# The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

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

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## Abstract

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.


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## Body

## 1. Introduction

Boundary value problems in 2D domains with cracks (double-sided open arcs) are very important for applications [1].It is known that if the Dirichlet problem for the Laplace equation is considered in a 2D domain bounded by sufficiently smooth closed curves, and if the function specified in the boundary condition is smooth enough, then existence of a classical solution follows from existence of a weak solution. In the present paper, we consider the Dirichlet problem for the Laplace equation in both interior and exterior 2D domain bounded by closed curves and double-sided open arcs (cracks) of an arbitrary shape. The Dirichlet condition is specified on the whole boundary, that is, on both closed curves and on the cracks, so that different functions may be specified on opposite sides of the cracks. The case of this problem, when the Dirichlet boundary data satisfies compatibility conditions at the tips of the cracks, has been previously studied in [2–6], where theorems on existence and uniqueness of a classical solution have been proved, and the integral representation for a classical solution has been obtained. In the present paper, this problem is considered in case when the Dirichlet boundary data may not satisfy the compatibility conditions at the tips of the cracks. However if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. We prove that there exists a unique classical solution to our problem and obtain an integral representation for the classical solution. In addition, we prove that a weak solution to our problem may not exist even if both cracks and functions in the boundary conditions are sufficiently smooth. This result follows from the fact that the square of the gradient of a classical solution, basically, is not integrable near the ends of the cracks, since singularities of the gradient are rather strong there. This result is very important for numerical analysis, it shows that finite elements and finite difference methods cannot be applied to numerical treatment of the Dirichlet problem in question directly, since all these methods imply existence of a weak solution. To use difference methods for numerical analysis, one has to localize all strong singularities first and next to use difference method in a domain excluding the neighbourhoods of the singularities. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.


## 2.   Formulation of the Problem

By an open curve we mean a simple smooth nonclosed arc of finite length without self-intersections [7].LetΓ be a set of curves, which may be both closed and open. We say that Γ∈C2,λ (or Γ∈C1,λ) if curves Γ belong to the class C2,λ (or C1,λ) with the Hölder exponent λ∈(0,1].In a plane in Cartesian coordinatesx=(x1,x2)∈R2, we consider a multiply connected domain bounded by simple open curves Γ11,…,ΓN11∈C2,λ and simple closed curves Γ12,…,ΓN22∈C2,λ,  λ∈(0,1], in such a way that all curves do not have common points, in particular, endpoints. We will consider both the case of an exterior domain and the case of an interior domain when the curve Γ12 encloses all others. Set
(2.1)Γ1=⋃n=1N1‍Γn1,Γ2=⋃n=1N2‍Γn2,Γ=Γ1∪Γ2.
The connected domain bounded by Γ2 and containing curves Γ1 will be called 𝒟, so that ∂𝒟=Γ2,Γ1⊂𝒟. We assume that each curve Γnj is parametrized by the arc length s:
(2.2)Γnj={x:x=x(s)=(x1(s),x2(s)),s∈[anj,bnj]},n=1,…,Nj,j=1,2,
so that,
(2.3)a11<b11<⋯<aN11<bN11<a12<b12<⋯<aN22<bN22,
and the domain 𝒟 is placed to the right when the parameter s increases on Γn2. The points x∈Γ and values of the parameter s are in one-to-one correspondence except the points an2,  bn2, which correspond to the same point x for n=1,…,N2. Further on, the sets of the intervals
(2.4)⋃n=1N1‍[an1,bn1],⋃n=1N2‍[an2,bn2],⋃j=12‍⋃n=1Nj‍[anj,bnj]
on the Os-axis will be denoted by Γ1,  Γ2, and Γ also.SetC0(Γn2)={ℱ(s):ℱ(s)∈C0[an2,bn2],  ℱ(an2)=ℱ(bn2)}, and C0(Γ2)=⋂n=1N2C0(Γn2). The tangent vector to Γ in the point x(s), in the direction of the increment of s, will be denoted by τx=(cosα(s),sinα(s)), while the normal vector coinciding with τx after rotation through an angle of π/2 in the counterclockwise direction will be denoted by nx=(sinα(s),-cosα(s)). According to the chosen parametrization cosα(s)=x1′(s),   sinα(s)=x2′(s). Thus, nx is an inward normal to 𝒟 on Γ2. By X we denote the point set consisting of the endpoints of Γ1:X=⋃n=1N1(x(an1)∪x(bn1)).We considerΓ1 as a set of cracks (or double-sided open arcs). The side of the crack Γ1, which is situated on the left when the parameter s increases, will be denoted by (Γ1)+, while the opposite side will be denoted by (Γ1)-.We say that the functionu(x) belongs to the smoothness class K1, if (1)
u∈C0(𝒟∖Γ1¯∖X)∩C2(𝒟∖Γ1),∇u∈C0(𝒟∖Γ1¯∖Γ2∖X),(2)
in the neighbourhood of any pointx(d)∈X, both the inequality
(2.5)|u(x)|<Const
and the equality
(2.6)limr→+0∫∂S(d,r)‍|∂u(x)∂nx|dl=0
hold, where the curvilinear integral of the first kind is taken over a circumference ∂S(d,r) of a radius r with the center in the point x(d), in addition, nx is a normal in the point x∈∂S(d,r), directed to the center of the circumference and d=an1 or d=bn1,  n=1,…,N1.Remark 2.1.
ByC0(𝒟∖Γ1¯∖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. To obtain the definition of the class C0(𝒟∖Γ1¯∖Γ2∖X) we have to replace C0(𝒟∖Γ1¯∖X) by C0(𝒟∖Γ1¯∖Γ2∖X) and 𝒟¯∖Γ1 by 𝒟∖Γ1 in the previous sentence.Let us formulate the Dirichlet problem for the Laplace equation in a domain𝒟∖Γ1 (interior or exterior, see Figures 1 and 2).Figure 1
An example of an interior domain.Figure 2
An example of an exterior domain.ProblemD1
Find a functionu(x) from the class K1, so that u(x) obeys the Laplace equation
(2.7)ux1x1(x)+ux2x2(x)=0
in 𝒟∖Γ1 and satisfies the boundary conditions
(2.8)u(x)|x(s)∈(Γ1)+=F+(s),u(x)|x(s)∈(Γ1)-=F-(s),u(x)|x(s)∈Γ2=F(s).
If 𝒟 is an exterior domain, then we add the following condition at infinity:
(2.9)|u(x)|≤const,|x|=x12+x22→∞.
All conditions of the problemD1 must be satisfied in a classical sense. The boundary conditions (2.8) on Γ1 must be satisfied in the interior points of Γ1, their validity at the ends of Γ1 is not required.Theorem 2.2.
IfΓ∈C2,λ,λ∈(0,1], then there is no more than one solution to the problem D1.Proof.
It is sufficient to prove that the homogeneous problemD1 admits the trivial solution only. Let u0(x) be a solution to the homogeneous problem D1 with F+(s)≡F-(s)≡0,F(s)≡0. Let S(d,ϵ) be a disc of a small enough radius ϵ with the center in the point x(d)  (d=an1 or d=bn1,  n=1,...,N1). Let Γn,ϵ1 be a set consisting of such points of the curve Γn1 which do not belong to discs S(an1,ϵ) and S(bn1,ϵ). We choose a number ϵ0 so small that the following conditions are satisfied:(1)
for any0<ϵ≤ϵ0 the set of points Γn,ϵ1 is a unique non-closed arc for each n=1,...,N1,(2)
the points belonging toΓ∖Γn1 are placed outside the discs S(an1,ϵ0), S(bn1,ϵ0) for any n=1,...,N1,(3)
discs of radiusϵ0 with centers in different ends of Γ1 do not intersect.
SetΓ1,ϵ=∪n=1N1Γn,ϵ1,  Sϵ=(∪n=1N1[S(an1,ϵ)∪S(bn1,ϵ)]),  𝒟ϵ=𝒟∖Γ1,ϵ∖Sϵ. If 𝒟 is an exterior domain, then we set 𝒟ϵ,R=𝒟ϵ∩SR, where SR is a disc with a center in the origin and with sufficiently large radius R.
SinceΓ2∈C2,λ,  u0(x)∈C0(𝒟¯∖Γ1)  (remind that u0(x)∈K1), and since u0|Γ2=0∈C2,λ(Γ2), and owing to the lemma on regularity of solutions of elliptic equations near the boundary [8, lemma 6.18], we obtain u0(x)∈C1(𝒟¯∖Γ1). Since u0(x)∈K1, we observe that u0(x)∈C1(𝒟¯ϵ) for any ϵ∈(0,ϵ0]. By C1(𝒟¯ϵ) we mean C1(𝒟ϵ∪Γ2∪(Γ1,ϵ)+∪(Γ1,ϵ)-∪∂Sϵ). Analogously, in case of exterior domain 𝒟:u0(x)∈C1(𝒟¯ϵ,R) for ϵ∈(0,ϵ0]. Let 𝒟 be an interior domain. Since the boundary of a domain 𝒟ϵ is piecewise smooth, we write out Green's formula [9, page 328] for the function u0(x):
(2.10)∥∇u0∥L2(𝒟ϵ)2=∫Γ1,ϵ‍(u0)+(∂u0∂nx)+ds-∫Γ1,ϵ‍(u0)-(∂u0∂nx)-ds-∫Γ2‍u0∂u0∂nxds+∫∂Sϵ‍u0∂u0∂nxdl.
By nx the exterior (with respect to 𝒟ϵ) normal on ∂Sϵ at the point x∈∂Sϵ is denoted. 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 (2.8) on Γ, we observe that u0|Γ2=0 and (u0)±|Γ1,ϵ=0 for any ϵ∈(0,ϵ0]. Therefore, identity (2.10) takes the form
(2.11)∥∇u0∥L2(𝒟ϵ)2=∫∂Sϵ‍u0∂u0∂nxdl,ϵ∈(0,ϵ0].
Setting ϵ→+0 in (2.11), taking into account that u0(x)∈K1 and using the relationships (2.6), (2.5) we obtain
(2.12)∥∇u0∥L2(𝒟∖Γ1)2=limϵ→+0∥∇u0∥L2(𝒟ϵ)2=0.
From the homogeneous boundary conditions (2.8) we conclude that u0(x)≡0 in 𝒟∖Γ1, where 𝒟 is an interior domain.
Let𝒟 be an exterior domain. Since the boundary of a domain 𝒟ϵ,R is piecewise smooth and since u0(x)∈C1(𝒟¯ϵ,R) for any ϵ∈(0,ϵ0], we may write Green's formula in a domain 𝒟ϵ,R for a harmonic function u0(x)  [9, c.328]:
(2.13)∥∇u0∥L2(𝒟ϵ,R)2=∫Γ1,ϵ‍(u0)+(∂u0∂nx)+ds-∫Γ1,ϵ‍(u0)-(∂u0∂nx)-ds-∫Γ2‍u0∂u0∂nxds+∫∂Sϵ‍u0∂u0∂nxdl+∫∂SR‍u0∂u0∂|x|dl.
By nx on ∂Sϵ we denote an outward (with respect to 𝒟ϵ,R) normal in the point x∈∂Sϵ. It follows from condition (2.9) and from the theorem on behaviour of a gradient of a harmonic function at infinity [9, page 373] that
(2.14)∂u0(x)∂|x|=O(1|x|2),as|x|→∞.
Consequently,
(2.15)limR→∞∫∂SR‍u0(x)∂u0(x)∂|x|dl=0,
and formula (2.13) transforms to the formula (2.10) as R→∞. Repeating all arguments, presented above for the case of an interior domain 𝒟, we arrive to formula (2.12). Taking into account homogeneous boundary conditions (2.8), we obtain from (2.12) that u0(x)≡0 in 𝒟∖Γ1, where 𝒟 is an exterior domain. Thus, in all cases u0(x)≡0 in 𝒟∖Γ1. Theorem is proved.Remark 2.3.
The maximum principle cannot be used for the proof of the theorem even in the case of an interior domain𝒟, since the solution to the problem may not satisfy the boundary condition (2.8) at the ends of the cracks, and it may not be continuous at the ends of the cracks.


## 3. Properties of the Double Layer Potential on the Open Curve

Letγ be an open curve of class C1,λ,λ∈(0,1]. Assume that γ is parametrized by the arc length s:γ={x:x(s)=(x1(s),x2(s)),s∈[a,b]}. The points x∈Γ and values of the parameter s are in one-to-one correspondence, so the segment [a,b] will be also denoted by γ. The tangent vector to γ in the point x(s), in the direction of the increment of s, will be denoted by τx=(cosα(s),sinα(s)), while the normal vector to γ in the point x(s) will be denoted by nx=(sinα(s),-cosα(s)). According to the chosen parametrization cosα(s)=x1′(s),  sinα(s)=x2′(s). The side of the crack γ, which is situated on the left when the parameter s increases, will be denoted by γ+, while the opposite side will be denoted by γ-. Let Xγ=x(a)∪x(b) be a set of the ends of γ.Setμ(s)∈C0,λ[a,b], and consider the double layer potential for Laplace equation in a plane
(3.1)𝒲[μ](x)=-12π∫ab‍μ(σ)∂∂nyln|x-y(σ)|dσ.
Set z=x1+ix2, t=t(σ)=(y1(σ)+iy2(σ))∈γ, μ^(t)=μ^(t(σ))=μ(σ). If μ(s)∈C0,λ[a,b], then μ^(t)∈C0,λ(γ), since
(3.2)|μ^(t2)-μ^(t1)|=|μ^(t(σ2))-μ^(t(σ1))|=|μ(σ2)-μ(σ1)|≤c|σ2-σ1|λ=c(|σ2-σ1||t(σ2)-t(σ1)|)λ|t(σ2)-t(σ1)|λ=c·c0λ|t2-t1|λ,
where c and c0 are constants, t2=t(σ2)∈γ,  t1=t(σ1)∈γ. We took into account in deriving the latter inequality that
(3.3)|σ2-σ1||t(σ2)-t(σ1)|∈C0([a,b]×[a,b]),
(see lemma 1 in [10]), whence
(3.4)|σ2-σ1||t(σ2)-t(σ1)|≤c0.Consider the integral of the Cauchy type with the real densityμ^(t):
(3.5)Φ(z)=12πi∫γ‍μ^(t)dtt-z,
then 𝒲[μ](x)=-ReΦ(z). It follows from properties of the Cauchy type integral that if μ(σ)∈C0,λ[a,b], then 𝒲[μ](x)∈C0(R2∖γ¯∖Xγ). This means that the potential 𝒲[μ](x) is continuously extensible to γ from the left and from the right in interior points (though its values on γ from the left and from the right may be different). If, in addition, μ(d)=0, then the potential 𝒲[μ](x) is continuously extensible to the end x(d), where d=a or d=b  (see [7, Section   15.2]). Set
(3.6)cosψ(x,y)=x1-y1|x-y|=-|x-y|y1′,sinψ(x,y)=x2-y2|x-y|=-|x-y|y2′,
then ψ(x,y) is a polar angle of the coordinate system with the origin in the point y. Formulae for cosψ(x,y),sinψ(x,y) define the angle ψ(x,y) with indeterminacy up to 2πm  (m is an integer number). Let S(d,ϵ) be a disc of a sufficiently small radius ϵ with the center in x(d)  (d=a or d=b). From asymptotic formulae describing behavior of Φ(z) at the ends of γ [7, Section   22], we may derive the asymptotic formulae for 𝒲[μ](x)=-ReΦ(z) at the ends of γ. Namely, for any x∈S(d,ϵ) and x∉γ, the formula holds:
(3.7)𝒲[μ](x)=±μ(d)2πψ(x,x(d))+Ω(x).
Here by ψ(x,x(d)) we mean some fixed branch of this function, so that the branch varies continuously in x in a neighbourhood of the point x(d), cut along γ. The upper sign is taken if d=a, while the lower sign is taken if d=b. The function Ω(x) is continuous as x→x(d). Moreover, Ω(x) is continuous in S(d,ϵ) outside γ and is continuously extensible from the left and from the right to the part of γ lying in S(d,ϵ). It follows from formula (3.7) that for any x∈S(d,ϵ) and x∉γ the inequality holds
(3.8)|𝒲[μ](x)|≤const.Now we will study properties of derivatives of the double layer potential. It follows from Cauchy-Riemann relations that(3.9)dΦ(z)dz=(ReΦ)x1′-i(ReΦ)x2′=-𝒲x1′+i𝒲x2′.
On the other hand, if μ(σ)∈C1,λ[a,b], then for z∉γ:
(3.10)dΦ(z)dz=12πi∫γ‍μ^(t)ddz1t-zdt=-12πi∫γ‍μ^(t)(ddt1t-z)dt=-12πi(μ(b)t(b)-z-μ(a)t(a)-z-∫γ‍μ^′(t)t-zdt),
where
(3.11)dμ^(t)dt=dμ(σ)dσdσdt=μ'(σ)t'(σ)=e-iα(σ)μ'(σ).
Since γ∈C1,λ, then e-iα(σ)∈C0,λ[a,b], so one can show that μ^′(t)∈C0,λ(γ) (the proof repeats the given above proof of the fact that μ^(t)∈C0,λ(γ), if μ(σ)∈C0,λ[a,b]). From (3.9) and (3.10) and from properties of the Cauchy type integral [7, Section   15], it follows that if μ(σ)∈C1,λ[a,b], then ∇𝒲[μ](x)∈C0(R2∖γ¯∖Xγ), that is, ∇𝒲[μ](x) is continuously extensible to γ from the left and from the right in interior points, though the limiting values of ∇𝒲[μ](x) on γ from the left and from the right can be different. We can write (3.10) in the form
(3.12)dΦ(z)dz=12πi(μ(b)e-iψ(x,x(b))|x-x(b)|-μ(a)e-iψ(x,x(a))|x-x(a)|)+Ω0(z),
where
(3.13)Ω0(z)=12πi∫γ‍μ^′(t)t-zdt.
It follows from [7, § 22] that for all z∈S(d,ϵ) (d=a or d=b), such that z∉γ, the inequality holds
(3.14)|Ω0(z)|≤c0(|μ'(d)|ln1|x-x(d)|+1)≤cln1|x-x(d)|,
where c0 and c are constants.Comparing formulae (3.9) and (3.12), we obtain that for x∈S(d,ϵ) and x∉γ, the formulae hold
(3.15)∂𝒲[μ](x)∂x1=12π∓μ(d)|x-x(d)|sinψ(x,x(d))+Ω1(x),∂𝒲[μ](x)∂x2=12π±μ(d)|x-x(d)|cosψ(x,x(d))+Ω2(x),
where
(3.16)|Ωj(x)|≤c1(|μ'(d)|ln1|x-x(d)|+1)≤c2ln1|x-x(d)|,j=1,2,c1,c2 are constants. The upper sign in formulae is taken if d=a, while the lower sign is taken if d=b. It follows from [7, Section   15.2] that if μ'(d)=0, then the functions Ω1(x) and Ω2(x) are continuously extensible to the end x(d). Moreover, if x∈S(d,ϵ) and x∉γ, then for the functions Ω1(x) and Ω2(x), the formulae hold:
(3.17)Ω1(x)=-ReΩ0(z)=∓μ′(d)2π{sin(α(d))ln|x-x(d)|-cos(α(d))ψ(x,x(d))}+Ω10(x),Ω2(x)=ImΩ0(z)=±μ'(d)2π{cos(α(d))ln|x-x(d)|+sin(α(d))ψ(x,x(d))}+Ω20(x),
which can be derived, using the asymptotics for Ω0(z) from [7, § 22]. The upper sign in formulae is taken if d=a, while the lower sign is taken if d=b. Functions Ω10(x) and Ω20(x) are continuously extensible to the end x(d). By ψ(x,x(d)) we mean some fixed branch of this function, which varies continuously in x in a neighbourhood of a point x(d), cut along γ.Letμ(σ)∈C1,λ[a,b], and let nx be a normal in the point x∈∂S(d,ϵ), directed to the center of the circumference ∂S(d,ϵ), that is, nx=(-cosψ(x,x(d)),-sinψ(x,x(d))), then we obtain from (3.15) for x∉γ(3.18)∂𝒲[μ](x)∂nx|∂S(d,ϵ)=-Ω1(x)cosψ(x,x(d))-Ω2(x)sinψ(x,x(d)).
Therefore, according to (3.16):
(3.19)|∂𝒲[μ](x)∂nx||∂S(d,ϵ)≤constln1|x-x(d)||∂S(d,ϵ)=constln1ϵ,
since |x-x(d)|=ϵ on ∂S(d,ϵ). From here we obtain that
(3.20)∫∂S(d,ϵ)‍|∂𝒲[μ](x)∂nx|dl=∫02π‍|∂𝒲[μ](x)∂nx|ϵdψ≤2πconstϵln1ϵ→0,
if ϵ→+0, so
(3.21)limϵ→+0∫∂S(d,ϵ)‍|∂𝒲[μ](x)∂nx|dl=0.
Now let ϵ be a fixed positive number (sufficiently small). Using formulae (3.15) and setting r=|x-x(d)|,  ψ=ψ(x,x(d)), we consider the integral over the disc S(d,ϵ):
(3.22)∫S(d,ϵ)‍|∇𝒲[μ](x)|2dx=∫02π‍∫0ϵ‍{(μ(d)2πr)2+μ(d)πr((μ(d)2πr)2∓Ω1(x)sinψ±Ω2(x)cosψ)+Ω12(x)+Ω22(x)(μ(d)2πr)2}rdrdψ=I1+I2,(3.23)I1=12π∫0ϵ‍1rμ2(d)dr,I2=∫02π‍∫0ϵ‍{μ(d)π(∓Ω1(x)sinψ±Ω2(x)cosψ)+r(Ω12(x)+Ω22(x))}drdψ.
The integral I2 converges according to estimates (3.16):
(3.24)|I2|≤4c2∫0ϵ‍ln1r(|μ(d)|+c2πrln1r)dr≤const.
Hence, if integral (3.22) converges, then the integral I1 converges as well (as a difference of two convergent integrals), but the integral I1 converges if and only if μ(d)=0, while in other cases I1 diverges. Thus, the integral (3.22) converges if and only if μ(d)=0. Consequently |∇𝒲[μ](x)| belongs to L2(S(d,ϵ)) with small ϵ>0 if and only if μ(d)=0. Let us formulate obtained results in the form of the theorem.Theorem 3.1.
Letγ be an open curve of class C1,λ,  λ∈(0,1]. Let S(d,ϵ) be a disc of a sufficiently small radius ϵ with the center in the point x(d)∈Xγ  (d=a or d=b). (i)
Ifμ(s)∈C0,λ[a,b], then 𝒲[μ](x)∈C0(R2∖γ¯∖Xγ) and for any x∈S(d,ϵ), such that x∉γ, the relationships (3.7) and (3.8) hold.(ii)
Ifμ(s)∈C1,λ[a,b], then
(1)
∇𝒲[μ](x)∈C0(R2∖γ¯∖Xγ);(2)
for anyx∈S(d,ϵ), such that x∉γ, the formulae (3.15) hold, in which the functions Ω1(x) and  Ω2(x) satisfy relationships (3.16) and (3.17);(3)
for𝒲[μ](x) the property (3.21) holds;(4)
|∇𝒲[μ](x)| belongs to L2(S(d,ϵ)) for sufficiently small ϵ>0 if and only if μ(d)=0.Remark 3.2.
Each function of classC0(R2∖γ¯∖Xγ) is continuous in R2∖γ, is continuously extensible to γ∖Xγ from the left and from the right, but limiting values of such a function on γ∖Xγ from the left and from the right can be different, that is, the function may have a jump on γ∖Xγ.
Let us study smoothness of the direct value of the double layer potential on the open curve.Lemma 3.3.
Letγ be an open curve of class C2,λ,  λ∈(0,1], and let μ(s)∈C0[a,b]. Let
(3.25)I1(s)=-12π∫γ‍μ(σ)∂ln|x(s)-y(σ)|∂nydσ
be the direct value of the double layer potential 𝒲[μ](x) on γ. Then
(3.26)I1(s)∈C1,λ/4[a,b].Proof.
Let us prove thatI1(s)∈C1,λ/4[a,b]. Taking into account that ny=(y2′(σ),-y1′(σ)), we find
(3.27)∂ln|x(s)-y(σ)|∂ny=T(s,σ)g(s,σ),g(s,σ)=|x(s)-y(σ)|2(s-σ)2,T(s,σ)=[x2(s)-y2(σ)]y1′(σ)-[x1(s)-y1(σ)]y2′(σ)(s-σ)2.
Note that y(σ) is a point on Γ corresponding to s=σ. So, we may put x(σ)=y(σ). For j=1,2, we have [10, § 3]
(3.28)xj(s)-xj(σ)=(s-σ)Zj1(s,σ)=-xj′(σ)(σ-s)+(σ-s)2Zj2(σ,s),
where
(3.29)Zj1(s,σ)=∫01‍xj′(σ+ξ(s-σ))dξ∈C1,λ([a,b]×[a,b]),Zj2(σ,s)=∫01‍ξxj′′(s+ξ(σ-s))dξ∈C0,λ([a,b]×[a,b]).
Note that the function
(3.30)g(s,σ)=|x(s)-x(σ)|2(s-σ)2={[Z11(s,σ)]2+[Z21(s,σ)]2}∈C1,λ([a,b]×[a,b])
does not equal zero anywhere on Γ and g(s,s)=1; therefore,
(3.31)1g(s,σ)∈C1([a,b]×[a,b]).
Further,
(3.32)∂∂s1g(s,σ)=∂∂s(s-σ)2|x(s)-x(σ)|2=-gs′(s,σ)g2(s,σ)=-2Z11(s,σ)[Z11(s,σ)]s′+Z21(s,σ)[Z21(s,σ)]s′g2(s,σ)∈C0,λ([a,b]×[a,b]).
Consequently, 1/g(s,σ)∈C1,λ([a,b]×[a,b]). Similarly,
(3.33)T(s,σ)=[x2(s)-x2(σ)]x1′(σ)-[x1(s)-x1(σ)]x2′(σ)(s-σ)2=[Z22(σ,s)x1′(σ)-Z12(σ,s)x2′(σ)]∈C0,λ([a,b]×[a,b]).
Consider ∂T(s,σ)/∂s=J1(s,σ)-2J2(s,σ), where
(3.34)J1(s,σ)=x2′(s)x1′(σ)-x1′(s)x2′(σ)(s-σ)2=[x2′(s)-x2′(σ)]x1′(σ)-[x1′(s)-x1′(σ)]x2′(σ)(s-σ)2=1s-σ{x1′(σ)∫01‍x2′′(s+ξ(σ-s))dξ-x2′(σ)∫01‍x1′′(s+ξ(σ-s))dξ};J2(s,σ)=[x2(s)-x2(σ)]x1′(σ)-[x1(s)-x1(σ)]x2′(σ)(s-σ)3=1s-σ{x1′(σ)∫01‍ξx2′′(s+ξ(σ-s))dξ-x2′(σ)∫01‍ξx1′′(s+ξ(σ-s))dξ}.
Then
(3.35)∂T(s,σ)∂s=1s-σ{x1′(σ)∫01‍(1-2ξ)x2′′(s+ξ(σ-s))dξ-x2′(σ)∫01(1-2ξ)x1′′(s+ξ(σ-s))dξ}=K(s,σ)s-σ,
where K(s,σ)∈C0,λ([a,b]×[a,b]) and K(s,s)=0. According to [7, Section   5.7], the following representation holds
(3.36)∂T(s,σ)∂s=K*(s,σ)|s-σ|1-λ/4,K*(s,σ)∈C0,3λ/4([a,b]×[a,b]). Using properties of Hölder functions [7], we obtain the representation
(3.37)∂∂s∂ln|x(s)-y(σ)|∂ny=1g(s,σ)∂T(s,σ)∂s+T(s,σ)∂∂s1g(s,σ)=K1(s,σ)|s-σ|1-λ/4+K2(s,σ),
where K1(s,σ)∈C0,3λ/4([a,b]×[a,b]),  K2(s,σ)∈C0,λ([a,b]×[a,b]). By formal differentiation under the integral, we find
(3.38)dI1(s)ds=-12π∫γ‍μ(σ)∂∂s∂ln|x(s)-y(σ)|∂nydσ=-12π∫γ‍μ(σ)K1(s,σ)|s-σ|1-λ/4dσ-12π∫γ‍μ(σ)K2(s,σ)dσ.
The validity of differentiation under the integral can be proved in the same way as at the end of   [9, Section   1.6] (Fubini theorem on change of integration order is used). Taking into account the obtained representation for dI1(s)/ds and applying results of [7, Section   51.1], we obtain that dI1(s)/ds∈C0,λ/4[a,b]. Lemma is proved.


## 4. Existence of A Classical Solution

We will construct the solution to the problemD1 in assumption that F+(s),F-(s)∈C1,λ(Γ1),  λ∈(0,1],  F(s)∈C0(Γ2). Note that we do not require compatibility conditions at the tips of the cracks, that is, we we do not require that F+(d)=F-(d) for any x(d)∈X. We will look for a solution to the problem D1 in the form
(4.1)u(x)=-w[F+-F-](x)+v(x),
where
(4.2)w[F+-F-](x)=-12π∫Γ1‍(F+(σ)-F-(σ))∂∂nyln|x-y(σ)|dσ
is the double layer potential. The potential w[F+-F-](x) satisfies the Laplace equation (2.7) in 𝒟∖Γ1 and belongs to the class K1 according to Theorem 3.1. Limiting values of the potential w[F+-F-](x) on (Γ1)± are given by the formula
(4.3)w[F+-F-](x)|x(s)∈(Γ1)±=∓F+(s)-F-(s)2+w[F+-F-](x(s)),
where w[F+-F-](x(s)) is the direct value of the potential on Γ1.The functionv(x) in (4.1) must be a solution to the following problem.ProblemD
Find a functionv(x)∈C0(𝒟¯)∩C2(𝒟∖Γ1), which obeys the Laplace equation (2a) in the domain 𝒟∖Γ1 and satisfies the boundary conditions
(4.4)v(x)|x(s)∈Γ1=F+(s)+F-(s)2+w[F+-F-](x(s))=f(s),v(x)|x(s)∈Γ2=F(s)+w[F+-F-](x(s))=f(s).
If x(s)∈Γ1, then w[F+-F-](x(s)) is the direct value of the potential on Γ1.
If𝒟 is an exterior domain, then we add the following condition at infinity:
(4.5)|v(x)|≤const,|x|=x12+x22→∞.
All conditions of the problemD have to be satisfied in a classical sense. Obviously, w[F+-F-](x(s))∈C0(Γ2). It follows from Lemma 3.3 that w[F+-F-](x(s))∈C1,λ/4(Γ1)  (here by w[F+-F-](x(s)) we mean the direct value of the potential on Γ1). So, f(s)∈C1,λ/4(Γ1) and f(s)∈C0(Γ2).
We will look for the functionv(x) in the smoothness class K.
We say that the functionv(x) belongs to the smoothness class K if (1)
v(x)∈C0(𝒟¯)∩C2(𝒟∖Γ1),∇v∈C0(𝒟∖Γ1¯∖Γ2∖X), where X is a pointset consisting of the endpoints of Γ1.(2)
in a neighbourhood of any pointx(d)∈X for some constants 𝒞>0,  δ>-1 the inequality |∇v|≤𝒞|x-x(d)|δ holds, where x→x(d) and d=an1 or d=bn1,  n=1,…,N1.The definition of the functional classC0(𝒟∖Γ1¯∖Γ2∖X) is given in the remark to the definition of the smoothness class K1. Clearly, K⊂K1, that is, if v(x)∈K, then v(x)∈K1.It can be verified directly that ifv(x) is a solution to the problem  D in the class K, then the function (4.1) is a solution to the problem D1.Theorem 4.1.
LetΓ∈C2,λ/4,  f(s)∈C1,λ/4(Γ1),  λ∈(0,1],  f(s)∈C0(Γ2). Then the solution to the problem  D in the smoothness class K exists and is unique.Theorem4.1 has been proved in the following papers: (1)
in [2, 3], if 𝒟 is an interior domain;(2)
in [4], if 𝒟 is an exterior domain and Γ2≠∅;(3)
in [5, 6], if Γ2=∅ and so 𝒟=R2 is an exterior domain.In all these papers, the integral representations for the solution to the problemD in the class K are obtained in the form of potentials, densities in which are defined from the uniquely solvable Fredholm integroalgebraic equations of the second kind and index zero. Uniqueness of a solution to the problem D is proved either by the maximum principle or by the method of energy (integral) identities. In the latter case, we take into account that a solution to the problem belongs to the class K. Note that the problem D is a particular case of more general boundary value problems studied in [3–6].Note that conditions of Theorem4.1 hold if Γ∈C2,λ,  F+(s)∈C1,λ(Γ1),  F-(s)∈C1,λ(Γ1),  λ∈(0,1],  F(s)∈C0(Γ2). From Theorems 3.1 and 4.1 we obtain the solvability of the problem D1.Theorem 4.2.
LetΓ∈C2,λ,  F+(s)∈C1,λ(Γ1),  F-(s)∈C1,λ(Γ1),  λ∈(0,1],  F(s)∈C0(Γ2). Then a solution to the problem D1 exists and is given by the formula (4.1), where v(x) is a unique solution to the problem D in the class  K ensured by Theorem 4.1.Uniqueness of a solution to the problemD1 follows from Theorem 2.2. In fact, the solution to the problem D1 found in Theorem 4.2 is a classical solution. Let us discuss, under which conditions this solution to the problem D1 is not a weak solution.


## 5. Nonexistence of a Weak Solution

Letu(x) be a solution to the problem D1 defined in Theorem 4.2 by the formula (4.1). Consider a disc S(d,ϵ) with the center in the point x(d)∈X and of radius ϵ>0  (d=an1 or d=bn1,  n=1,…,N1). In doing so, ϵ is a fixed positive number, which can be taken small enough. Since v(x)∈K, we have v(x)∈L2(S(d,ϵ)) and |∇v(x)|∈L2(S(d,ϵ)) (this follows from the definition of the smoothness class K). Let x∈S(d,ϵ) and x∉Γ1. It follows from (4.1) that |∇w[μ](x)|≤|∇u(x)|+|∇v(x)|, whence
(5.1)|∇w[μ](x)|2≤|∇u(x)|2+|∇v(x)|2+2|∇u(x)|·|∇v(x)|≤2(|∇u(x)|2+|∇v(x)|2),
since 2|∇u(x)|·|∇v(x)|≤|∇u(x)|2+|∇v(x)|2. Assume that |∇u(x)| belongs to L2(S(d,ϵ)), then, integrating this inequality over S(d,ϵ), we obtain ∥∇w∥2|L2(S(d,ϵ))≤2(∥∇u∥2|L2(S(d,ϵ))+∥∇v∥2|L2(S(d,ϵ))). Consequently, if |∇u(x)|∈L2(S(d,ϵ)), then |∇w|∈L2(S(d,ϵ)). However, according to Theorem 3.1, if F+(d)-F-(d)≠0, then |∇w| does not belong to L2(S(d,ϵ)). Therefore, if F+(d)≠F-(d), then our assumption that |∇u|∈L2(S(d,ϵ)) does not hold, that is, |∇u|∉L2(S(d,ϵ)). Thus, if among numbers a11,…,aN11,b11,…,bN11 there exists such a number d that F+(d)≠F-(d), then for some ϵ>0, we have |∇u|∉L2(S(d,ϵ))=L2(S(d,ϵ)∖Γ1), so u∉H1(S(d,ϵ)∖Γ1), where H1 is a Sobolev space of functions from L2, which have generalized derivatives from L2. We have proved the following theorem.Theorem 5.1.
Let conditions of Theorem4.2 hold and among numbers a11,…,aN11,b11,…,bN11 there exists such a number d, that F+(d)≠F-(d) (i.e., compatibility condition does not hold at the tip x(d)∈X). Then the solution to the problem D1, ensured by Theorem 4.2, does not belong to H1(S(d,ϵ)∖Γ1) for some ϵ>0, whence it follows that it does not belong to Hloc1(𝒟∖Γ1). Here S(d,ϵ) is a disc of a radius ϵ with the center in the point x(d)∈X.ByHloc1(𝒟∖Γ1) we denote a class of functions, which belong to H1 on any bounded subdomain of 𝒟∖Γ1. If conditions of Theorem 5.1 hold, then the unique solution to the problem D1, constructed in Theorem 4.2, does not belong to Hloc1(𝒟∖Γ1), and so it is not a weak solution. We arrive to the following corollary.Corollary 5.2.
Let conditions of Theorem5.1 hold, then a weak solution to the problem D1 in the class of functions Hloc1(𝒟∖Γ1) does not exist.Remark 5.3.
It should be stressed that even if closed curves and cracks are very smooth and if boundary data is very smooth as well, but if there exists a tip of the crack, where the compatibility condition does not hold, then a weak solution of the problemD1 in the class of functions Hloc1(𝒟∖Γ1) does not exist.Remark 5.4.
Even if the numberd, mentioned in Theorem 5.1, does not exist, then the solution u(x) to the problem D1, ensured by Theorem 4.2, may not be a weak solution to the problem D1. 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 [11, Section   12.5] (the classical solution exists in this example).Clearly,L2(𝒟∖Γ1)=L2(𝒟), since Γ1 is a set of zero measure.


## 6.  Singularities of the Gradient of the Solution at the Endpoints of the Cracks

It follows from Theorems4.2 and 4.1 that the gradient of the solution of problem D1 given by formula (4.1) can be unbounded at the endpoints of the cracks Γ1.Letv(x) be a solution of the Problem D ensured by Theorem 4.1. Let u(x) be a solution of the Problem D1 ensured by Theorem 4.2 and given by formula (4.1). Let x(d)∈X be one of the endpoints of Γ1. In the neighbourhood of x(d), we introduce the system of polar coordinates
(6.1)x1=x1(d)+|x-x(d)|cosψ(x,x(d)),x2=x2(d)+|x-x(d)|sinψ(x,x(d)).
We assume that ψ(x,x(d))∈(α(d),α(d)+2π) if d=an1, and ψ(x,x(d))∈(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 angleψ(x,x(d)) varies continuously in the neighbourhood of the endpoint x(d), cut along Γ1.Letμ(s) be a solution of the integral equation ensured by Theorem 4 in [2] or by Theorem 4.4 in [3] or by Theorem 4 in [4]. The integral representation for the solution v(x) of the Problem D is constructed in [2–4] on the basis of the function μ(s) that is a solution of the certain integral equation.Alternatively, one can assume thatμ(s) is an element of a solution to equations ensured by Corollary 3.2 in [5] or by Theorem 4 in [6]. The solution v(x) of the Problem D is constructed in [5, 6] with the help of the function μ(s), which is an element of a solution to certain equations.We will use the notationμ1(s)=μ(s)|s-d|1/2, and put μ1(d)=μ1(an1)=μ1(an1+0) if d=an1,andμ1(d)=μ1(bn1)=μ1(bn1-0) if d=bn1.At first, we study the behaviour of the gradient of a solutionv(x) of the problem D at the tips of the cracks. Using the representation of the derivatives of harmonic potentials in terms of Cauchy type integrals (see [10]) and using the properties of these integrals near the endpoints of the integration line, presented in [7], we can prove the following assertion.Theorem 6.1.
Letv(x) be a solution of the problem D ensured by Theorem 4.1. 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 of the problem D in the neighbourhood of x(d) have the following asymptotic behaviour.
Ifd=an1, then
(6.2)∂∂x1v(x)=μ1(an1)2|x-x(an1)|1/2sin(ψ(x,x(an1))+α(an1)2)+O(1),∂∂x2v(x)=-μ1(an1)2|x-x(an1)|1/2cos(ψ(x,x(an1))+α(an1)2)+O(1),
Ifd=bn1, then
(6.3)∂∂x1v(x)=-μ1(bn1)2|x-x(bn1)|1/2cos(ψ(x,x(bn1))+α(bn1)2)+O(1),∂∂x2v(x)=-μ1(bn1)2|x-x(bn1)|1/2sin(ψ(x,x(bn1))+α(bn1)2)+O(1).
By O(1) we denote functions which are continuous at the endpoint x(d). Moreover, the functions denoted as 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 of problemD near the endpoint x(d) of cracks Γ1 behave as O(|x-x(d)|-1/2). However, if μ1(d)=0, then ∇v(x) is bounded and even continuous at the endpoint x(d)∈X.On the basis of Theorem3.1, Theorem 6.1 and formula (4.1), we may study the behaviour of the gradient of a solution u(x) of the problem D1 at the tips of the cracks. Using notations introduced above and notations from Section 3, we arrive at the following assertion.Theorem 6.2.
Letu(x) be a solution of the problem D1 ensured by Theorem 4.2. 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 of the problem D1 in the neighbourhood of x(d) have the following asymptotic behaviour.
Ifd=an1, then
(6.4)∂∂x1u(x)=12πF+(an1)-F-(an1)|x-x(an1)|sinψ(x,x(an1))+μ1(an1)2|x-x(an1)|1/2sin(ψ(x,x(an1))+α(an1)2)+F'+(an1)-F'-(an1)2π(sinα(an1)ln|x-x(an1)|-ψ(x,x(an1))cosα(an1))+O(1),∂∂x2u(x)=-12πF+(an1)-F-(an1)|x-x(an1)|cosψ(x,x(an1))-μ1(an1)2|x-x(an1)|1/2cos(ψ(x,x(an1))+α(an1)2)-F′+(an1)-F′-(an1)2π(cosα(an1)ln|x-x(an1)|+ψ(x,x(an1))sinα(an1))+O(1).
Ifd=bn1, then
(6.5)∂∂x1u(x)=-12πF+(bn1)-F-(bn1)|x-x(bn1)|sinψ(x,x(bn1))-μ1(bn1)2|x-x(bn1)|1/2cos(ψ(x,x(bn1))+α(bn1)2)-F'+(bn1)-F'-(bn1)2π(sinα(bn1)ln|x-x(bn1)|-ψ(x,x(bn1))cosα(bn1))+O(1),∂∂x2u(x)=12πF+(bn1)-F-(bn1)|x-x(d)|cosψ(x,x(bn1))-μ1(bn1)2|x-x(bn1)|1/2sin(ψ(x,x(bn1))+α(bn1)2)+F'+(bn1)-F'-(bn1)2π(cosα(bn1)ln|x-x(bn1)|+ψ(x,x(bn1))sinα(bn1))+O(1).
By O(1) we denote functions which are continuous at the endpoint x(d). Moreover, the functions denoted as 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.In the formulation of the theorem we use the notationF'±(s)=dF±(s)/ds. The formulas of the theorem demonstrate the following curious fact. In the general case, the derivatives of the solution of problem D1 near the endpoint x(d) of cracks Γ1 behave as
(6.6)O(1|x-x(d)|)+O(1|x-x(d)|1/2)+O(ln1|x-x(d)|).
However, if F+(d)-F-(d)=μ1(d)=F'+(d)-F'-(d)=0, then ∇u(x) is bounded and even continuous at the endpoint x(d)∈X.



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