16c) that satisfies the homogeneous equation (3.16). On the basis of Theorem 3.1, is a solution to the homogeneous Problem . According to Theorem 2.2, for . Using the limit formulas for normal derivatives of a single-layer potential on , we have

Hence, for , and satisfies (4.16a), which can be written as

It is shown in [5, page 502–504] and [6, page 187–189] that is the unique solution of (4.18) in .

Consequently, if , then and . It follows from the homogeneous identity (4.1) for and that . Hence, . Thus, the homogeneous Fredholm equation (4.13) has only a trivial solution in .

We have proved the following assertion.

Theorem 4.2. If , then (4.13) is a Fredholm equation of the second kind and of index zero in the space . Moreover, (4.13) has a unique solution for any

As a consequence of Theorem 4.2 and the Lemma 4.1 we obtain the following corollary.

Corollary 4.3. If , then equation (4.13) has a unique solution: for any where .

We recall that belongs to the class of smoothness required in Corollary 4.3 if conditions (3.1a) and (3.1b) hold. Besides, (4.13) is equivalent to the system of (4.4) and (4.6). As mentioned before, if is a solution of the system of (4.4) and (4.6), and , then the function is a solution of the system of (3.19a), (3.19b) and (3.22), and therefore satisfies (3.16). We obtain the following statement.

Theorem 4.4. If , and if conditions (3.1a) and (3.1b) hold, then equation (3.16) has a solution . This solution is expressed by the formula , where the function is found by solving the Fredholm equation (4.13), which is uniquely solvable according to Corollary 4.3.

On the basis of Theorem 3.1 we arrive at the solvability theorem for the Problem .

Theorem 4.5. If , and if conditions (3.1a) and (3.1b) hold, then the solution to the Problem exists and is given by (3.8), where is defined in (3.13) and is a solution of equation (3.16) in ensured by Theorem 4.4.

It can be checked directly that the solution to the Problem satisfies condition (2.3) with . Explicit expressions for singularities of the solution gradient at the endpoints of the cracks will be presented in the next section.

Theorem 4.5 ensures existence of a classical solution to the Problem when , , and conditions (3.1a) and (3.1b) hold. The uniqueness of the classical solution follows from Theorem 2.2. On the basis of our consideration we suggest the following scheme for solving the Problem . First, we find the unique solution of the Fredholm equation (4.13) from . This solution automatically belongs to , . Second, we construct the solution of (3.16) from by the formula . Finnaly, substituting from (3.13) and in (3.8) we obtain the solution to the Problem .

Remark 4.6. It is important to stress that the solution to the Problem , ensured by Theorem 4.5, is a classical solution, but it may be not a weak solution to the Problem . In other words, classical solution to the Problem exists and is ensured by Theorem 4.5, but weak solution to the Problem may not exist in space. This follows from the fact that Dirichlet data on the closed curves is assumed to be continuous only. Continuity of a Dirichlet boundary data on closed curves is not sufficient for existence of a weak solution in space. The Hadamard example of a nonexistence of a weak solution to a harmonic Dirichlet problem in a disc with continuous boundary data is given in the book [7, section 12.5] by Sobolev himself (the classical solution exists in this example).

5. Singularities of the Gradient of the Solution at the Endpoints of the Cracks

As noted at the end of Section 4, the gradient of the solution to Problem can be unbounded at the endpoints of the cracks , so that the gradient of the solution to the Problem satisfies estimate (2.3) with the exponent . We will now make a detailed analysis of the behaviour of at the endpoints of .

Let be a solution to the Problem ensured by Theorem 4.5 and given by (3.8). Let be one of the endpoints of . In the neighbourhood of , we introduce the system of polar coordinates: We will assume that if and if . We recall that is the angle between the direction of the axis and the tangent vector to at the point .

Hence, if and if .

Thus, the angle varies continuously in the neighbourhood of the endpoint , cut along .

We will use the notation and put if , and if .

Using the representation of the derivatives of harmonic potentials in terms of Cauchy type integrals (see [4]) and using the properties of these integrals near the endpoints of the integration line, presented in [15], we can prove the following assertion.

Theorem 5.1. Let be a solution to the Problem ensured by Theorem 4.5. Let be an arbitrary endpoint of the cracks , that is, and or for some . Then the derivatives of the solution to the Problem in the neighbourhood of have the following asymptotic behaviour.
If , then If , then By one denotes functions which are continuous at the endpoint . Moreover, the functions denoted by are continuous in the neighbourhood of the endpoint cut along and are continuously extensible to and to from this neighbourhood.

The formulas of the theorem demonstrate the following curious fact. In the general case, the derivatives of the solution to the Problem near the endpoint of cracks behave as However, if , then will be bounded and even continuous at the endpoint .