# The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

Authors: P. A. Krutitskii
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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.


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## Body

## 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 inHloc1 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 x=(x1,x2)∈R2 we consider an exterior multiply connected domain bounded by simple open curves Γ11,…,ΓN11∈C2,λ and simple closed curves Γ12,…,ΓN22∈C2,λ,λ∈(0,1], so that the curves do not have common points; in particular, they do not have common endpoints. Suppose that N1+N2>0. We set Γ1=⋃n=1N1Γn1, Γ2=⋃n=1N2Γn2, and Γ=Γ1∪Γ2. The exterior connected domain bounded by Γ2 will be denoted by 𝒟. We assume that each curve Γnk is parametrized by the arc length s:
(2.1)Γnk={x:x=x(s)=(x1(s),x2(s)),s∈[ank,bnk]},n=1,…,Nk,k=1,2,
so that a11<b11<⋯<aN11<bN11<a12<b12<⋯<aN22<bN22 and the domain 𝒟 is to the right when the parameter s increases on Γn2. Therefore points x∈Γ and values of the parameter s are in one-to-one correspondence except for an2 and bn2, which correspond to the same point x for n=1,…,N2. Below the sets of the intervals on the Os axis ⋃n=1N1[an1,bn1],⋃n=1N2[an2,bn2],⋃k=12⋃n=1Nk[ank,bnk] will be denoted by Γ1,Γ2, and Γ also.We setC0(Γn2)={ℱ(s):ℱ(s)∈C0[an2,bn2],ℱ(an2)=ℱ(bn2)}, and
(2.2)C0(Γ2)=⋂n=1N2‍C0(Γn2).The tangent vector toΓ at the point x(s) is denoted by τx=(cosα(s),sinα(s)), where cosα(s)=x1′(s), and sinα(s)=x2′(s). Let nx=(sinα(s),-cosα(s)) be the normal vector to Γ at x(s). The direction of nx is chosen such that it will coincide with the direction of τx if nx is rotated anticlockwise through an angle of π/2.We considerΓ1 as a set of cracks. The side of Γ1 which is on the left, when the parameter s increases, will be denoted by (Γ1)+ and the opposite side will be denoted by (Γ1)-.We say that the functionw(x) belongs to the smoothness class  K if(1)
w∈C0(𝒟∖Γ1¯)∩C2(𝒟∖Γ1),(2)
∇w∈C0(𝒟∖Γ1¯∖Γ2∖X), where X is a point-set, consisting of the endpoints of Γ1:X=⋃n=1N1(x(an1)∪x(bn1)),(3)
in the neighbourhood of any endpointx(d)∈X for some constants 𝒞>0,ϵ>-1 the inequality
(2.3)|∇w|≤C|x-x(d)|ϵ
holds, where x→x(d) and d=an1 or d=bn1 for n=1,…,N1.Remark 2.1.
ByC0(𝒟∖Γ1¯∖Γ2∖X) we denote the class of continuous in 𝒟∖Γ1 functions, which are continuously extensible to the sides of the cracks Γ1∖X from the left and from the right, but their limiting values on Γ1∖X can be different from the left and from the right, so that these functions may have a jump on Γ1∖X. The functions of class C0(𝒟∖Γ1¯∖Γ2∖X) belong to the class C0(𝒟∖Γ1¯) if they are continuously extensible to Γ2 from 𝒟 and if they are continuously extensible to the tips of the cracks Γ1.Let us formulate the Dirichlet problem for the Helmholtz equation in the exterior domain𝒟∖Γ1 (see Figure 1).Figure 1
An example of an exterior domain.Problem
U. Find a function u(x) of the class  K which satisfies the Helmholtz equation:(2.4a)ux1x1(x)+ux2x2(x)+β2u(x)=0,x∈D∖Γ1,β=const>0,
the boundary conditions:
(2.4b)u(x(s))|(Γ1)+=F+(s),u(x(s))|(Γ1)-=F-(s),u(x(s))|Γ2=F(s),
and the radiating conditions at infinity:
(2.4c)u=O(|x|-1/2),∂u∂|x|-iβu=o(|x|-1/2),|x|=x12+x22⟶∞.All conditions of the ProblemU must be satisfied in the classical sense. Problem U includes two particular cases. In the first particular case, there are no cracks Γ1 (i.e., Γ1=∅), then we get the Dirichlet problem for the Helmholtz equation in the exterior of several closed curves Γ2 in a plane (see [1–3]). In another particular case, there are no closed curves Γ2 (i.e., Γ2=∅), and we obtain the Dirichlet problem for the Helmholtz equation in the exterior of several curvilinear cracks Γ1 in a plane [4].By∫Γk⋯dσ we mean ∑n=1Nk∫ankbnk⋯dσ. 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 Γ2 (see [16, lemma 6.18]), we can easily prove the following assertion.Theorem 2.2.
IfΓ∈C2,λ,λ∈(0,1], then the Problem U has at most one solution.Proof.
It is sufficiently to prove that the homogeneous ProblemU admits the trivial solution only. Let u0(x) be a solution to the homogeneous Problem U with F+(s)≡F-(s)≡0 and F(s)≡0. Let Sr be an open disc with a center in the origin and with sufficiently large radius r. Assume that Γ⊂Sr0 for some r0 and assume that r>r0.
SinceΓ2∈C2,λ, u0(x)∈C0(𝒟¯∖Γ1) (remind that u0(x)∈K), and since u0|Γ2=0∈C2,λ(Γ2), and owing to the lemma on regularity of solutions of elliptic equations near the boundary [16, lemma 6.18], we obtain u0(x)∈C1(𝒟¯∖Γ1). Since u0(x)∈K, we observe that u0(x)∈C1(𝒟∖Γ1¯∖X). We envelope each crack Γn1(n=1,…,N1) by a closed contour and write first Green's formula for u0(x) in a domain, bounded by these contours, by Γ2 and by ∂Sr. Then we allow to shrink closed contours onto cracks Γ1 and use smoothness of the function u0(x). In this way we arrive at the identity for u0(x) in the domain 𝒟∩Sr∖Γ1(*)‖∇u0‖L2(D∩Sr∖Γ1)2-β2‖u0‖L2(D∩Sr∖Γ1)2=∫Γ1‍(u-0)+(∂u0∂nx)+ds-∫Γ1‍(u-0)-(∂u0∂nx)-ds-∫Γ2‍u-0∂u0∂nxds+∫∂Sr‍u-0∂u0∂|x|dl,
which is true for any r>r0, where r0 is some constant. The curviliniear integral of the 1st kind is taken over ∂Sr. By u-0(x) the complex conjugate function to u0(x) is denoted. Clearly, u-0(x) belongs to class  K and satisfies homogeneous boundary conditions.
By the superscripts + and − we denote the limiting values of functions on(Γ1)+ and on (Γ1)-, respectively. Since u0(x) satisfies the homogeneous boundary condition (2.4b) on Γ, we rewrite identity (*) in the following form:
(2.5)‖∇u0‖L2(D∩Sr∖Γ1)2-β2‖u0‖L2(D∩Sr∖Γ1)2=∫∂Sru-0(x)∂u0(x)∂|x|dl.
Taking the imaginary part, we obtain the identity
(2.6)Im⁡∫∂Sru-0(x)∂u0(x)∂|x|dl=0,
which is true for any r>r0. Tending r→∞ in this identity and taking into account conditions (2.4c) at infinity, we obtain
(2.7)lim⁡r→∞Im⁡∫∂Sru-0(x)∂u0(x)∂|x|dl=βlim⁡r→∞∫∂Sr|u0|2dl=0.
Since β>0, we have
(2.8)lim⁡r→∞∫∂Sr|u0|2dl=0,
whence u0(x)≡0 in 𝒟∖Γ1 on the basis of the Rellich lemma [1, section 229]. Thus, u0(x) is a trivial solution to the homogeneous Problem U. Consequently, the homogeneous Problem U has only the trivial solution, and the theorem is proved owing to the linearity of the Problem U.


## 3. Integral Equations at the Boundary

To prove existence of a solution to the ProblemU,  we assume that (3.1a)F+(s)∈C1,λ(Γ1),F-(s)∈C1,λ(Γ1),F(s)∈C0(Γ2),λ∈(0,1],(3.1b)F+(an1)=F-(an1),F+(bn1)=F-(bn1),n=1,…,N1.The conditions (3.1b) are compatibility conditions for functions F+(s) and F-(s) at the tips of the cracks. To solve Problem U we discuss some preliminary matter.Ifℬ1(Γ1) and ℬ2(Γ2) are Banach spaces of functions given on Γ1 and Γ2, then for functions given on Γ we introduce the Banach space ℬ1(Γ1)∩ℬ2(Γ2) with the norm ∥·∥ℬ1(Γ1)∩ℬ2(Γ2)=∥·∥ℬ1(Γ1)+∥·∥ℬ2(Γ2).We consider the angular potential from [4] for (2.4a) on Γ1:
(3.2)v1[ν](x)=i4∫Γ1ν(σ)V(x,σ)dσ.
The kernel V(x,σ) is defined on each curve Γn1,n=1,…,N1, by
(3.3)V(x,σ)=∫an1σ‍∂∂nyH0(1)(β|x-y(ξ)|)dξ,σ∈[an1,bn1],
where ℋ0(1)(z) is the Hankel function of the first kind [3]: (3.4)H0(1)(z)=2exp⁡⁡(iz-iπ/4)πz∫0∞‍exp⁡(-t)t-1/2(1+it2z)-1/2dt,y=y(ξ)=(y1(ξ),y2(ξ)),|x-y(ξ)|=(x1-y1(ξ))2+(x2-y2(ξ))2.Here in after we suppose thatν(σ) belongs to C0,λ(Γ1) and satisfies the following additional conditions: (3.5)∫an1bn1‍ν(σ)dσ=0,n=1,…,N1.As shown in [4], for such ν(σ) the angular potential v1[ν](x) belongs to the class K. In particular, the condition (2.3) is satisfied for any ϵ∈(0,1). Moreover, integrating v1[ν](x) by parts and using (3.5) we express the angular potential in terms of a double-layer potential:
(3.6)v1[ν](x)=-i4∫Γ1ρ(σ)∂∂nyH0(1)(β|x-y(σ)|)dσ
with the density
(3.7)ρ(σ)=∫an1σ‍ν(ξ)dξ,σ∈[an1,bn1],n=1,…,N1.
Consequently, v1[ν](x) satisfies both equation (2.4a) outside Γ1 and the conditions at infinity (2.4c).Let us construct a solution to the ProblemU. 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:
(3.8)u[ν,μ](x)=v1[ν](x)+w[μ](x),
where v1[ν](x) is given by (3.2), (3.6), and
(3.9)w[μ](x)=w1[μ](x)+w2[μ](x),w1[μ](x)=i4∫Γ1μ(σ)H0(1)(β|x-y(σ)|)dσ,w2[μ](x)=i4∫Γ2μ(σ)(∂∂ny-i)H0(1)(β|x-y(σ)|)dσ.The densityν(σ) must belong to C0,λ(Γ1) and must satisfy conditions (3.5).We will look forμ(s) in the Banach space Cqω(Γ1)∩C0(Γ2),ω∈(0,1],q∈[0,1) with the norm ∥·∥Cqω(Γ1)∩C0(Γ2)=∥·∥Cqω(Γ1)+∥·∥C0(Γ2). We say that μ(s) belongs to the Banach space Cqω(Γ1) with some ω∈(0,1] and q∈[0,1) if
(3.10)μ(s)∏n=1N1‍|s-an1|q|s-bn1|q∈C0,ω(Γ1),
where C0,ω(Γ1) is a Hölder space with the exponent ω. The norm in the Banach space Cqω(Γ1) is defined by
(3.11)‖μ(⋅)‖Cqω(Γ1)=‖μ(⋅)∏n=1N1‍|⋅-an1|q|⋅-bn1|q‖C0,ω(Γ1).It can be checked directly with the help of [4] that for such μ(s) the function w1[μ](x) satisfies (2.4a) and belongs to the class K. In particular, the inequality (2.3) holds with ϵ=-q if q∈(0,1). The potential w2[μ](x) satisfies (2.4a) and belongs to C0(𝒟¯)∩C2(𝒟). It is clear that the function (3.8) satisfies conditions at infinity (2.4c). So, the function (3.8) with densities μ(s) and ν(s) subject to requirements described before satisfies all conditions of the Problem U 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 μ(s) and ν(s):(3.12a)±12ρ(s)+i4∫Γ1ν(σ)V(x(s),σ)dσ+i4∫Γ1μ(σ)H0(1)(β|x(s)-y(σ)|)dσ+i4∫Γ2μ(σ)(∂∂ny-i)H0(1)(β|x(s)-y(σ)|)dσ=F±(s),s∈Γ1,(3.12b)i4∫Γ1ν(σ)V(x(s),σ)dσ+i4∫Γ1μ(σ)H0(1)(β|x(s)-y(σ)|)dσ+12μ(s)+i4∫Γ2μ(σ)(∂∂ny-i)H0(1)(β|x(s)-y(σ)|)dσ=F(s),s∈Γ2,where ρ(s) is defined in terms of ν(s) 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 x→x(s)∈(Γ1)± and comprises two integral equations. The upper sign denotes the integral equation on (Γ1)+, and the lower sign denotes the integral equation on (Γ1)-.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
(3.13)ρ(s)=(F+(s)-F-(s))∈C1,λ(Γ1),ν(s)=(F′+(s)-F′-(s))∈C0,λ(Γ1),F′±(s)=ddsF±(s).We note thatν(s) 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 functionf(s) on Γ by
(3.14)f(s)=F(s)-i4∫Γ1(F′+(σ)-F′-(σ))V(x(s),σ)dσ,s∈Γ,
where F(s) is specified on Γ2 in (2.4b) and F(s)=(1/2)(F+(s)+F-(s)) if s∈Γ1. As shown in [4], if s∈Γ1, then f(s)∈C1,p0(Γ1) where p0=λ if 0<λ<1 and p0=1-ϵ0 for any ϵ0∈(0,1) if λ=1. Consequently, f(s)∈C1,p0(Γ1)∩C0(Γ2).We set(3.15)δ(s)={0ifs∈Γ1,1ifs∈Γ2.
Adding the integral equations (3.12a) and taking into account (3.12b) we obtain the integral equation for μ(s) on Γ:
(3.16)i4∫Γ1μ(σ)H0(1)(β|x(s)-y(σ)|)dσ+12δ(s)μ(s)+i4∫Γ2μ(σ)(∂∂ny-i)H0(1)(β|x(s)-y(σ)|)dσ=f(s),s∈Γ,
where f(s) is given in (3.14).Thus, ifμ(s) is a solution of (3.16) in the space Cqω(Γ1)∩C0(Γ2),ω∈(0,1],q∈[0,1), then the potential (3.8) with ν(s) from (3.13) satisfies all conditions of the Problem U. We arrive at the following statement.Theorem 3.1.
IfΓ∈C2,λ, if conditions (3.1a) and (3.1b) hold, and if equation (3.16) has a solution μ(s) from the Banach space Cqω(Γ1)∩C0(Γ2),ω∈(0,1],q∈[0,1),  then a solution to the Problem U exists and is given by formula (3.8), where ν(s) is defined in (3.13).Ifs∈Γ2, then (3.16) is an equation of the second kind. If s∈Γ1, then (3.16) is an equation of the first kind and its kernel has a logarithmic singularity, because(3.17)H0(1)(z)=2iπln⁡⁡zβ+h(z),
where h(z) is a smooth function. Indeed, as z→0+0,
(3.18)h(z)=const+O(z2ln⁡⁡z),h′(z)=O(zln⁡⁡z),h′′(z)=O(ln⁡⁡z).Our further treatment will be aimed to the proof of the solvability of equation (3.16) in the Banach space Cqω(Γ1)∩C0(Γ2). 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 Γ1 we reduce it to the following singular integral equation on Γ1: (3.19a)∂∂sw[μ](x(s))=12π∫Γ1μ(σ)sin⁡φ0(x(s),y(σ))|x(s)-y(σ)|dσ+i4∫Γ1μ(σ)∂∂sh(β|x(s)-y(σ)|)dσ+i4∫Γ2μ(σ)(∂∂ny-i)∂∂sH0(1)(β|x(s)-y(σ)|)dσ=f′(s),s∈Γ1,
where the function h(z) is defined by (3.17), and φ0(x,y) is the angle between the vector xy→ and the direction of the normal nx. The angle φ0(x,y) is taken to be positive if it is measured anticlockwise from nx and negative if it is measured clockwise from nx. Besides, φ0(x,y) is continuous in x,y∈Γ if x≠y.Equation (3.16) on Γ2 we rewrite in the following form:
(3.19b)μ(s)+∫Γμ(σ)A2(s,σ)dσ=2f(s),s∈Γ2,where
(3.20)A2(s,σ)={i2(1-δ(σ))H0(1)(β|x(s)-y(σ)|)+i2δ(σ)(∂∂ny-i)H0(1)(β|x(s)-y(σ)|)}=iπδ(σ)ln⁡|x(s)-y(σ)|+I1(x(s),y(σ))=iπδ(σ)I2(x(s),y(σ))|x(s)-y(σ)|1/3+I1(x(s),y(σ)).
Here I1(x(s),y(σ))∈C0(Γ2×Γ) (see [3, page 339]),
(3.21)I2(x(s),y(σ))=|x(s)-y(σ)|1/3ln⁡⁡|x(s)-y(σ)|∈C0(Γ2×Γ2).
We note that (3.19a) is equivalent to (3.16) on Γ1 if and only if (3.19a) is accompanied by the following additional conditions:
(3.22)w[μ](x(an1))=f(an1),n=1,…,N1.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
(3.23)(sin⁡⁡φ0(x(s),y(σ))|x(s)-y(σ)|-1σ-s)∈C0,λ(Γ1×Γ1).
Therefore we can rewrite (3.19a) in the following form:
(3.24)2∂∂sw[μ](x(s))=1π∫Γ1μ(σ)dσσ-s+∫Γμ(σ)Y(s,σ)dσ=2f′(s),s∈Γ1,
where (see [4])
(3.25)Y(s,σ)={(1-δ(σ))[1π(sin⁡φ0(x(s),y(σ))|x(s)-y(σ)|-1σ-s)+i2∂∂sh(β|x(s)-y(σ)|)]+i2δ(σ)(∂∂ny-i)∂∂sH0(1)(β|x(s)-y(σ)|)}∈C0,p0(Γ1×Γ),p0=λif0<λ<1,andp0=1-ϵ0foranyϵ0∈(0,1)ifλ=1.


## 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]:
(4.1)μ(s)+1Q1(s)∫Γμ(σ)A1(s,σ)dσ+1Q1(s)∑n=0N1-1‍Gnsn=1Q1(s)Φ1(s),s∈Γ1,
where
(4.2)A1(s,σ)=-1π∫Γ1Y(ξ,σ)ξ-sQ1(ξ)dξ,Q1(s)=∏n=1N1‍|s-an1bn1-s|sign⁡(s-an1),Φ1(s)=-1π∫Γ12Q1(σ)f′(σ)σ-sdσ,
and G0,…,GN1-1 are arbitrary constants. We set sign(s-an1)=1 as s=an1; then the function sign(s-an1) belongs to C∞(Γ1) in s variable for n=1,…,N1.It can be shown using the properties of singular integrals [15] that Φ1(s) and A1(s,σ) are Hölder continuous functions if s∈Γ1 and σ∈Γ. Consequently, any solution of (4.1) belongs to C1/2ω(Γ1) with some ω∈(0,1], and here in after we look for μ(s) on Γ1 in this space.We set(4.3)Q(s)=(1-δ(s))Q1(s)+δ(s),s∈Γ.Instead ofμ(s)∈C1/2ω(Γ1)∩C0(Γ2) we introduce the new unknown function μ*(s)=μ(s)Q(s)∈C0,ω(Γ1)∩C0(Γ2) and rewrite (4.1) and (3.19b) in the form of one equation:
(4.4)μ*(s)+∫Γμ*(σ)Q-1(σ)A(s,σ)dσ+(1-δ(s))∑n=0N1-1‍Gnsn=Φ(s),s∈Γ,
where
(4.5)A(s,σ)=(1-δ(s))A1(s,σ)+δ(s)A2(s,σ),Φ(s)=(1-δ(s))Φ1(s)+2δ(s)f(s).To derive equations forG0,…,GN1-1 we substitute μ(s) from (4.1) and (3.19b) in the conditions (3.22); then in terms of μ*(s) we obtain
(4.6)∫ΓQ-1(ξ)μ*(ξ)ln(ξ)dξ+∑m=0N1-1‍BnmGm=Hn,n=1,…,N1,
where
(4.7)ln(ξ)=-w[Q-1(⋅)A(⋅,ξ)](an1),Hn=-w[Q-1(⋅)Φ(⋅)](an1)+f(an1),Bnm=-w[Q-1(⋅)(1-δ(⋅))(⋅)m](an1).
By · we denote the variable of integration in the potential w[μ](x) in (3.9).Thus, the system of (3.19a), (3.19b), and (3.22) for μ(s) has been reduced to the system of (4.4) and (4.6) for the function μ*(s) and constants G0,…,GN1-1. 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,Φ1(s) and A1(s,σ) are Hölder continuous functions if s∈Γ1, and σ∈Γ. More precisely (see [4]), Φ1(s)∈C0,p(Γ1),p=min{1/2,λ}, and A1(s,σ) belongs to C0,p(Γ1) in s uniformly with respect to σ∈Γ. Using these properties we can prove the following.Lemma 4.1.
IfΓ∈C2,λ,λ∈(0,1],Φ(s)∈C0,p(Γ1)∩C0(Γ2),p=min{λ,1/2}, and if μ*(s) from C0(Γ) satisfies equation (4.4), then μ*(s)∈C0,p(Γ1)∩C0(Γ2).The conditionΦ(s)∈C0,p(Γ1)∩C0(Γ2) holds if conditions (3.1a) and (3.1b) hold.Hence here in after we will look forμ*(s) from C0(Γ).SinceA1(s,σ)∈C0(Γ1×Γ), and due to the special representation for A2(s,σ) from (3.19b), the integral operator from (4.4)
(4.8)Aμ*=∫Γμ*(σ)Q-1(σ)A(s,σ)dσ={∫Γμ*(σ)Q-1(σ)A1(s,σ)dσifs∈Γ1,∫Γμ*(σ)Q-1(σ)A2(s,σ)dσifs∈Γ2,
is a compact operator mapping C0(Γ) into itself. Indeed, one can check using Arzela theorem [17] that the integral operator with the kernel A1(s,σ) is a compact operator mapping C0(Γ) into C0(Γ1), while the integral operator with the kernel A2(s,σ) is a compact operator mapping C0(Γ1) into C0(Γ2). Moreover, it can be verified directly with the help of the Arzela theorem that the integral operator ∫Γμ*(σ)Q-1(σ)A2(s,σ)dσ is a compact operator mapping C0(Γ2) into C0(Γ2). To verify equicontinuity in the Arzela theorem, we may use the property of uniform continuity in x on Γ2 for the functions |x-y|1/3 and |x-y|1/3ln|x-y|. In doing so, we may use the Cauchy-Bunyakovski inequality for estimates.We rewrite (4.4) in the following operator form:
(4.9)(I+A)μ*+PG=Φ,
where P is the operator of multiplication of the row P=(1-δ(s))(s0,…,sN1-1) by the column G=(G0,…,GN1-1)T. The operator P is finite-dimensional from EN1 into C0(Γ) and therefore compact [18].Now we rewrite (4.6) in the following form:
(4.10)IN1G+Lμ*+(B-IN1)G=H,
where H=(H1,…,HN1)T is a column of N1 elements, IN1 is the identity operator in EN1, B is a  N1×N1 matrix consisting of the elements Bnm from (4.7). The operator L acts from C0(Γ) into EN1, so that Lμ*=(L1μ*,…,LN1μ*)T, where(4.11)Lnμ*=∫ΓQ-1(ξ)μ*(ξ)ln(ξ)dξ.
The operators (B-IN1) and L are finite-dimensional and therefore compact [19].We consider the columns(4.12)μ-=(μ*G),Φ-=(ΦH)
in the Banach space C0(Γ)×EN1 with the norm ∥μ-∥C0(Γ)×EN1=∥μ*∥C0(Γ)+∥G∥EN1.We write the system of (4.9) and (4.10) in the form of one equation:
(4.13)(I+R)μ-=Φ-,R=(APLB-IN1),
where I is the identity operator in the space C0(Γ)×EN1. It is clear that R is a compact operator mapping C0(Γ)×EN1 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 C0(Γ)×EN1 (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
(4.14)μ-0=(μ*0G0)∈C0(Γ)×EN1
be an arbitrary solution of the homogeneous equation (4.13). According to Lemma 4.1,
(4.15)μ-0=(μ*0G0)∈C0,p(Γ1)∩C0(Γ2)×EN1,p=min⁡⁡{λ,12}.
Therefore the function μ0(s)=μ*0(s)Q-1(s)∈C1/2p(Γ1)∩C0(Γ2) and the column G0 convert the homogeneous equations (4.1), (3.19b) and (4.6) into identities. For instance, (3.19b) takes the following form:(4.16a)lim⁡x→x(s)∈Γ2w[μ0](x)=0,x∈D.
Using the homogeneous identities (4.1) and (3.19b), we check that the homogeneous identities (4.6) are equivalent to
(4.16b)w[μ0](an1)=0,n=1,…,N1.
Besides, acting on the homogeneous identity (4.1) with a singular operator with the kernel (s-t)-1 we observe that μ0(s) satisfies the homogeneous equation (3.24):
(4.16c)∂∂sw[μ0](x(s))|Γ1=0.It follows from (4.16a), (4.16b) and (4.16c) that μ0(s) satisfies the homogeneous equation (3.16). On the basis of Theorem 3.1, u[0,μ0](x)≡w[μ0](x) is a solution to the homogeneous Problem U. According to Theorem 2.2, w[μ0](x)≡0 for x∈𝒟∖Γ1. Using the limit formulas for normal derivatives of a single-layer potential on Γ1, we have
(4.17)lim⁡x→x(s)∈(Γ1)+∂∂nxw[μ0](x)-lim⁡x→x(s)∈(Γ1)-∂∂nxw[μ0](x)=μ0(s)≡0,s∈Γ1.Hence,w[μ0](x)=w2[μ0](x)≡0 for x∈𝒟, and μ0(s) satisfies (4.16a), which can be written as
(4.18)12μ0(s)+i4∫Γ2μ0(σ)(∂∂ny-i)H0(1)(β|x(s)-y(σ)|)dσ=0,s∈Γ2.It is shown in [5, page 502–504] and [6, page 187–189] that μ0(s)≡0 is the unique solution of (4.18) in C0(Γ2).Consequently, ifs∈Γ, then μ0(s)≡0 and μ*0(s)=μ0(s)Q-1(s)≡0. It follows from the homogeneous identity (4.1) for μ0(s) and G00,…,GN1-10 that G0=(G00,…,GN1-10)T≡0. Hence, μ-0≡0. Thus, the homogeneous Fredholm equation (4.13) has only a trivial solution in C0(Γ)×EN1.We have proved the following assertion.Theorem 4.2.
IfΓ∈C2,λ,λ∈(0,1], then (4.13) is a Fredholm equation of the second kind and of index zero in the space C0(Γ)×EN1. Moreover, (4.13) has a unique solution(4.19)μ-=(μ*G)∈C0(Γ)×EN1
for any(4.20)Φ-=(ΦH)∈C0(Γ)×EN1.As a consequence of Theorem4.2 and the Lemma 4.1 we obtain the following corollary.Corollary 4.3.
IfΓ∈C2,λ,λ∈(0,1], then equation (4.13) has a unique solution:
(4.21)μ-=(μ*G)∈C0,p(Γ1)∩C0(Γ2)×EN1
for any
(4.22)Φ-=(ΦH)∈C0,p(Γ1)∩C0(Γ2)×EN1,
where p=min{λ,1/2}.We recall thatΦ- 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 {μ*(s),G0,…,GN1-1} is a solution of the system of (4.4) and (4.6), and μ*(s)∈C0,p(Γ1)∩C0(Γ2), then the function μ(s)=μ*(s)Q-1(s)∈C1/2p(Γ1)∩C0(Γ2) is a solution of the system of (3.19a), (3.19b) and (3.22), and therefore μ(s) satisfies (3.16). We obtain the following statement.Theorem 4.4.
IfΓ∈C2,λ, and if conditions (3.1a) and (3.1b) hold, then equation (3.16) has a solution μ(s)∈C1/2p(Γ1)∩C0(Γ2),p=min{1/2,λ}. This solution is expressed by the formula μ(s)=μ*(s)Q-1(s), where the function μ*(s)∈C0,p(Γ1)∩C0(Γ2) is found by solving the Fredholm equation (4.13), which is uniquely solvable according to Corollary 4.3.On the basis of Theorem3.1 we arrive at the solvability theorem for the Problem U.Theorem 4.5.
IfΓ∈C2,λ, and if conditions (3.1a) and (3.1b) hold, then the solution to the Problem U exists and is given by (3.8), where ν(s) is defined in (3.13) and μ(s) is a solution of equation (3.16) in C1/2p(Γ1)∩C0(Γ2),p=min{1/2,λ} ensured by Theorem 4.4.It can be checked directly that the solution to the ProblemU satisfies condition (2.3) with ϵ=-1/2. Explicit expressions for singularities of the solution gradient at the endpoints of the cracks will be presented in the next section.Theorem4.5 ensures existence of a classical solution to the Problem U when Γ∈C2,λ, λ∈(0,1], 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 U. First, we find the unique solution of the Fredholm equation (4.13) from C0(Γ)×EN1. This solution automatically belongs to C0,p(Γ1)∩C0(Γ2)×EN1, p=min{λ,1/2}. Second, we construct the solution of (3.16) from C1/2p(Γ1)∩C0(Γ2) by the formula μ(s)=μ*(s)Q-1(s). Finnaly, substituting ν(s) from (3.13) and μ(s) in (3.8) we obtain the solution to the Problem U.Remark 4.6.
It is important to stress that the solutionu(x) to the Problem U, ensured by Theorem 4.5, is a classical solution, but it may be not a weak solution to the Problem U. In other words, classical solution to the Problem U exists and is ensured by Theorem 4.5, but weak solution to the Problem U may not exist in Hloc1(𝒟∖Γ1) space. This follows from the fact that Dirichlet data on the closed curves Γ2 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 Hloc1(𝒟∖Γ1) 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 Section4, the gradient of the solution to Problem U can be unbounded at the endpoints of the cracks Γ1, so that the gradient of the solution to the Problem U satisfies estimate (2.3) with the exponent ϵ=-1/2. We will now make a detailed analysis of the behaviour of ∇u(x) at the endpoints of Γ1.Letu(x) be a solution to the Problem U ensured by Theorem 4.5 and given by (3.8). Let x(d)∈X be one of the endpoints of Γ1. In the neighbourhood of x(d), we introduce the system of polar coordinates: (5.1)x1=x1(d)+|x-x(d)|cos⁡⁡φ,x2=x2(d)+|x-x(d)|sin⁡⁡φ.
We will assume that φ∈(α(d),α(d)+2π) if d=an1 and φ∈(α(d)-π,α(d)+π) if d=bn1. We recall that α(s) is the angle between the direction of the Ox1 axis and the tangent vector τx to Γ1 at the point x(s).Hence,α(d)=α(an1+0) if d=an1 and α(d)=α(bn1-0) if d=bn1.Thus, the angleφ varies continuously in the neighbourhood of the endpoint x(d), cut along Γ1.We will use the notationμ1(s)=μ(s)|s-d|1/2=Q-1(s)μ*(s)|s-d|1/2 and put μ1(d)=μ1(an1)=μ1(an1+0) if d=an1, and μ1(d)=μ1(bn1)=μ1(bn1-0) if d=bn1.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.
Letu(x) be a solution to the Problem U ensured by Theorem 4.5. Let x(d) be an arbitrary endpoint of the cracks Γ1, that is, x(d)∈X and d=an1 or d=bn1 for some n=1,…,N1. Then the derivatives of the solution to the Problem U in the neighbourhood of x(d) have the following asymptotic behaviour.
Ifd=an1, then
(5.2)∂∂x1u(x)=μ1(an1)2|x-x(an1)|1/2sin⁡(φ+α(an1)2)-ν(an1)2π(-sin⁡α(an1)ln⁡|x-x(an1)|+φcos⁡α(an1))+O(1),∂∂x2u(x)=-μ1(an1)2|x-x(an1)|1/2cos⁡(φ+α(an1)2)-ν(an1)2π(cos⁡α(an1)ln⁡|x-x(an1)|+φsin⁡α(an1))+O(1).
If d=bn1, then
(5.3)∂∂x1u(x)=-μ1(bn1)2|x-x(bn1)|1/2cos⁡(φ+α(bn1)2)+ν(bn1)2π(-sin⁡α(bn1)ln⁡|x-x(bn1)|+φcos⁡α(bn1))+O(1),(5.4)∂∂x2u(x)=-μ1(bn1)2|x-x(bn1)|1/2sin⁡(φ+α(bn1)2)+ν(bn1)2π(cos⁡α(bn1)ln⁡|x-x(bn1)|+φsin⁡α(bn1))+O(1).
By O(1) one denotes functions which are continuous at the endpoint x(d). Moreover, the functions denoted by O(1) are continuous in the neighbourhood of the endpoint x(d) cut along Γ1 and are continuously extensible to (Γ1)+ and to (Γ1)- from this neighbourhood.The formulas of the theorem demonstrate the following curious fact. In the general case, the derivatives of the solution to the ProblemU near the endpoint x(d) of cracks Γ1 behave as
(5.5)O(|x-x(d)|-1/2)+O(ln⁡(1|x-x(d)|)).
However, if μ1(d)=ν(d)=0, then ∇u(x) will be bounded and even continuous at the endpoint x(d)∈X.



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