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

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.

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].

Let Γ be a set of curves, which may be both closed and open. We say that Γ∈𝐶^2,𝜆 (or Γ∈𝐶^1,𝜆) if curves Γ belong to the class 𝐶^2,𝜆 (or 𝐶^1,𝜆) with the Hölder exponent 𝜆∈(0,1].

In a plane in Cartesian coordinates 𝑥=(𝑥₁,𝑥₂)∈𝑅², we consider a multiply connected domain bounded by simple open curves Γ¹₁,…,Γ¹_𝑁1∈𝐶^2,𝜆 and simple closed curves Γ²₁,…,Γ²_𝑁2∈𝐶^2,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 Γ²₁ encloses all others. Set Γ¹=^𝑁1_𝑛=1Γ¹_𝑛,Γ²=^𝑁2_𝑛=1Γ²_𝑛,Γ=Γ¹∪Γ². (2.1) 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 Γ^𝑗_𝑛 is parametrized by the arc length 𝑠: Γ^𝑗_𝑛=𝑥∶𝑥=𝑥(𝑠)=𝑥₁(𝑠),𝑥₂(𝑠),𝑠∈𝑎^𝑗_𝑛,𝑏^𝑗_𝑛,𝑛=1,…,𝑁_𝑗,𝑗=1,2, (2.2) so that, 𝑎¹₁<𝑏¹₁<⋯<𝑎¹_𝑁1<𝑏¹_𝑁1<𝑎²₁<𝑏²₁<⋯<𝑎²_𝑁2<𝑏²_𝑁2, (2.3) and the domain 𝒟 is placed to the right when the parameter 𝑠 increases on Γ²_𝑛. The points 𝑥∈Γ and values of the parameter 𝑠 are in one-to-one correspondence except the points 𝑎²_𝑛, 𝑏²_𝑛, which correspond to the same point 𝑥 for 𝑛=1,…,𝑁₂. Further on, the sets of the intervals ^𝑁1_𝑛=1𝑎¹_𝑛,𝑏¹_𝑛,^𝑁2_𝑛=1𝑎²_𝑛,𝑏²_𝑛,^2𝑁_𝑗=1^𝑗_𝑛=1𝑎^𝑗_𝑛,𝑏^𝑗_𝑛 (2.4) on the 𝑂𝑠-axis will be denoted by Γ¹, Γ², and Γ also.

Set 𝐶⁰(Γ²_𝑛)={ℱ(𝑠)∶ℱ(𝑠)∈𝐶⁰[𝑎²_𝑛,𝑏²_𝑛],  ℱ(𝑎²_𝑛)=ℱ(𝑏²_𝑛)}, and 𝐶⁰(Γ²)=⋂^𝑁2_𝑛=1𝐶⁰(Γ²_𝑛). The tangent vector to Γ in the point 𝑥(𝑠), in the direction of the increment of 𝑠, will be denoted by 𝜏_𝑥=(cos𝛼(𝑠),sin𝛼(𝑠)), while the normal vector coinciding with 𝜏_𝑥 after rotation through an angle of 𝜋/2 in the counterclockwise direction will be denoted by 𝐧_𝑥=(sin𝛼(𝑠),−cos𝛼(𝑠)). According to the chosen parametrization cos𝛼(𝑠)=𝑥^′₁(𝑠),   sin𝛼(𝑠)=𝑥^′₂(𝑠). Thus, 𝐧_𝑥 is an inward normal to 𝒟 on Γ². By 𝑋 we denote the point set consisting of the endpoints of Γ¹∶𝑋=⋃^𝑁1_𝑛=1(𝑥(𝑎¹_𝑛)∪𝑥(𝑏¹_𝑛)).

We consider Γ¹ 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 function 𝑢(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 point 𝑥(𝑑)∈𝑋, both the inequality |𝑢(𝑥)|<Const (2.5) and the equality lim_{𝑟→+0𝜕𝑆(𝑑,𝑟)}𝜕𝑢(𝑥)𝜕𝐧_𝑥𝑑𝑙=0 (2.6) 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, 𝐧_𝑥 is a normal in the point 𝑥∈𝜕𝑆(𝑑,𝑟), directed to the center of the circumference and 𝑑=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛,  𝑛=1,…,𝑁₁.

Remark 2.1. By 𝐶⁰(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 domain 𝒟⧵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.

Problem 𝐃₁
Find a function 𝑢(𝑥) from the class 𝐊₁, so that 𝑢(𝑥) obeys the Laplace equation 𝑢_𝑥1𝑥1(𝑥)+𝑢_𝑥2𝑥2(𝑥)=0 (2.7) in 𝒟⧵Γ¹ and satisfies the boundary conditions 𝑢(𝑥)|_{𝑥(𝑠)∈(Γ1)}⁺=𝐹⁺(𝑠),𝑢(𝑥)|_{𝑥(𝑠)∈(Γ}¹₎⁻=𝐹⁻(𝑠),𝑢(𝑥)|_{𝑥(𝑠)∈Γ}²=𝐹(𝑠). (2.8) If 𝒟 is an exterior domain, then we add the following condition at infinity: |𝑢(𝑥)|≤const,|𝑥|=𝑥²₁+𝑥²₂→∞. (2.9)
All conditions of the problem 𝐃₁ 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. If Γ∈𝐶^2,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 problem 𝐃₁ 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=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛,  𝑛=1,...,𝑁₁). Let Γ¹_𝑛,𝜖 be a set consisting of such points of the curve Γ¹_𝑛 which do not belong to discs 𝑆(𝑎¹_𝑛,𝜖) and 𝑆(𝑏¹_𝑛,𝜖). We choose a number 𝜖₀ so small that the following conditions are satisfied:(1)for any 0<𝜖≤𝜖₀ the set of points Γ¹_𝑛,𝜖 is a unique non-closed arc for each 𝑛=1,...,𝑁₁, (2)the points belonging to Γ⧵Γ¹_𝑛 are placed outside the discs 𝑆(𝑎¹_𝑛,𝜖₀), 𝑆(𝑏¹_𝑛,𝜖₀) for any 𝑛=1,...,𝑁₁,(3)discs of radius 𝜖₀ with centers in different ends of Γ¹ do not intersect.
Set Γ^1,𝜖=∪^𝑁1_𝑛=1Γ¹_𝑛,𝜖,  𝑆_𝜖=(∪^𝑁1_𝑛=1[𝑆(𝑎¹_𝑛,𝜖)∪𝑆(𝑏¹_𝑛,𝜖)]),  𝒟_𝜖=𝒟⧵Γ^1,𝜖⧵𝑆_𝜖. If 𝒟 is an exterior domain, then we set 𝒟_𝜖,𝑅=𝒟_𝜖∩𝑆^𝑅, where 𝑆^𝑅 is a disc with a center in the origin and with sufficiently large radius 𝑅.
Since Γ²∈𝐶^2,𝜆,  𝑢⁰(𝑥)∈𝐶⁰(𝒟⧵Γ¹) (remind that 𝑢⁰(𝑥)∈𝐊₁), and since 𝑢⁰|_Γ²=0∈𝐶^2,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(𝒟_𝜖∪Γ²∪(Γ^1,𝜖)⁺∪(Γ^1,𝜖)⁻∪𝜕𝑆_𝜖). 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 𝒟_𝜖 is piecewise smooth, we write out Green's formula [9, page 328] for the function 𝑢⁰(𝑥): ∇𝑢⁰²_{𝐿2(𝒟𝜖)}=_Γ1,𝜖𝑢⁰⁺𝜕𝑢⁰𝜕𝐧_𝑥⁺𝑑𝑠−_Γ1,𝜖𝑢⁰⁻𝜕𝑢⁰𝜕𝐧_𝑥⁻𝑑𝑠−_Γ2𝑢⁰𝜕𝑢⁰𝜕𝐧_𝑥𝑑𝑠+_𝜕𝑆𝜖𝑢⁰𝜕𝑢⁰𝜕𝐧_𝑥𝑑𝑙. (2.10) By 𝐧_𝑥 the exterior (with respect to 𝒟_𝜖) normal on 𝜕𝑆_𝜖 at the point 𝑥∈𝜕𝑆_𝜖 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)^±|_Γ^1,𝜖=0 for any 𝜖∈(0,𝜖₀]. Therefore, identity (2.10) takes the form ∇𝑢⁰²_{𝐿2𝒟𝜖}=_𝜕𝑆𝜖𝑢⁰𝜕𝑢⁰𝜕𝐧_𝑥𝑑𝑙,𝜖∈0,𝜖₀. (2.11) Setting 𝜖→+0 in (2.11), taking into account that 𝑢⁰(𝑥)∈𝐊₁ and using the relationships (2.6), (2.5) we obtain ∇𝑢⁰²_{𝐿2𝒟⧵Γ1}=lim_𝜖→+0∇𝑢⁰²_{𝐿2(𝒟𝜖)}=0. (2.12) From the homogeneous boundary conditions (2.8) we conclude that 𝑢⁰(𝑥)≡0 in 𝒟⧵Γ¹, where 𝒟 is an interior domain.
Let 𝒟 be an exterior domain. Since the boundary of a domain 𝒟_𝜖,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 𝒟_𝜖,𝑅 for a harmonic function 𝑢⁰(𝑥) [9, c.328]: ∇𝑢⁰²_{𝐿2𝒟𝜖,𝑅}=_Γ1,𝜖𝑢⁰⁺𝜕𝑢⁰𝜕𝐧_𝑥⁺𝑑𝑠−_Γ1,𝜖𝑢⁰⁻𝜕𝑢⁰𝜕𝐧_𝑥⁻𝑑𝑠−_Γ2𝑢⁰𝜕𝑢⁰𝜕𝐧_𝑥𝑑𝑠+_𝜕𝑆𝜖𝑢⁰𝜕𝑢⁰𝜕𝐧_𝑥𝑑𝑙+_𝜕𝑆𝑅𝑢⁰𝜕𝑢⁰𝜕|𝑥|𝑑𝑙. (2.13) By 𝐧_𝑥 on 𝜕𝑆_𝜖 we denote an outward (with respect to 𝒟_𝜖,𝑅) normal in the point 𝑥∈𝜕𝑆_𝜖. 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 𝜕𝑢⁰(𝑥)𝜕|𝑥|1=𝑂|𝑥|²,as|𝑥|→∞. (2.14) Consequently, lim_{𝑅→∞𝜕𝑆𝑅}𝑢⁰(𝑥)𝜕𝑢⁰(𝑥)𝜕|𝑥|𝑑𝑙=0, (2.15) 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 domain 𝒟, 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

Let 𝛾 be an open curve of class 𝐶^1,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 𝜏_𝑥=(cos𝛼(𝑠),sin𝛼(𝑠)), while the normal vector to 𝛾 in the point 𝑥(𝑠) will be denoted by 𝐧_𝑥=(sin𝛼(𝑠),−cos𝛼(𝑠)). According to the chosen parametrization cos𝛼(𝑠)=𝑥^′₁(𝑠),  sin𝛼(𝑠)=𝑥^′₂(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 𝑋_𝛾=𝑥(𝑎)∪𝑥(𝑏) be a set of the ends of 𝛾.

Set 𝜇(𝑠)∈𝐶^0,𝜆[𝑎,𝑏], and consider the double layer potential for Laplace equation in a plane 𝒲𝜇𝑥)1(=−2𝜋^𝑏_𝑎𝜇(𝜎)𝜕𝜕𝐧_𝑦ln𝑥−𝑦(𝜎)𝑑𝜎. (3.1) Set 𝑧=𝑥₁+𝑖𝑥₂, 𝑡=𝑡(𝜎)=(𝑦₁(𝜎)+𝑖𝑦₂(𝜎))∈𝛾, 𝜇(𝑡)=𝜇(𝑡(𝜎))=𝜇(𝜎). If 𝜇(𝑠)∈𝐶^0,𝜆[𝑎,𝑏], then 𝜇(𝑡)∈𝐶^0,𝜆(𝛾), since 𝜇𝑡₂−𝜇𝑡₁=𝜇𝑡𝜎₂−𝜇𝑡𝜎₁=𝜇𝜎₂−𝜇𝜎₁≤𝑐𝜎₂−𝜎₁^𝜆=𝑐|𝜎₂−𝜎₁||𝑡(𝜎₂)−𝑡(𝜎₁)|^𝜆𝑡𝜎₂−𝑡𝜎₁^𝜆=𝑐⋅𝑐^𝜆₀𝑡₂−𝑡₁^𝜆, (3.2) where 𝑐 and 𝑐₀ are constants, 𝑡₂=𝑡(𝜎₂)∈𝛾,  𝑡₁=𝑡(𝜎₁)∈𝛾. We took into account in deriving the latter inequality that 𝜎₂−𝜎₁𝑡𝜎₂−𝑡𝜎₁∈𝐶⁰([𝑎,𝑏]×[𝑎,𝑏]), (3.3) (see lemma 1 in [10]), whence 𝜎₂−𝜎₁𝑡𝜎₂−𝑡𝜎₁≤𝑐₀. (3.4)

Consider the integral of the Cauchy type with the real density 𝜇(𝑡): Φ(𝑧)=12𝜋𝑖_𝛾𝜇(𝑡)𝑑𝑡,𝑡−𝑧 (3.5) then 𝒲[𝜇](𝑥)=−ReΦ(𝑧). It follows from properties of the Cauchy type integral that if 𝜇(𝜎)∈𝐶^0,𝜆[𝑎,𝑏], then 𝒲[𝜇](𝑥)∈𝐶⁰(𝑅²⧵𝛾⧵𝑋_𝛾). 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 =𝑥cos𝜓𝑥,𝑦¹−𝑦₁𝑥−𝑦=−𝑥−𝑦^′_𝑦1=𝑥,sin𝜓𝑥,𝑦²−𝑦₂𝑥−𝑦=−𝑥−𝑦^′_𝑦2, (3.6) 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: 𝒲𝜇𝑥)=(𝑑)(±𝜇𝜓((𝑑(𝑥).2𝜋𝑥,𝑥))+Ω (3.7) 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 𝒲𝜇(𝑥)≤const. (3.8)

Now we will study properties of derivatives of the double layer potential. It follows from Cauchy-Riemann relations that 𝑑Φ(𝑧)=𝑑𝑧ReΦ^′_𝑥1−𝑖ReΦ^′_𝑥2=−𝒲^′_𝑥1+𝑖𝒲^′_𝑥2. (3.9) On the other hand, if 𝜇(𝜎)∈𝐶^1,𝜆[𝑎,𝑏], then for 𝑧∉𝛾: 𝑑Φ(𝑧)=1𝑑𝑧2𝜋𝑖_𝛾𝜇(𝑡)𝑑1𝑑𝑧1𝑡−𝑧𝑑𝑡=−2𝜋𝑖_𝛾𝜇(𝑡)𝑑1𝑑𝑡1𝑡−𝑧𝑑𝑡=−𝜇(𝑏)2𝜋𝑖𝑡(𝑏)−𝜇(𝑎)−𝑧𝑡(𝑎)−−𝑧_𝛾𝜇^′(𝑡),𝑡−𝑧𝑑𝑡 (3.10) where 𝑑𝜇(𝑡)=(𝜎)𝑑𝑡𝑑𝜇𝑑𝜎𝑑𝜎=(𝜎)𝑑𝑡𝜇′𝑡′(𝜎)=𝑒^{−𝑖𝛼(𝜎)}𝜇′(𝜎). (3.11) Since 𝛾∈𝐶^1,𝜆, then 𝑒^{−𝑖𝛼(𝜎)}∈𝐶^0,𝜆[𝑎,𝑏], so one can show that 𝜇^′(𝑡)∈𝐶^0,𝜆(𝛾) (the proof repeats the given above proof of the fact that 𝜇(𝑡)∈𝐶^0,𝜆(𝛾), if 𝜇(𝜎)∈𝐶^0,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)∈𝐶^1,𝜆[𝑎,𝑏], then ∇𝒲[𝜇](𝑥)∈𝐶⁰(𝑅²⧵𝛾⧵𝑋_𝛾), 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 𝑑Φ(𝑧)=1𝑑𝑧𝜇(𝑏)𝑒2𝜋𝑖|(𝑏^{−𝑖𝜓(𝑥,𝑥(𝑏))}−𝜇(𝑎)𝑒𝑥−𝑥)||𝑥−𝑥(𝑎)|^{−𝑖𝜓(𝑥,𝑥(𝑎))}+Ω₀(𝑧), (3.12) where Ω₀(𝑧)=12𝜋𝑖_𝛾𝜇^′(𝑡)𝑡−𝑧𝑑𝑡. (3.13) It follows from [7, § 22] that for all 𝑧∈𝑆(𝑑,𝜖) (𝑑=𝑎 or 𝑑=𝑏), such that 𝑧∉𝛾, the inequality holds Ω₀(𝑧)≤𝑐₀(𝑑1𝜇′)ln|(𝑑1𝑥−𝑥)|+1≤𝑐ln|(𝑑,𝑥−𝑥)| (3.14) 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 𝜕𝒲𝜇(𝑥)𝜕𝑥₁=1(𝑑)2𝜋∓𝜇|(𝑑((𝑑𝑥−𝑥)|sin𝜓𝑥,𝑥))+Ω₁(𝑥),𝜕𝒲𝜇(𝑥)𝜕𝑥₂=1(𝑑)2𝜋±𝜇|(𝑑((𝑑𝑥−𝑥)|cos𝜓𝑥,𝑥))+Ω₂(𝑥), (3.15) where Ω_𝑗(𝑥)≤𝑐₁(𝑑1𝜇′)ln|𝑥−𝑥(𝑑)|+1≤𝑐₂1ln|𝑥−𝑥(𝑑)|,𝑗=1,2, (3.16)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: Ω₁(𝑥)=−ReΩ₀(𝑧)𝜇=∓^′(𝑑)(𝛼(𝑑|(𝑑(𝛼(𝑑𝜓((𝑑2𝜋sin))ln𝑥−𝑥)|−cos))𝑥,𝑥))+Ω₁₀(𝑥),Ω₂(𝑥)=ImΩ₀(𝑧)(𝑑)=±𝜇′(𝛼(𝑑|(𝑑(𝛼(𝑑𝜓((𝑑2𝜋cos))ln𝑥−𝑥)|+sin))𝑥,𝑥))+Ω₂₀(𝑥), (3.17) 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.

Let 𝜇(𝜎)∈𝐶^1,𝜆[𝑎,𝑏], and let 𝐧_𝑥 be a normal in the point 𝑥∈𝜕𝑆(𝑑,𝜖), directed to the center of the circumference 𝜕𝑆(𝑑,𝜖), that is, 𝐧_𝑥=(−cos𝜓(𝑥,𝑥(𝑑)),−sin𝜓(𝑥,𝑥(𝑑))), then we obtain from (3.15) for 𝑥∉𝛾𝜕𝒲[𝜇](𝑥)𝜕𝐧_{𝑥𝜕𝑆(𝑑,𝜖)}=−Ω₁(𝑥)cos𝜓(𝑥,𝑥(𝑑))−Ω₂(𝑥)sin𝜓(𝑥,𝑥(𝑑)). (3.18) Therefore, according to (3.16): 𝜕𝒲[𝜇](𝑥)𝜕𝐧_{𝑥𝜕𝑆(𝑑,𝜖)}1≤constln|𝑥−𝑥(𝑑)|_{𝜕𝑆(𝑑,𝜖)}1=constln𝜖, (3.19) since |𝑥−𝑥(𝑑)|=𝜖 on 𝜕𝑆(𝑑,𝜖). From here we obtain that 𝜕𝒲𝜇_{𝜕𝑆(𝑑,𝜖)}(𝑥)𝜕𝐧_𝑥𝑑𝑙=₀^2𝜋𝜕𝒲𝜇(𝑥)𝜕𝐧_𝑥1𝜖𝑑𝜓≤2𝜋const𝜖ln𝜖→0, (3.20) if 𝜖→+0, so lim_{𝜖→+0𝜕𝑆(𝑑,𝜖)}𝜕𝒲𝜇(𝑥)𝜕𝐧_𝑥𝑑𝑙=0. (3.21) 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): ∇𝒲𝜇_{𝑆(𝑑,𝜖)}(𝑥)²𝑑𝑥=₀^2𝜋𝜖₀𝜇(𝑑)2𝜋𝑟²+𝜇(𝑑)𝜋𝑟∓Ω₁(𝑥)sin𝜓±Ω₂(𝑥)cos𝜓+Ω²₁(𝑥)+Ω²₂(𝑥)𝑟𝑑𝑟𝑑𝜓=𝐼₁+𝐼₂,𝐼₁=12𝜋^𝜖₀1𝑟𝜇²(𝑑)𝐼𝑑𝑟,₂=₀^2𝜋𝜖₀𝜇(𝑑)𝜋∓Ω₁(𝑥)sin𝜓±Ω₂(𝑥)cos𝜓+𝑟Ω²₁(𝑥)+Ω²₂(𝑥)𝑑𝑟𝑑𝜓. (3.22)(3.23) The integral 𝐼₂ converges according to estimates (3.16): 𝐼₂≤4𝑐₂^𝜖₀1ln𝑟𝜇(𝑑)+𝑐₂1𝜋𝑟ln𝑟𝑑𝑟≤const. (3.24) 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. Let 𝛾 be an open curve of class 𝐶^1,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)∈𝑋_𝛾 (𝑑=𝑎 or 𝑑=𝑏). (i) If 𝜇(𝑠)∈𝐶^0,𝜆[𝑎,𝑏], then 𝒲[𝜇](𝑥)∈𝐶⁰(𝑅²⧵𝛾⧵𝑋_𝛾) and for any 𝑥∈𝑆(𝑑,𝜖), such that 𝑥∉𝛾, the relationships (3.7) and (3.8) hold.(ii) If 𝜇(𝑠)∈𝐶^1,𝜆[𝑎,𝑏], then(1)∇𝒲[𝜇](𝑥)∈𝐶⁰(𝑅²⧵𝛾⧵𝑋_𝛾);(2)for any 𝑥∈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)for 𝒲[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 class 𝐶⁰(𝑅²⧵𝛾⧵𝑋_𝛾) is continuous in 𝑅²⧵𝛾, is continuously extensible to 𝛾⧵𝑋_𝛾 from the left and from the right, but limiting values of such a function on 𝛾⧵𝑋_𝛾 from the left and from the right can be different, that is, the function may have a jump on 𝛾⧵𝑋_𝛾.
Let us study smoothness of the direct value of the double layer potential on the open curve.

Lemma 3.3. Let 𝛾 be an open curve of class 𝐶^2,𝜆, 𝜆∈(0,1], and let 𝜇(𝑠)∈𝐶⁰[𝑎,𝑏]. Let 𝐼₁(𝑠)1=−2𝜋_𝛾𝜇(𝜎)𝜕ln𝑥(𝑠)−𝑦(𝜎)𝜕𝐧_𝑦𝑑𝜎 (3.25) be the direct value of the double layer potential 𝒲[𝜇](𝑥) on 𝛾. Then 𝐼₁(𝑠)∈𝐶^1,𝜆/4[𝑎,𝑏]. (3.26)

Proof. Let us prove that 𝐼₁(𝑠)∈𝐶^1,𝜆/4[𝑎,𝑏]. Taking into account that 𝐧_𝑦=(𝑦^′₂(𝜎),−𝑦^′₁(𝜎)), we find 𝜕ln𝑥(𝑠)−𝑦(𝜎)𝜕𝐧_𝑦=𝑇(𝑠,𝜎)𝑔()()=𝑥(𝑠)(𝜎𝑠,𝜎,𝑔𝑠,𝜎−𝑦)²(𝑠−𝜎)²,𝑇()=𝑥𝑠,𝜎²(𝑠)−𝑦₂(𝜎)𝑦^′₁(𝜎)−𝑥₁(𝑠)−𝑦₁(𝜎)𝑦^′₂(𝜎)(𝑠−𝜎)². (3.27) 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] 𝑥_𝑗(𝑠)−𝑥_𝑗(𝜎)=(𝑠−𝜎)𝑍¹_𝑗(𝑠,𝜎)=−𝑥^′_𝑗(𝜎)(𝜎−𝑠)+(𝜎−𝑠)²𝑍²_𝑗(𝜎,𝑠), (3.28) where 𝑍¹_𝑗(𝑠,𝜎)=¹₀𝑥^′_𝑗(𝜎+𝜉(𝑠−𝜎))𝑑𝜉∈𝐶^1,𝜆]×[,𝑍([𝑎,𝑏𝑎,𝑏])_2𝑗(𝜎,𝑠)=¹₀𝜉𝑥_𝑗^′′(𝑠+𝜉(𝜎−𝑠))𝑑𝜉∈𝐶^0,𝜆([𝑎,𝑏]×[𝑎,𝑏]). (3.29) Note that the function 𝑔()=|𝑥(𝑠)(𝜎𝑠,𝜎−𝑥)|²(𝑠−𝜎)²=𝑍¹₁(𝑠,𝜎)²+𝑍¹₂(𝑠,𝜎)²∈𝐶^1,𝜆([𝑎,𝑏]×[𝑎,𝑏]) (3.30) does not equal zero anywhere on Γ and 𝑔(𝑠,𝑠)=1; therefore, 1𝑔(𝑠,𝜎)∈𝐶¹([𝑎,𝑏]×[𝑎,𝑏]). (3.31) Further, 𝜕1𝜕𝑠𝑔()=𝜕𝑠,𝜎()𝜕𝑠𝑠−𝜎²|𝑥(𝑠)−𝑥(𝜎)|²𝑔=−^′𝑠(𝑠,𝜎)𝑔²()𝑍𝑠,𝜎=−2¹¹(𝑠,𝜎)𝑍¹₁(𝑠,𝜎)^′_𝑠+𝑍¹₂(𝑠,𝜎)𝑍¹₂(𝑠,𝜎)^′_𝑠𝑔²(𝑠,𝜎)∈𝐶^0,𝜆([𝑎,𝑏]×[𝑎,𝑏]). (3.32) Consequently, 1/𝑔(𝑠,𝜎)∈𝐶^1,𝜆([𝑎,𝑏]×[𝑎,𝑏]). Similarly, 𝑇()=𝑥𝑠,𝜎²(𝑠)−𝑥₂(𝜎)𝑥^′₁(𝜎)−𝑥₁(𝑠)−𝑥₁(𝜎)𝑥^′₂(𝜎)(𝑠−𝜎)²=𝑍²₂(𝜎,𝑠)𝑥^′₁(𝜎)−𝑍²₁(𝜎,𝑠)𝑥^′₂(𝜎)∈𝐶^0,𝜆([𝑎,𝑏]×[𝑎,𝑏]). (3.33) Consider 𝜕𝑇(𝑠,𝜎)/𝜕𝑠=𝐽₁(𝑠,𝜎)−2𝐽₂(𝑠,𝜎), where 𝐽₁()=𝑥𝑠,𝜎^′2(𝑠)𝑥^′₁(𝜎)−𝑥^′₁(𝑠)𝑥^′₂(𝜎)(𝑠−𝜎)²=𝑥^′₂(𝑠)−𝑥^′₂(𝜎)𝑥^′₁(𝜎)−𝑥^′₁(𝑠)−𝑥^′₁(𝜎)𝑥^′₂(𝜎)(𝑠−𝜎)²=1𝑥𝑠−𝜎^′1(𝜎)¹₀𝑥₂^′′(𝑠+𝜉(𝜎−𝑠))𝑑𝜉−𝑥^′₂(𝜎)¹₀𝑥₁^′′((;𝐽𝑠+𝜉𝜎−𝑠))𝑑𝜉₂()=𝑥𝑠,𝜎²(𝑠)−𝑥₂(𝜎)𝑥^′₁(𝜎)−𝑥₁(𝑠)−𝑥₁(𝜎)𝑥^′₂(𝜎)(𝑠−𝜎)³=1𝑥𝑠−𝜎^′1(𝜎)¹₀𝜉𝑥₂^′′(𝑠+𝜉(𝜎−𝑠))𝑑𝜉−𝑥^′₂(𝜎)¹₀𝜉𝑥₁^′′(𝑠+𝜉(𝜎−𝑠))𝑑𝜉. (3.34) Then 𝜕𝑇(𝑠,𝜎)=1𝜕𝑠𝑥𝑠−𝜎^′1(𝜎)¹₀(1−2𝜉)𝑥₂^′′(𝑠+𝜉(𝜎−𝑠))𝑑𝜉−𝑥^′₂(𝜎)¹₀(1−2𝜉)𝑥₁^′′((=𝐾()𝑠+𝜉𝜎−𝑠))𝑑𝜉𝑠,𝜎,𝑠−𝜎 (3.35) where 𝐾(𝑠,𝜎)∈𝐶^0,𝜆([𝑎,𝑏]×[𝑎,𝑏]) and 𝐾(𝑠,𝑠)=0. According to [7, Section   5.7], the following representation holds 𝜕𝑇(𝑠,𝜎)=𝐾𝜕𝑠^∗(𝑠,𝜎)|𝑠−𝜎|^1−𝜆/4, (3.36)𝐾^∗(𝑠,𝜎)∈𝐶^0,3𝜆/4([𝑎,𝑏]×[𝑎,𝑏]). Using properties of Hölder functions [7], we obtain the representation 𝜕𝑥(𝑠)(𝜎𝜕𝑠𝜕ln−𝑦)𝜕𝐧_𝑦=1𝑔()()𝑠,𝜎𝜕𝑇𝑠,𝜎()𝜕𝜕𝑠+𝑇𝑠,𝜎1𝜕𝑠𝑔()=𝐾𝑠,𝜎₁(𝑠,𝜎)|𝑠−𝜎|^1−𝜆/4+𝐾₂(𝑠,𝜎), (3.37) where 𝐾₁(𝑠,𝜎)∈𝐶^0,3𝜆/4([𝑎,𝑏]×[𝑎,𝑏]),  𝐾₂(𝑠,𝜎)∈𝐶^0,𝜆([𝑎,𝑏]×[𝑎,𝑏]). By formal differentiation under the integral, we find 𝑑𝐼₁(𝑠)1𝑑𝑠=−2𝜋_𝛾𝜇(𝜎)𝜕𝑥(𝑠)(𝜎𝜕𝑠𝜕ln−𝑦)𝜕𝐧_𝑦1𝑑𝜎=−2𝜋_𝛾𝜇(𝜎)𝐾₁(𝑠,𝜎)|𝑠−𝜎|^1−𝜆/41𝑑𝜎−2𝜋_𝛾𝜇(𝜎)𝐾₂(𝑠,𝜎)𝑑𝜎. (3.38) 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∈𝐶^0,𝜆/4[𝑎,𝑏]. Lemma is proved.

4. Existence of A Classical Solution

We will construct the solution to the problem 𝐃₁ in assumption that 𝐹⁺(𝑠),𝐹⁻(𝑠)∈𝐶^1,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 𝑢(𝑥)=−𝑤𝐹⁺−𝐹⁻(𝑥)+𝑣(𝑥), (4.1) where 𝑤𝐹⁺−𝐹⁻𝑥)1(=−2𝜋_Γ¹𝐹⁺(𝜎)−𝐹⁻(𝜎)𝜕𝜕𝐧_𝑦ln𝑥−𝑦(𝜎)𝑑𝜎 (4.2) 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 𝑤𝐹⁺−𝐹⁻(𝑥)_{𝑥(𝑠)∈(Γ1)}^±𝐹=∓⁺(𝑠)−𝐹⁻(𝑠)2+𝑤𝐹⁺−𝐹⁻(𝑥(𝑠)), (4.3) where 𝑤[𝐹⁺−𝐹⁻](𝑥(𝑠)) is the direct value of the potential on Γ¹.

The function 𝑣(𝑥) in (4.1) must be a solution to the following problem.

Problem D
Find a function 𝑣(𝑥)∈𝐶⁰(𝒟)∩𝐶²(𝒟⧵Γ¹), which obeys the Laplace equation (2a) in the domain 𝒟⧵Γ¹ and satisfies the boundary conditions 𝑣(𝑥)|_{𝑥(𝑠)∈Γ}¹=𝐹⁺(𝑠)+𝐹⁻(𝑠)2+𝑤𝐹⁺−𝐹⁻𝑥(𝑠(𝑠),𝑣(𝑥())=𝑓)|_{𝑥(𝑠)∈Γ}²=𝐹(𝑠)+𝑤𝐹⁺−𝐹⁻(𝑥(𝑠))=𝑓(𝑠). (4.4) If 𝑥(𝑠)∈Γ¹, then 𝑤[𝐹⁺−𝐹⁻](𝑥(𝑠)) is the direct value of the potential on Γ¹.
If 𝒟 is an exterior domain, then we add the following condition at infinity: |𝑣(𝑥)|≤const,|𝑥|=𝑥²₁+𝑥²₂→∞. (4.5)
All conditions of the problem 𝐃 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))∈𝐶^1,𝜆/4(Γ¹) (here by 𝑤[𝐹⁺−𝐹⁻](𝑥(𝑠)) we mean the direct value of the potential on Γ¹). So, 𝑓(𝑠)∈𝐶^1,𝜆/4(Γ¹) and 𝑓(𝑠)∈𝐶⁰(Γ²).
We will look for the function 𝑣(𝑥) in the smoothness class 𝐊.
We say that the function 𝑣(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 point 𝑥(𝑑)∈𝑋 for some constants 𝒞>0, 𝛿>−1 the inequality |∇𝑣|≤𝒞|𝑥−𝑥(𝑑)|^𝛿 holds, where 𝑥→𝑥(𝑑) and 𝑑=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛, 𝑛=1,…,𝑁₁.

The definition of the functional class 𝐶⁰(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 if 𝑣(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. Let Γ∈𝐶^2,𝜆/4, 𝑓(𝑠)∈𝐶^1,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.

Theorem 4.1 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 problem D 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 Theorem 4.1 hold if Γ∈𝐶^2,𝜆,  𝐹⁺(𝑠)∈𝐶^1,𝜆(Γ¹),  𝐹⁻(𝑠)∈𝐶^1,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. Let Γ∈𝐶^2,𝜆,  𝐹⁺(𝑠)∈𝐶^1,𝜆(Γ¹),  𝐹⁻(𝑠)∈𝐶^1,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 problem 𝐃₁ 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

Let 𝑢(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=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛,  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 ∇𝑤𝜇(𝑥)²≤|∇𝑢(𝑥)|²+|∇𝑣(𝑥)|²|(𝑥⋅|(𝑥|(𝑥+2∇𝑢)|∇𝑣)|≤2∇𝑢)|²+|∇𝑣(𝑥)|², (5.1) 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,…,𝑎¹_𝑁1,𝑏¹₁,…,𝑏¹_𝑁1 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 Theorem 4.2 hold and among numbers 𝑎¹₁,…,𝑎¹_𝑁1,𝑏¹₁,…,𝑏¹_𝑁1 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 𝐻¹_loc(𝒟⧵Γ¹). Here 𝑆(𝑑,𝜖) is a disc of a radius 𝜖 with the center in the point 𝑥(𝑑)∈𝑋.

By 𝐻¹_loc(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 𝐻¹_loc(𝒟⧵Γ¹), and so it is not a weak solution. We arrive to the following corollary.

Corollary 5.2. Let conditions of Theorem 5.1 hold, then a weak solution to the problem 𝐃₁ in the class of functions 𝐻¹_loc(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 problem 𝐃₁ in the class of functions 𝐻¹_loc(𝒟⧵Γ¹) does not exist.

Remark 5.4. Even if the number 𝑑, 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 Theorems 4.2 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.

Let 𝑣(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 𝑥₁=𝑥₁(𝑑)+|𝑥−𝑥(𝑑)|cos𝜓(𝑥,𝑥(𝑑)),𝑥₂=𝑥₂(𝑑)+|𝑥−𝑥(𝑑)|sin𝜓(𝑥,𝑥(𝑑)). (6.1) We assume that 𝜓(𝑥,𝑥(𝑑))∈(𝛼(𝑑),𝛼(𝑑)+2𝜋) if 𝑑=𝑎¹_𝑛, and 𝜓(𝑥,𝑥(𝑑))∈(𝛼(𝑑)−𝜋,𝛼(𝑑)+𝜋) if 𝑑=𝑏¹_𝑛. We recall that 𝛼(𝑠) is the angle between the direction of the 𝑂𝑥₁ axis and the tangent vector 𝜏_𝑥 to Γ¹ at the point 𝑥(𝑠).

Hence, 𝛼(𝑑)=𝛼(𝑎¹_𝑛+0) if 𝑑=𝑎¹_𝑛, and 𝛼(𝑑)=𝛼(𝑏¹_𝑛−0) if 𝑑=𝑏¹_𝑛.

Thus, the angle 𝜓(𝑥,𝑥(𝑑)) varies continuously in the neighbourhood of the endpoint 𝑥(𝑑), cut along Γ¹.

Let 𝜇(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 that 𝜇(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 notation 𝜇₁(𝑠)=𝜇(𝑠)|𝑠−𝑑|^1/2, and put 𝜇₁(𝑑)=𝜇₁(𝑎¹_𝑛)=𝜇₁(𝑎¹_𝑛+0) if 𝑑=𝑎¹_𝑛,and𝜇₁(𝑑)=𝜇₁(𝑏¹_𝑛)=𝜇₁(𝑏¹_𝑛−0) if 𝑑=𝑏¹_𝑛.

At first, we study the behaviour of the gradient of a solution 𝑣(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. Let 𝑣(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=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛 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.
If 𝑑=𝑎¹_𝑛, then 𝜕𝜕𝑥₁𝑣(𝑥)=𝜇₁𝑎¹_𝑛2𝑥−𝑥(𝑎¹_𝑛)^1/2𝜓𝑎sin𝑥,𝑥¹_𝑛+𝛼𝑎¹_𝑛2(1),𝜕+𝑂𝜕𝑥₂𝑣(𝑥)𝜇=−¹𝑎¹_𝑛2𝑥−𝑥(𝑎¹_𝑛)^1/2𝜓𝑎cos𝑥,𝑥¹_𝑛+𝛼𝑎¹_𝑛2+𝑂(1), (6.2)
If 𝑑=𝑏¹_𝑛, then 𝜕𝜕𝑥₁𝑣(𝑥)𝜇=−¹𝑏¹_𝑛2𝑥−𝑥(𝑏¹_𝑛)^1/2𝜓𝑏cos𝑥,𝑥¹_𝑛+𝛼𝑏¹_𝑛2(1),𝜕+𝑂𝜕𝑥₂𝑣(𝑥)𝜇=−¹𝑏¹_𝑛2𝑥−𝑥(𝑏¹_𝑛)^1/2𝜓𝑏sin𝑥,𝑥¹_𝑛+𝛼𝑏¹_𝑛2+𝑂(1). (6.3) 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 problem 𝐃 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 Theorem 3.1, 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. Let 𝑢(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=𝑎¹_𝑛 or 𝑑=𝑏¹_𝑛 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.
If 𝑑=𝑎¹_𝑛, then 𝜕𝜕𝑥₁𝑢(𝑥)=1𝐹2𝜋⁺𝑎¹_𝑛−𝐹⁻𝑎¹_𝑛𝑥−𝑥𝑎¹_𝑛sin𝜓𝑥,𝑥𝑎¹_𝑛+𝜇₁𝑎¹_𝑛2𝑥−𝑥(𝑎¹_𝑛)^1/2𝜓𝑎sin𝑥,𝑥¹_𝑛+𝛼𝑎¹_𝑛2+𝐹′⁺𝑎¹_𝑛−𝐹′⁻𝑎¹_𝑛𝑎2𝜋sin𝛼¹_𝑛ln𝑥−𝑥𝑎¹_𝑛−𝜓𝑥,𝑥𝑎¹_𝑛cos𝛼𝑎¹_𝑛(1),𝜕+𝑂𝜕𝑥₂𝑢(𝑥)1=−𝐹2𝜋⁺𝑎¹_𝑛−𝐹⁻𝑎¹_𝑛𝑥−𝑥𝑎¹_𝑛cos𝜓𝑥,𝑥𝑎¹_𝑛−𝜇₁𝑎¹_𝑛2𝑥−𝑥(𝑎¹_𝑛)^1/2𝜓𝑎cos𝑥,𝑥¹_𝑛+𝛼𝑎¹_𝑛2−𝐹^′+𝑎¹_𝑛−𝐹^′−𝑎¹_𝑛𝑎2𝜋cos𝛼¹_𝑛ln𝑥−𝑥𝑎¹_𝑛+𝜓𝑥,𝑥𝑎¹_𝑛sin𝛼𝑎¹_𝑛+𝑂(1). (6.4)
If 𝑑=𝑏¹_𝑛, then 𝜕𝜕𝑥₁𝑢(𝑥)1=−𝐹2𝜋⁺𝑏¹_𝑛−𝐹⁻𝑏¹_𝑛𝑥−𝑥𝑏¹_𝑛sin𝜓𝑥,𝑥𝑏¹_𝑛−𝜇₁𝑏¹_𝑛2𝑥−𝑥(𝑏¹_𝑛)^1/2𝜓𝑏cos𝑥,𝑥¹_𝑛+𝛼𝑏¹_𝑛2−𝐹′⁺𝑏¹_𝑛−𝐹′⁻𝑏¹_𝑛𝑏2𝜋sin𝛼¹_𝑛ln𝑥−𝑥𝑏¹_𝑛−𝜓𝑥,𝑥𝑏¹_𝑛cos𝛼𝑏¹_𝑛(1),𝜕+𝑂𝜕𝑥₂𝑢(𝑥)=1𝐹2𝜋⁺𝑏¹_𝑛−𝐹⁻𝑏¹_𝑛|(𝑑𝑏𝑥−𝑥)|cos𝜓𝑥,𝑥¹_𝑛−𝜇₁𝑏¹_𝑛2𝑥−𝑥(𝑏¹_𝑛)^1/2𝜓𝑏sin𝑥,𝑥¹_𝑛+𝛼𝑏¹_𝑛2+𝐹′⁺𝑏¹_𝑛−𝐹′⁻𝑏¹_𝑛𝑏2𝜋cos𝛼¹_𝑛ln𝑥−𝑥𝑏¹_𝑛+𝜓𝑥,𝑥𝑏¹_𝑛sin𝛼𝑏¹_𝑛+𝑂(1). (6.5) 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 notation 𝐹′^±(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 𝑂1|(𝑑1𝑥−𝑥)|+𝑂|𝑥−𝑥(𝑑)|^1/21+𝑂ln|(𝑑.𝑥−𝑥)| (6.6) 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.