Heat distribution in a plane with a crack with a variable coefficient of thermal conductivity

Abstract

We prove the existence of a solution to a problem modeling the stationary heat distribution in an inhomogeneous plane with a crack and derive explicit representations of singular terms of an asymptotic expansion of the heat flow in the vicinity of the crack tips.

Keywords

crack heat flow Macdonald function singularity asymptotics

1. Introduction

The aim of this paper is to study an asymptotic solution to the thermal problem for a non-homogeneous material with a crack. The study of asymptotic properties of solutions to boundary-value problems and initially boundary-value problems of mathematical physics is the subject of many researches. In the case when it is impossible to construct an explicit solution to the problem under study, methods based on a priori estimates of the solutions in Sobolev type spaces are invoked, see [6,9,10]. Of special interest are the problems where the asymptotics on smoothness are investigated, i.e. the discontinuous or singular parts of the solution are extracted. This can happen in the case of a collapse of intrusion spots in fluids when the behavior of the jump of density of a fluid is studied (see [6,7,10]). Many investigations of non-homogeneous materials with cracks, e.g., thermal, elastic and thermo-elastic boundary-value problems for non-homogeneous materials, are also related to the abovementioned class of problems.

We consider here the thermal problem for a functionally graded material (FGM), the thermal conductivity of this material being a continuous function of one variable. FGMs are a new class of composites which consist of two or more materials with a continually varying composition in a spatial direction and, accordingly, with continually varying properties in this direction. FGMs are widely used in different engineering applications and numerous papers are devoted to problems of their modeling and analysis, see reviews [2,18]. Finite element methods, boundary integral analysis and their different modifications are applied for this purpose [5,16,17,20]. Boundary integral methods have the advantage that only boundary discretization is needed, in comparison with volume discretization in finite element methods. However, they are based on knowing the fundamental solutions of the corresponding partial differential equations with variable coefficients and have therefore limitations. One of the fundamental solutions is the Green’s function which is constructed only for simplest cases. For example, the Green’s function for a FGM with exponentially varying properties has been obtained in closed form for heat conduction problems and elasticity problems [3,14].

Asymptotic fields ahead of a crack tip in homogeneous materials are well known and extensively used in fracture mechanics. In particular, a mathematical analysis of heat conduction in a homogeneous media with lines of discontinuities showed that heat fluxes possess an inverse square-root singularity in terms of the radial distance from the point of the line of discontinuity [19].

Many researchers have investigated the asymptotic of fields (stresses and heat fluxes) near crack tips in FGMs, see [1,4,11,12]. In [12] the stress singularity of a two-dimensional non-homogeneous material with a crack under thermal loading was discussed and it was concluded that the near-tip field singularity and the angular distribution of stresses around the crack tip in non-homogeneous materials are identical to those in homogeneous materials when the variation modulus is expressed as a continuous and piecewise differentiable function of the spatial position. The problem of an arbitrarily-oriented partially-insulated crack in an infinite FGM plane, subjected to uniform heat flux along the direction of material gradient, was solved in [4] by using Fourier transforms and singular integral equations methods. The square-root singularity for near-tip heat fluxes was used for determining the heat flux intensity factor at crack tips. It was shown that the inhomogeneity in the thermal conductivity around a crack tip gives rise to a higher temperature gradient, and that the heat flux intensity factor is strongly influenced by the crack orientation as well as by material inhomogeneities. In [1] the extent of dominance of asymptotic crack tip fields (K-dominant regions) in elastic FGMs was evaluated by comparing the stress field calculated by finite element analysis to that calculated by asymptotic equations. It was noted that it is important for purposes of experimental measurements to identify the extent of such K-regions in the body.

In this paper we consider the problem of modeling the stationary temperature distribution in the plane with a crack with a variable coefficient of internal thermal conductivity. The plane is under a heat flux normal to the crack line. In contrast to the previous paper [8] where the similar problem was considered for FGM with the thermal conductivity coefficient given by $e^{\hat{k} x_{2} + {\hat{k}}_{1}}$ ( $\hat{k} \equiv const$ , ${\hat{k}}_{1} \equiv const$ ), in the present work the thermal conductivity coefficient is given by a more general exponential function, i.e. $G (x_{2}) = e^{k (x_{2})}$ . Here k is a continuous function (not necessarily a linear function, $k x_{2} + k_{1}$ , $k \equiv const \neq 0$ , $k_{1} \equiv const$ ). As an example of this material gradation can serve the problem in [13] where the effect of the material inhomogeneity on the stress concentration factor due to a circular hole in functionally graded panels, is numerically investigated.

The paper is organized as follows. Section 2 contains the general statement of the problem. Section 3 is devoted to auxiliary statements and description of the basic method to obtain a solution. In Section 4 we consider the original problem and a sketch of its research. The proof of the existence of a solution and the construction of the asymptotics of the heat flow in the vicinity of the crack tips are the subject of Section 5. Some final conclusions are given in Section 6.

2. Statement of the problem

We consider the partial differential equation $\begin{matrix} (1) & \frac{\partial^{2} U (x_{1}, x_{2})}{\partial x_{1}^{2}} + \frac{\partial^{2} U (x_{1}, x_{2})}{\partial x_{2}^{2}} + k^{'} (x_{2}) \frac{\partial U (x_{1}, x_{2})}{\partial x_{2}} = 0, x = (x_{1}, x_{2}) \in R^{2} ∖ l, \end{matrix}$ where $l = [- 1; 1] \times {0}$ , supplemented by the boundary conditions $\begin{array}{l} (2) & U (x_{1}, + 0) - U (x_{1}, - 0) = e^{- \frac{k (0)}{2}} q_{0} (x_{1}), \\ (3) & \frac{\partial U (x_{1}, + 0)}{\partial x_{2}} + \frac{k^{'} (0)}{2} U (x_{1}, + 0) - \frac{\partial U (x_{1}, - 0)}{\partial x_{2}} - \frac{k^{'} (0)}{2} U (x_{1}, - 0) = e^{- \frac{k (0)}{2}} q_{1} (x_{1}), \end{array}$ where $x_{1} \in (- 1; 1)$ and $q_{0} (x_{1})$ and $q_{1} (x_{1})$ are some given functions.

Definition 1.
A function $U = U (x_{1}, x_{2})$ is called solution to problem (1)–(3) if it satisfies the following three conditions:
U belongs to $C^{2} (R^{2} ∖ l)$ and satisfies equation (1) in the domain $R^{2} ∖ l$ .

The boundary condition (2) is fulfilled by U by continuity for $x_{1} \in (- 1; 1)$ , i.e., if U is continuous for $x_{2}$ at the point $\pm 0$ from right and left sides of $x_{1}$ and ${lim}_{ξ \to + 0} (U (x_{1}, ξ) - U (x_{1}, - ξ)) = e^{- \frac{k (0)}{2}} q_{0} (x_{1})$ .

The boundary condition (3) is fulfilled by U in the sense of the principal value at a point $x_{1} \in (- 1; 1)$ , i.e., the following limit exists: $\begin{matrix} lim_{ξ \to + 0} (\frac{\partial U (x_{1}, + ξ)}{\partial x_{2}} + \frac{k^{'} (0)}{2} U (x_{1}, + ξ) - \frac{\partial U (x_{1}, - ξ)}{\partial x_{2}} - \frac{k^{'} (0)}{2} U (x_{1}, - ξ)) = e^{- \frac{k (0)}{2}} q_{1} (x_{1}) . \end{matrix}$

The functions $U (x_{1}, x_{2})$ , $\sqrt{{(1 \pm x_{1})}^{2} + x_{2}^{2}} \frac{\partial U (x_{1}, x_{2})}{\partial x_{2}}$ , $\frac{\partial U (x_{1}, x_{2})}{\partial x_{2}} - \frac{\partial U (x_{1}, - x_{2})}{\partial x_{2}}$ and $(1 \pm x_{1}) \frac{\partial U (x_{1}, x_{2})}{\partial x_{1}}$ are bounded in the vicinity of the crack l.

The segment $l = [- 1; 1] \times {0}$ models the crack, conditions (2) and (3) describe the temperature jump and the normal heat flow on the cracks surfaces, respectively. The necessity of such boundary conditions for this problem was discussed in [8].

The aim of our work is to prove the existence of solutions to problem (1)–(3) and give the construction of exact representations of the singular terms of the asymptotic expansions of the solution U and of its derivatives $\frac{\partial U}{\partial x_{i}}$ , $i = 1, 2$ in the vicinity of the crack tips, i.e. the asymptotic expansions at the points $(\pm 1, 0) \in R^{2}$ .
3. Auxiliary statements

The method we use here to obtain the solution to problem (1)–(3) is based on the relationship between this solution and the solution to the auxiliary problem with constant coefficient that will be described below.

The auxiliary problem consists of the equation $\begin{matrix} (4) & \frac{\partial^{2} \tilde{u} (x_{1}, x_{2})}{\partial x_{1}^{2}} + \frac{\partial^{2} \tilde{u} (x_{1}, x_{2})}{\partial x_{2}^{2}} + \hat{k} \frac{\partial \tilde{u} (x_{1}, x_{2})}{\partial x_{2}} = 0, x = (x_{1}, x_{2}) \in R^{2} ∖ l, \end{matrix}$ supplemented by the boundary conditions $\begin{array}{l} (5) & \tilde{u} (x_{1}, + 0) - \tilde{u} (x_{1}, - 0) = q_{0} (x_{1}), \\ (6) & \frac{\partial \tilde{u} (x_{1}, + 0)}{\partial x_{2}} + \frac{\hat{k}}{2} \tilde{u} (x_{1}, + 0) - \frac{\partial \tilde{u} (x_{1}, - 0)}{\partial x_{2}} - \frac{\hat{k}}{2} \tilde{u} (x_{1}, - 0) = q_{1} (x_{1}), \end{array}$ where $x_{1} \in (- 1; 1)$ .

Problem (4)–(6) models the stationary temperature distribution in an inhomogeneous plane with a crack, the heterogeneity of the material being described by the function $k (x_{2}) = G_{0} e^{\hat{k} x_{2}}$ , where $G_{0} \equiv const \neq 0$ , $\hat{k} \equiv const \neq 0$ . This problem has been studied in [8]. Since the results of this paper develop and make use of those from [8], we present a brief sketch of the study of problem (4)–(6).

By setting in (4) $\tilde{u} (x_{1}, x_{2}) = e^{- \frac{\hat{k} x_{2}}{2}} \tilde{V} (x_{1}, x_{2})$ , the equation is reduced to the following one: $\begin{matrix} (7) & Δ \tilde{V} (x_{1}, x_{2}) - \frac{{\hat{k}}^{2}}{4} \tilde{V} (x_{1}, x_{2}) = 0, x \in R^{2} ∖ l . \end{matrix}$

Boundary conditions (5), (6) take the form $\begin{array}{l} (8) & \tilde{V} (x_{1}, + 0) - \tilde{V} (x_{1}, - 0) = q_{0} (x_{1}), \\ (9) & \frac{\partial \tilde{V} (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial \tilde{V} (x_{1}, - 0)}{\partial x_{2}} = q_{1} (x_{1}), \end{array}$ where $x_{1} \in (- 1; 1)$ .

Definition 2.
Let the function q belong to the space $C ([- 1; 1])$ . By $q (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})$ we will denote the generalized function from $D^{'} (R^{2})$ , operating according to the following rule: for any function φ, belonging to the space $D (R^{2})$ , $\begin{matrix} (q (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2}), φ (x_{1}, x_{2})) = \int_{- 1}^{1} q (σ_{1}) φ (σ_{1}, 0) d σ_{1} . \end{matrix}$

Note that the generalized function $q (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})$ is finite and $supp q (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2}) \subset [- 1; 1]$ .

If $q_{0}$ and $q_{1}$ belong to $C ([- 1; 1])$ , then problem (7)–(9) may be reduced to the following generalized problem in $D^{'} (R^{2})$ (see [21]): $\begin{matrix} (10) & Δ \tilde{V} (x_{1}, x_{2}) - \frac{{\hat{k}}^{2}}{4} \tilde{V} (x_{1}, x_{2}) = q_{1} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2}) + \frac{\partial}{\partial x_{2}} (q_{0} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})) . \end{matrix}$
Remark 1.
The fundamental solution to the operator $Δ - {\frac{\hat{k}}{4}}^{2}$ in $R^{2}$ is the function $E (x_{1}, x_{2}) = - \frac{1}{2 π} K_{0} (\frac{| \hat{k} |}{2} | x |)$ , where $K_{0} (z)$ is the Macdonald function (see [22]).

We will give several statements related to the problem (7)–(9), most of which are contained in [8] (however some results have been improved compared with [8]). Statement 1.
The solution to the generalized problem ( 10 ) exists in $D^{'} (R^{2})$ and the following representation holds: $\begin{matrix} \tilde{V} (x_{1}, x_{2}) = {\tilde{V}}_{1} (x_{1}, x_{2}) + {\tilde{V}}_{0} (x_{1}, x_{2}), \end{matrix}$ where ${\tilde{V}}_{1} (x_{1}, x_{2}) = E (x_{1}, x_{2}) * q_{1} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})$ is a stationary surface heat potential of simple layer, and $\begin{matrix} {\tilde{V}}_{0} (x_{1}, x_{2}) = E (x_{1}, x_{2}) * \frac{\partial}{\partial x_{2}} (q_{0} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})) \end{matrix}$ is a stationary surface heat potential of double layer.
Proof.
The representation $\tilde{V}$ follows directly from the convolution theorem with a finite functional, and from the properties of the convolution of generalized functions (see [21]). □
Statement 2.
Let the functions $q_{1}$ and $q_{0}$ belong to the space $C^{2} ([- 1; 1])$ . Then the following assertions hold:

(i) Let us consider a stationary simple layer potential of the form $\begin{matrix} (11) & {\tilde{V}}_{1} (x_{1}, x_{2}) = - \frac{1}{2 π} \int_{- 1}^{1} K_{0} (\frac{| \hat{k} |}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{1} (σ_{1}) d σ_{1} . \end{matrix}$ Then at $x_{1} \in (- 1; 1)$ by continuity (see Definition 2 (ii)) the following conditions are fulfilled: $\begin{array}{l} (12) & \frac{\partial {\tilde{V}}_{1} (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial {\tilde{V}}_{1} (x_{1}, - 0)}{\partial x_{2}} = q_{1} (x_{1}), \\ (13) & {\tilde{V}}_{1} (x_{1}, + 0) - {\tilde{V}}_{1} (x_{1}, - 0) = 0 . \end{array}$

If we additionally demand that $q_{1} (\pm 1) = 0$ , then the boundary conditions ( 12 ) and ( 13 ) are fulfilled at the points $x_{1} = \pm 1$ by continuity.

(ii) Let us consider a stationary heat potential of double layer in the form $\begin{matrix} (14) & {\tilde{V}}_{0} (x_{1}, x_{2}) = \int_{- 1}^{1} K_{1} (\frac{| \hat{k} |}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) \frac{q_{0} (σ_{1})}{\sqrt{x_{2}^{2} + {(x_{1} - σ_{1})}^{2}}} d σ_{1} . \end{matrix}$ Then at $x_{1} \in (- 1; 1)$ by continuity the following conditions are fulfilled: $\begin{array}{l} (15) & {\tilde{V}}_{0} (x_{1}, + 0) - {\tilde{V}}_{0} (x_{1}, - 0) = q_{0} (x_{1}), \\ (16) & \frac{\partial {\tilde{V}}_{0} (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial {\tilde{V}}_{0} (x_{1}, - 0)}{\partial x_{2}} = 0 . \end{array}$

If we additionally ask that $q_{0} (\pm 1) = 0$ , then condition ( 16 ) is fulfilled at the points $x_{1} = \pm 1$ in the sense of a principal value, and condition ( 15 ) is fulfilled by continuity. For the stronger requirement $q_{0}^{'} (\pm 1) = 0$ , conditions ( 15 ) and ( 16 ) are fulfilled at the points $x_{1} = \pm 1$ by continuity.

(iii) The solution $\tilde{V}$ to the problem ( 7 )–( 9 ) has the form $\begin{array}{l} \tilde{V} (x_{1}, x_{2}) = & \frac{| \hat{k} | x_{2}}{4 π} \int_{- 1}^{1} K_{1} (\frac{| \hat{k} |}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) \frac{q_{0} (σ_{1})}{\sqrt{x_{2}^{2} + {(x_{1} - σ_{1})}^{2}}} d σ_{1} \\ (17) & - \frac{1}{2 π} \int_{- 1}^{1} K_{0} (\frac{| \hat{k} |}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{1} (σ_{1}) d σ_{1} \end{array}$ and belongs to $C^{\infty} (R^{2} ∖ l)$ .
Proof.
We show first that for the function ${\tilde{V}}_{0}$ the representation (14) holds.

Let $ε > 0$ . Let us introduce the function $ξ (x) \in C^{\infty} (R)$ such that $\begin{matrix} ξ (x) = \{\begin{matrix} 1, & if x ⩾ - ε, \\ 0, & if x ⩽ - 2 ε . \end{matrix} \end{matrix}$

Apply the functional ${\tilde{V}}_{0}$ to an arbitrary basic function φ, belonging to the space $D (R^{2})$ . Using the definition and the properties of the convolution of the generalized functions, we obtain the chain of equalities $\begin{array}{l} ({\tilde{V}}_{0} (x_{1}, x_{2}), φ (x_{1}, x_{2})) \\ = (E (x_{1}, x_{2}) * \frac{\partial}{\partial x_{2}} (q_{0} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})), φ (x_{1}, x_{2})) \\ = (\frac{\partial}{\partial x_{2}} (q_{0} (x_{1}) δ_{[- 1; 1]} (x_{1}, x_{2})) E (x_{1}, x_{2}), ξ (x_{2}) ξ (- x_{2}) ξ (1 - x_{1}) ξ (1 + x_{1}) φ (x_{1} + y_{1}, x_{2} + y_{2})) \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} K_{0} (\frac{\hat{k}}{2} \sqrt{y_{1}^{2} + y_{2}^{2}}) \frac{\partial}{\partial y_{2}} (\int_{- 1}^{1} q_{0} (x_{1}) φ (x_{1} + y_{1}, y_{2}) d x_{1}) d y_{2} d y_{1} \\ = - \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{\partial K_{0} (\frac{\hat{k}}{2} \sqrt{y_{1}^{2} + y_{2}^{2}})}{\partial y_{2}} (\int_{- 1}^{1} q_{0} (x_{1}) φ (x_{1} + y_{1}, y_{2}) d x_{1}) d y_{2} d y_{1} . \end{array}$

Using the change $x_{1} + y_{1} = η_{1}$ and applying the formulas for the derivatives of the Macdonald functions (see [23]), we obtain $\begin{array}{l} ({\tilde{V}}_{0} (x_{1}, x_{2}), φ (x_{1}, x_{2})) \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} K_{1} (\frac{\hat{k}}{2} \sqrt{y_{1}^{2} + y_{2}^{2}}) \frac{\hat{k} y_{2}}{2 \sqrt{y_{2}^{2} + η_{1}^{2}}} (\int_{- 1 + y_{1}}^{1 + y_{1}} q_{0} (η_{1} - y_{1}) φ (η_{1}, y_{2}) d η_{1}) d y_{2} \\ = \frac{\hat{k}}{4 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (\int_{- 1 + y_{1}}^{1 + y_{1}} K_{1} (\frac{\hat{k}}{2} \sqrt{η_{1}^{2} + y_{2}^{2}}) \frac{y_{2}}{\sqrt{y_{2}^{2} + η_{1}^{2}}} q_{0} (y_{1} - η_{1}) d η_{1}) φ (y_{1}, y_{2}) d y_{1} d y_{2} . \end{array}$ Thus, we have the integral representation $\begin{matrix} {\tilde{V}}_{0} (x_{1}, x_{2}) = \frac{\hat{k}}{4 π} \int_{- 1 + x_{1}}^{1 + x_{1}} K_{1} (\frac{\hat{k}}{2} \sqrt{η_{1}^{2} + x_{2}^{2}}) \frac{x_{2}}{\sqrt{x_{2}^{2} + η_{1}^{2}}} q_{0} (x_{1} - η_{1}) d η_{1} . \end{matrix}$

Setting $x_{1} - η_{1} = σ_{1}$ in the latter equality, we get the formula (14).

The representation formula (11) of the potential ${\tilde{V}}_{1}$ is obtained similarly.

Let us demonstrate the fulfillment of the boundary conditions (15) and (16).

Using the expansion of the Macdonald function [23], the potential ${\tilde{V}}_{0}$ is represented in the form of a sum of two functions, denoted by ${\tilde{V}}_{0}^{0}$ and ${\tilde{V}}_{0}^{1}$ , $\begin{array}{l} {\tilde{V}}_{0} (x_{1}, x_{2}) = & {\tilde{V}}_{0}^{0} (x_{1}, x_{2}) + {\tilde{V}}_{0}^{1} (x_{1}, x_{2}) \\ = & \frac{1}{2 π} \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} \\ + \frac{\hat{k}}{4 π} \int_{- 1}^{1} O (\frac{\hat{k}}{2} {({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{1 / 2}) \frac{x_{2}}{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} q_{0} (σ_{1}) d σ_{1}, \end{array}$ where the function ${\tilde{V}}_{0}^{1}$ satisfies the estimate $\begin{array}{l} | {\tilde{V}}_{0}^{1} (x_{1}, x_{2}) | & ⩽ \frac{c}{2 π} \int_{- 1}^{1} \frac{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}}{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} x_{2} q_{0} (σ_{1}) d σ_{1} ⩽ \frac{c}{2 π} max_{σ_{1} \in [- 1; 1]} | q_{0} (σ_{1}) | \int_{- 1}^{1} x_{2} d σ_{1} \to 0 \end{array}$ at $x_{2} \to \pm 0$ for each fixed $x_{1}$ .

Now let us study the integral ${\tilde{V}}_{0}^{0}$ . Let $x_{1}$ belong to the interval $(- 1; 1)$ .

Consider the auxiliary integral $\begin{matrix} I (x_{1}, x_{2}) = \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} = sgn x_{2} [arctan \frac{1 - x_{1}}{| x_{2} |} + arctan \frac{1 + x_{1}}{| x_{2} |}] \to π sgn x_{2} \end{matrix}$ at $| x_{2} | \to + 0$ , $x_{1} \in (- 1; 1)$ .

The difference of the functions ${\tilde{V}}_{0}^{0}$ and I is evaluated as $\begin{matrix} | {\tilde{V}}_{0}^{0} (x_{1}, x_{2}) - \frac{1}{2 π} I (x_{1}, x_{2}) q_{0} (x_{1}) | = \frac{1}{2 π} | \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} (q_{0} (σ_{1}) - q_{0} (x_{1})) d σ_{1} | . \end{matrix}$ We split the latter integral into two terms, denoted by $I_{1}$ and $I_{2}$ , respectively, i.e. $\begin{array}{l} \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} (q_{0} (σ_{1}) - q_{0} (x_{1})) d σ_{1} \\ = I_{1} + I_{2} \\ = \int_{(- 1; 1) \cap (x_{1} - δ; x_{1} + δ)} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} (q_{0} (σ_{1}) - q_{0} (x_{1})) d σ_{1} \\ + \int_{(- 1; 1) ∖ (x_{1} - δ; x_{1} + δ)} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} (q_{0} (σ_{1}) - q_{0} (x_{1})) d σ_{1} . \end{array}$

Note that the following estimate holds: $\begin{array}{l} | I_{2} | & ⩽ \int_{(- 1; 1) ∖ (x_{1} - δ; x_{1} + δ)} \frac{x_{2}}{δ^{2}} | q_{0} (σ_{1}) + q_{0} (x_{1}) | d σ_{1} \\ ⩽ \int_{(- 1; 1) ∖ (x_{1} - δ; x_{1} + δ)} 2 \frac{x_{2}}{δ^{2}} max_{σ_{1} \in [- 1; 1]} | q_{0} | d σ_{1} \to 0 \end{array}$ for $x_{2} \to \pm 0$ .

Using the Lagrange theorem and introducing the change of variables $σ_{1} - x_{1} = ξ$ , we obtain the following estimate for the integral $I_{1}$ : $\begin{array}{l} | I_{1} | ⩽ & max_{[- 1; 1]} | q_{0}^{'} | | x_{2} | ln (ξ + \sqrt{x_{2}^{2} + ξ^{2}}) |_{- 1 - x_{1}}^{1 - x_{1}} \\ = & max_{[- 1; 1]} | q_{0}^{'} | | x_{2} |^{1 - ε} {| x_{2} |^{ε} ln (1 - x_{1} + \sqrt{x_{2}^{2} + {(1 - x_{1})}^{2}}) \\ - | x_{2} |^{ε} ln (- 1 - x_{1} + \sqrt{x_{2}^{2} + {(1 + x_{1})}^{2}})} \to 0 for x_{2} \to \pm 0, 0 < ε < 1 . \end{array}$

Thus for $x_{1}$ in $(- 1; 1)$ the following equalities are fulfilled: $\begin{matrix} lim_{x_{2} \to + 0} {\tilde{V}}_{0} (x_{1}, x_{2}) = lim_{x_{2} \to + 0} (sgn x_{2} \frac{1}{2 π} [arctan \frac{1 - x_{1}}{| x_{2} |} + arctan \frac{1 + x_{1}}{| x_{2} |}] q_{0} (x_{1})) = \frac{q_{0} (x_{1})}{2}, \end{matrix}$ and $\begin{matrix} lim_{x_{2} \to - 0} {\tilde{V}}_{0} (x_{1}, x_{2}) = lim_{x_{2} \to - 0} (- \frac{1}{2 π} [arctan \frac{1 - x_{1}}{| x_{2} |} + arctan \frac{1 + x_{1}}{| x_{2} |}] q_{0} (x_{1})) = - \frac{q_{0} (x_{1})}{2} . \end{matrix}$

Similarly, we can show that $\begin{matrix} lim_{x_{2} \to \pm 0} {\tilde{V}}_{0} (1, x_{2}) = 0, lim_{x_{2} \to \pm 0} {\tilde{V}}_{0} (- 1, x_{2}) = 0 . \end{matrix}$

Thus, the fulfillment of the boundary condition (15) is proved.

Now we show the fulfillment of the condition (16).

Using the properties and the expansions of Macdonald functions [23], we see that the function $\frac{\partial {\tilde{V}}_{0}}{\partial x_{2}}$ takes the form $\begin{array}{l} \frac{\partial {\tilde{V}}_{0} (x_{1}, x_{2})}{\partial x_{2}} = & \frac{\hat{k}}{4 π} \int_{- 1}^{1} (- \frac{x_{2}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{\frac{3}{2}}} (\frac{1}{\frac{\hat{k}}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} \\ + O (\frac{\hat{k}}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}})) + \frac{1}{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} (\frac{2}{\hat{k} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} \\ + O (\frac{\hat{k}}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}})) - \frac{\hat{k} x_{2}^{2}}{4 ({(x_{1} - σ_{1})}^{2} + x_{2}^{2})} (ln (\frac{2}{\hat{k} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}}) \\ + O (1) + \frac{8}{{\hat{k}}^{2} ({(x_{1} - σ_{1})}^{2} + x_{2}^{2})} + O (1))) q_{0} (σ_{1}) d σ_{1} . \end{array}$

Discarding the terms which are obviously continuous at the boundary of the crack, we obtain the following expression: $\begin{matrix} \frac{\partial {\tilde{V}}_{0} (x_{1}, x_{2})}{\partial x_{2}} = \frac{1}{2 π} \int_{- 1}^{1} \frac{{(x_{1} - σ_{1})}^{2} - x_{2}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} + T (x_{1}, x_{2}), \end{matrix}$ where T is a continuous function at the boundary.

We introduce the notation $\begin{matrix} A = \frac{1}{2 π} \int_{- 1}^{1} \frac{{(x_{1} - σ_{1})}^{2} - x_{2}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} . \end{matrix}$

Let us study the integral A.

Let $x_{1} \in (- 1; 1)$ . In the integral A we make a double integration by parts and obtain $\begin{array}{l} A = & \frac{1}{2 π} \frac{x_{1} - 1}{{(x_{1} - 1)}^{2} + x_{2}^{2}} q_{0} (1) - \frac{1}{2 π} \frac{x_{1} + 1}{{(x_{1} + 1)}^{2} + x_{2}^{2}} q_{0} (- 1) + \frac{1}{4 π} ln ({(x_{1} - 1)}^{2} + x_{2}^{2}) q_{0}^{'} (1) \\ - \frac{1}{4 π} ln ({(x_{1} + 1)}^{2} + x_{2}^{2}) q_{0}^{'} (- 1) - \frac{1}{4 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0}^{″} (σ_{1}) d σ_{1} . \end{array}$

Observe that the function $ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2})$ has a unique integrable property at the point $(\pm 1, 0)$ and the other terms in the representation of the function A are continuous for $x_{1} \in (- 1; 1)$ . Consequently, the integral A is a continuous function of external variables and the condition $A (x_{1}, + 0) - A (x_{1}, - 0) = 0$ is fulfilled.

Now let $x_{1} = 1$ . Then, in a further study of the expression A two variants are possible.

(1) Consider the integral in the sense of a principal value.

If $x_{2} \to + 0$ and $x_{2} \to - 0$ at the same rate, then taking into account the parity of the integrand, we obtain the equality $\begin{matrix} \frac{1}{2 π} \int_{- 1}^{1} \frac{{(1 - σ_{1})}^{2} - {(+ 0)}^{2}}{{({(1 - σ_{1})}^{2} + {(+ 0)}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} - \frac{1}{2 π} \int_{- 1}^{1} \frac{{(1 - σ_{1})}^{2} - {(- 0)}^{2}}{{({(1 - σ_{1})}^{2} + {(- 0)}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} = 0 . \end{matrix}$

(2) Impose the additional conditions $q_{0} (\pm 1) = 0$ and $q_{0}^{'} (\pm 1) = 0$ .

By making in the integral A a double integration by parts, by virtue of these conditions, the non-integral terms are equal to zero, and the obtained integral has not jump at the boundary.

A similar proof can be carried out for $x_{1} = - 1$ .

Thus, we showed that the condition (16) is fulfilled.

The proof of the boundary conditions (12) and (13) is done in a similar way.

The explicit form (17) of the solution $\tilde{V}$ follows from the properties of the potentials and from Statement 1. Let us show that the function $\tilde{V}$ belongs to $C^{\infty} (R^{2} ∖ l)$ . Using the integral representation of Macdonald functions (see [23]), we get $\begin{array}{l} \tilde{V} (x_{1}, x_{2}) = & \frac{{\hat{k}}^{2} x_{2}}{16 π} \int_{- 1}^{1} q_{0} (σ_{1}) \int_{0}^{\infty} ξ^{- 2} e^{- ξ - \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} d ξ d σ_{1} \\ - \frac{1}{4 π} \int_{- 1}^{1} q_{1} (σ_{1}) \int_{0}^{\infty} ξ^{- 1} e^{- ξ - \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} d ξ d σ_{1} . \end{array}$

Let us introduce the notations $\begin{array}{l} L = \int_{0}^{\infty} ξ^{- 1} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} d ξ, \\ L_{1} = \int_{0}^{\infty} ξ^{- 2} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} d ξ . \end{array}$

To prove the infinite differentiability of the function $\tilde{V}$ , it is necessary to show the convergence of the integrals corresponding to derivatives $\frac{\partial^{m + s} L}{\partial x_{1}^{m} \partial x_{2}^{s}}$ , $\frac{\partial^{m + s} L_{1}}{\partial x_{1}^{m} \partial x_{2}^{s}}$ , where $m, s = 0, 1, 2, \dots$ .

Consider the expression $\begin{array}{l} \frac{\partial^{m + s} L}{\partial x_{1}^{m} \partial x_{2}^{s}} = & \frac{| \hat{k} |^{2 (s + m)} {(- 1)}^{s + m}}{8^{s + m}} \int_{0}^{\infty} ξ^{- 1 - s - m} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} {(x_{1} - σ_{1})}^{m} x_{2}^{s} d ξ \\ = & \frac{| \hat{k} |^{2 (s + m)} {(- 1)}^{s + m}}{8^{s + m}} \int_{0}^{1} ξ^{- 1 - s - m} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} {(x_{1} - σ_{1})}^{m} x_{2}^{s} d ξ \\ + \frac{| \hat{k} |^{2 (s + m)} {(- 1)}^{s + m}}{8^{s + m}} \int_{1}^{\infty} ξ^{- 1 - s - m} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} {(x_{1} - σ_{1})}^{m} x_{2}^{s} d ξ . \end{array}$

For $ξ \to \infty$ the factors $ξ^{- 1 - s - m}$ , $e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}}$ and ${(x_{1} - σ_{1})}^{m} x_{2}^{s}$ are bounded, and $e^{- ξ}$ tends to zero. Thus, the integral $\int_{1}^{\infty} ξ^{- 1 - s - m} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} {(x_{1} - σ_{1})}^{m} x_{2}^{s} d ξ$ converges.

Since $x_{1}$ , $x_{2}$ do not belong to the segment l, the expression ${(x_{1} - σ_{2})}^{2} + x_{2}^{2}$ does not vanish and consequently the factor $ξ^{- 1 - m - s} e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}}$ is bounded in the vicinity of the point $ξ = 0$ and the rest factors are also bounded. Thus, the integral $\begin{matrix} \int_{0}^{1} ξ^{- 1 - s - m} e^{- ξ} \cdot e^{- \frac{({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) {\hat{k}}^{2}}{16 ξ}} {(x_{1} - σ_{1})}^{m} x_{2}^{s} d ξ \end{matrix}$ converges.

Similarly we can prove the convergence of the integral $\frac{\partial^{m + s} L_{1}}{\partial x_{1}^{m} \partial x_{2}^{s}}$ . □
Statement 3.
The solution $\tilde{V}$ to the problem ( 7 )–( 9 ) is a bounded continuous function on $R^{2} ∖ l$ , and the functions $\frac{\partial \tilde{V}}{\partial x_{1}}$ , $\frac{\partial \tilde{V}}{\partial x_{2}}$ have the following expressions: $\begin{array}{l} (18) & \begin{matrix} \frac{\partial \tilde{V} (x_{1}, x_{2})}{\partial x_{1}} = & - \frac{q_{0} (1)}{2 π} \frac{x_{2}}{{(1 - x_{1})}^{2} + x_{2}^{2}} + \frac{q_{0} (- 1)}{2 π} \frac{x_{2}}{{(1 + x_{1})}^{2} + x_{2}^{2}} \\ - \frac{q_{1} (1)}{4 π} ln [{(1 - x_{1})}^{2} + x_{2}^{2}] + \frac{q_{1} (- 1)}{4 π} ln [{(1 + x_{1})}^{2} + x_{2}^{2}] + R_{1} (x_{1}, x_{2}), \end{matrix} \\ (19) & \begin{matrix} \frac{\partial \tilde{V} (x_{1}, x_{2})}{\partial x_{2}} = & - \frac{q_{0} (1)}{2 π} \frac{1 - x_{1}}{{(1 - x_{1})}^{2} + x_{2}^{2}} - \frac{q_{0} (- 1)}{2 π} \frac{1 + x_{1}}{{(1 + x_{1})}^{2} + x_{2}^{2}} \\ + \frac{q_{0}^{'} (1)}{4 π} ln [{(1 - x_{1})}^{2} + x_{2}^{2}] - \frac{q_{0}^{'} (- 1)}{4 π} ln [{(1 + x_{1})}^{2} + x_{2}^{2}] + R_{2} (x_{1}, x_{2}), \end{matrix} \end{array}$ where $R_{1}$ and $R_{2}$ are bounded functions on any compact set.
Proof.
Consider the function $\frac{\partial \tilde{V}}{\partial x_{2}}$ . Using the expansions for the Macdonald functions, it can be written as $\begin{array}{l} \frac{\partial \tilde{V}}{\partial x_{2}} = & I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7} + I_{8} + I_{9} + I_{10} + I_{11} + I_{12} + I_{13} + I_{14} \\ = & \frac{1}{2 π} [\frac{\hat{k}}{2} \int_{- 1}^{1} (ln \frac{4}{\hat{k}} - γ) q_{1} (σ_{1}) d σ_{1} - \frac{\hat{k}}{4} \int_{- 1}^{1} ln ({(σ_{1} - x_{1})}^{2} + x_{2}^{2}) d σ_{1} \\ + \frac{\hat{k}}{2} \int_{- 1}^{1} o (\sqrt{{(σ_{1} - x_{1})}^{2} + x_{2}^{2}}) q_{1} (σ_{1}) d σ_{1} - \frac{\hat{k}}{2} \int_{- 1}^{1} q_{0} (σ_{1}) \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} \\ - \int_{- 1}^{1} q_{0} (σ_{1}) O (x_{2}) d σ_{1} + \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{1} (σ_{1}) d σ_{1} + \frac{\hat{k}}{2} \int_{- 1}^{1} O (x_{2}) q_{1} (σ_{1}) d σ_{1} \\ - \frac{{\hat{k}}^{2}}{8} \int_{- 1}^{1} (ln \frac{4}{\hat{k}} - γ) q_{0} (σ_{1}) \frac{x_{2}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} - \frac{{\hat{k}}^{2}}{16} \int_{- 1}^{1} q_{0} (σ_{1}) ln ({(σ_{1} - x_{1})}^{2} + x_{2}^{2}) \\ \times \frac{x_{2}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} + \int_{- 1}^{1} O (\frac{x_{2}^{2}}{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}}) q_{0} (σ_{1}) d σ_{1} \\ - \int_{- 1}^{1} \frac{x_{2}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} - \frac{{\hat{k}}^{2}}{8} \int_{- 1}^{1} O (\frac{x_{2}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{0} (σ_{1}) d σ_{1} \\ + \int_{- 1}^{1} \frac{{(σ_{1} - x_{1})}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} + \frac{\hat{k}}{2} \int_{- 1}^{1} O (\frac{{(σ_{1} - x_{1})}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{0} (σ_{1}) d σ_{1}], \end{array}$ where the integrals $I_{1}$ – $I_{7}$ are evidently continuous and bounded for $x_{2} \to + 0$ , $x_{1} \in [- 1; 1]$ . Similar to the estimates in the proof of the fulfillment of the boundary conditions, we can show that the integrals $I_{8}$ , $I_{9}$ and $I_{12}$ tend to zero at $x_{2} \to + 0$ , $x_{1} \in [- 1; 1]$ .

For the integral $I_{10}$ one has the estimate $\begin{matrix} | I_{10} | ⩽ c | x_{2} | | \int_{- 1}^{1} \sqrt{\frac{x_{2}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}} q_{0} (σ_{1}) d σ_{1} | ⩽ c_{1} | x_{2} |, \end{matrix}$ so $I_{10}$ tends to zero at $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Now let us study the integral $I_{14}$ . From the explicit form of the integral is follows that $\begin{matrix} | I_{14} | ⩽ c | \int_{- 1}^{1} q_{0} (σ_{1}) \frac{{(σ_{1} - x_{1})}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} | . \end{matrix}$ In the last expression we make an integration by parts and obtain $\begin{array}{l} \int_{- 1}^{1} q_{0} (σ_{1}) \frac{{(σ_{1} - x_{1})}^{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} \\ = (σ_{1} - x_{1}) q_{0} (σ_{1}) |_{- 1}^{1} + x_{2} arctan \frac{(σ_{1} - x_{1})}{x_{2}} |_{- 1}^{1} \\ - \int_{- 1}^{1} q_{0}^{'} (σ_{1}) (σ_{1} - x_{1}) d σ_{1} - x_{2} \int_{- 1}^{1} q_{0}^{'} (σ_{1}) arctan \frac{(σ_{1} - x_{1})}{x_{2}} d σ_{1}, \end{array}$ whence, taking into account the estimate $\begin{matrix} | x_{2} | | \int_{- 1}^{1} q_{0}^{'} (σ_{1}) arctan \frac{(σ_{1} - x_{1})}{x_{2}} d σ_{1} | ⩽ c | x_{2} |, \end{matrix}$ the continuity and boundedness of the integral $I_{14}$ follow for $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Consider the integrals $I_{11} + I_{13}$ . Integrating by parts we get $\begin{array}{l} \frac{1}{2 π} \int_{- 1}^{1} \frac{{(σ_{1} - x_{1})}^{2} - x_{2}^{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} = & - \frac{1}{2 π} \frac{(σ_{1} - x_{1})}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) |_{- 1}^{1} \\ + \frac{1}{2 π} \int_{- 1}^{1} \frac{(σ_{1} - x_{1})}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0}^{'} (σ_{1}) d σ_{1} . \end{array}$ Then again by integration by parts in the last integral, the expression for $I_{11} + I_{13}$ takes the form $\begin{array}{l} \frac{1}{2 π} \int_{- 1}^{1} \frac{(σ_{1} - x_{1})}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0}^{'} (σ_{1}) d σ_{1} = & \frac{1}{4 π} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0}^{'} (σ_{1}) |_{- 1}^{1} \\ - \frac{1}{4 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0}^{″} (σ_{1}) d σ_{1} . \end{array}$ It was shown earlier that the integral of the form $\begin{matrix} \frac{1}{4 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0}^{″} (σ_{1}) d σ_{1} \end{matrix}$ is continuous and bounded at $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ , so that (19) holds.

Using the expansions of Macdonald functions, the derivative $\frac{\partial \tilde{V}}{\partial x_{1}}$ is written in the form $\begin{array}{l} \frac{\partial \tilde{V}}{\partial x_{1}} = & {\tilde{I}}_{1} + {\tilde{I}}_{2} + {\tilde{I}}_{3} + {\tilde{I}}_{4} + {\tilde{I}}_{5} + {\tilde{I}}_{6} + {\tilde{I}}_{7} + {\tilde{I}}_{8} + {\tilde{I}}_{9} \\ = & - \frac{1}{2 π} \int_{- 1}^{1} \frac{σ_{1} - x_{1}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{1} (σ_{1}) d σ_{1} + \int_{- 1}^{1} O (σ_{1} - x_{1}) q_{1} (σ_{1}) d σ_{1} \\ - \frac{{\hat{k}}^{2}}{16 π} \int_{- 1}^{1} (ln \frac{4}{\hat{k}} - γ) \frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} + \frac{{\hat{k}}^{2}}{32 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) \\ \times \frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} + \int_{- 1}^{1} O (\frac{(σ_{1} - x_{1}) x_{2}}{\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}}) q_{0} (σ_{1}) d σ_{1} \\ + \frac{1}{2 π} \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} + \int_{- 1}^{1} O (\frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{0} (σ_{1}) d σ_{1} \\ + \frac{1}{2 π} \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} + \int_{- 1}^{1} O (\frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{0} (σ_{1}) d σ_{1} . \end{array}$

Observe that the integral ${\tilde{I}}_{2}$ is continuous and bounded at $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Let us study the integrals ${\tilde{I}}_{3} + {\tilde{I}}_{7} + {\tilde{I}}_{9}$ . For this sum the following estimate holds: $\begin{matrix} | {\tilde{I}}_{3} + {\tilde{I}}_{7} + {\tilde{I}}_{9} | ⩽ c | \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} | . \end{matrix}$ In the latter expression we integrate by parts to get $\begin{array}{l} \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} = & \frac{1}{2} x_{2} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0} (σ_{1}) |_{- 1}^{1} \\ - \frac{1}{2} x_{2} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{0}^{'} (σ_{1}) d σ_{1}, \end{array}$ from which the continuity and boundedness of ${\tilde{I}}_{3} + {\tilde{I}}_{7} + {\tilde{I}}_{9}$ follows at $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Consider the integral ${\tilde{I}}_{4}$ . The following estimate holds: $\begin{array}{l} | {\tilde{I}}_{4} | & ⩽ c | x_{2}^{1 - ε} | | \int_{- 1}^{1} x_{2}^{ε} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) \frac{(σ_{1} - x_{1})}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} | \\ ⩽ c_{1} | x_{2}^{1 - ε} | | \int_{- 1}^{1} \frac{(σ_{1} - x_{1})}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} d σ_{1} | \\ = c_{1} | x_{2}^{1 - ε} | | \frac{1}{2} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) |_{- 1}^{1} | ⩽ \tilde{c} | x_{2}^{1 - 2 ε} |, \end{array}$ where $0 < ε < \frac{1}{3}$ . Consequently, ${\tilde{I}}_{4}$ is continuous and bounded for $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Consider the integral ${\tilde{I}}_{5}$ . One has $\begin{matrix} | {\tilde{I}}_{5} | ⩽ c | \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) d σ_{1} |, \end{matrix}$ from which, similar to the case of ${\tilde{I}}_{3} + {\tilde{I}}_{7} + {\tilde{I}}_{9}$ , and using the continuity of the function $\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}$ , we get that ${\tilde{I}}_{5}$ is continuous and bounded for $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ .

Consider now the integral ${\tilde{I}}_{1}$ . Integrating by parts yields $\begin{array}{l} {\tilde{I}}_{1} = & - \frac{1}{2 π} \int_{- 1}^{1} \frac{σ_{1} - x_{1}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{1} (σ_{1}) d σ_{1} \\ = & - \frac{1}{4 π} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{1} (σ_{1}) |_{- 1}^{1} + \frac{1}{4 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{1}^{'} (σ_{1}) d σ_{1} . \end{array}$ It was shown earlier that the integral of the form $\begin{matrix} \frac{1}{4 π} \int_{- 1}^{1} ln ({(x_{1} - σ_{1})}^{2} + x_{2}^{2}) q_{1}^{'} (σ_{1}) d σ_{1} \end{matrix}$ is continuous and bounded for $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ , so that ${\tilde{I}}_{1}$ has the same properties.

Consider finally ${\tilde{I}}_{6} + {\tilde{I}}_{8}$ where we integrate by parts to get $\begin{array}{l} {\tilde{I}}_{6} + {\tilde{I}}_{8} = & \frac{1}{π} \int_{- 1}^{1} \frac{(σ_{1} - x_{1}) x_{2}}{{({(x_{1} - σ_{1})}^{2} + x_{2}^{2})}^{2}} q_{0} (σ_{1}) d σ_{1} \\ = & - \frac{1}{2 π} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0} (σ_{1}) |_{- 1}^{1} \\ + \frac{1}{2 π} \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0}^{'} (σ_{1}) d σ_{1} . \end{array}$ It was shown earlier that the integral of the form $\begin{matrix} \frac{1}{2 π} \int_{- 1}^{1} \frac{x_{2}}{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}} q_{0}^{'} (σ_{1}) d σ_{1} \end{matrix}$ is continuous and bounded for $x_{2} \to + 0$ and $x_{1} \in [- 1; 1]$ , and so is ${\tilde{I}}_{6} + {\tilde{I}}_{8}$ .

Thus, we have proved that the asymptotic representations (18) and (19) hold true. □

4. Reduction procedure for problem (1)–(3)

Consider problem (1)–(3). Replacing U by $e^{- \frac{k (x_{2})}{2}} V (x_{1}, x_{2})$ , problem (1)–(3) is reduced to $\begin{array}{l} (20) & Δ V (x_{1}, x_{2}) - \frac{{\tilde{k}}^{2} (x_{2})}{4} V (x_{1}, x_{2}) = 0, x \in R^{2} ∖ l, \\ (21) & V (x_{1}, + 0) - V (x_{1}, - 0) = q_{0} (x_{1}), \\ (22) & \frac{\partial V (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial V (x_{1}, - 0)}{\partial x_{2}} = q_{1} (x_{1}), \end{array}$ where $x_{1} \in (- 1; 1)$ and ${\tilde{k}}^{2} (x_{2}) = {(k^{'} (x_{2}))}^{2} + 2 k^{″} (x_{2})$ .

Hereafter we will assume that the function k belongs to $C^{4} (R)$ ; then there exist $ε_{1}$ and $ε_{2}$ , such that $ε_{2} > {(k^{'} (x_{2}))}^{2} + 2 k^{″} (x_{2}) > ε_{1} > 0$ for $x_{2} \in R$ . We assure also that $q_{0}$ and $q_{1}$ belong to the space $C^{3} ([- 1; 1])$ .

Remark 2.
Obviously, the function $k (x_{2}) = \hat{k} x_{2}$ , where $\hat{k} = const \neq 0$ , satisfies the above condition. For such a k, problem (20)–(22) coincides with problem (7)–(9). Apart from this k, this condition can be satisfied, for example, by a function of the following form: $\begin{matrix} k (x_{2}) = \{\begin{matrix} c_{1} x_{2} + c_{2}, & x_{2} ⩽ - h, \\ F (x_{2}) + c_{1} x_{2} - F^{'} (- h) x_{2} + c_{2} - F (- h) - F^{'} (- h) h, & - h < x_{2} < h, \\ (F^{'} (h) + c_{1} - F^{'} (- h)) x_{2} + c_{2} + F (h) - F (- h) - F^{'} (- h) h - F^{'} (h) h, & x_{2} ⩾ h, \end{matrix} \end{matrix}$ where $h = const > 0$ , $c_{2} = const$ , $c_{1} = const > {max}_{x_{2} \in [- h; h]} (F^{'} (- h) - F^{'} (x_{2}))$ , $c_{1} \neq 0$ , and $\begin{matrix} F^{″} (x_{2}) = \{\begin{matrix} 0, & x_{2} ⩽ - h, \\ exp [- h^{2} {(h^{2} - x_{2}^{2})}^{- 1}], & - h < x_{2} < - h, \\ 0, & x_{2} ⩾ h . \end{matrix} \end{matrix}$

We will search the solution to problem (20)–(22) in the form of $V = u + W$ , where the function u is a solution to the problem $\begin{array}{l} (23) & Δ u (x_{1}, x_{2}) - \frac{{\tilde{k}}^{2} (0)}{4} u (x_{1}, x_{2}) = 0, x \in R^{2} ∖ l, \\ (24) & u (x_{1}, + 0) - u (x_{1}, - 0) = q_{0} (x_{1}), \\ (25) & \frac{\partial u (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial u (x_{1}, - 0)}{\partial x_{2}} = q_{1} (x_{1}), \end{array}$ for $x_{1} \in (- 1; 1)$ , and the function W is a solution to the problem $\begin{array}{l} (26) & Δ W (x_{1}, x_{2}) - \frac{{\tilde{k}}^{2} (x_{2})}{4} W (x_{1}, x_{2}) = 0, 25 ({\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)) u (x_{1}, x_{2}), x \in R^{2} ∖ l, \\ (27) & W (x_{1}, + 0) - W (x_{1}, - 0) = 0, \\ (28) & \frac{\partial W (x_{1}, + 0)}{\partial x_{2}} - \frac{\partial W (x_{1}, - 0)}{\partial x_{2}} = 0, \end{array}$ for $x_{1} \in (- 1; 1)$ .

Note that problem (23)–(25) coincides with problem (7)–(9) for $\hat{k} = \tilde{k} (0)$ .
5. Proof of the existence of problem (1)–(3), construction of the asymptotics of the heat flow in the vicinity of the crack tips

Lemma 1.
Let u be a solution to problem ( 23 )–( 25 ). Then $u \in W_{2}^{m} (Ω)$ , where $m \in N$ , and Ω is any domain from $R^{2}$ such that the distance from Ω to l is greater than a certain number $δ > 0$ .
Proof.
Let us show that $u \in W_{2}^{m} (R^{2} ∖ {\bar{B}}_{R} (0))$ for a sufficiently large R. From Statements 1–3 it follows that the solution to problem (23)–(25) is given by the formula $\begin{array}{l} u (x_{1}, x_{2}) = & - \frac{1}{2 π} \int_{- 1}^{1} K_{0} (\frac{\tilde{k} (0)}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{1} (σ_{1}) d σ_{1} \\ + \frac{\tilde{k} (0)}{4 π} \int_{- 1}^{1} K_{1} (\frac{\tilde{k} (0)}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) \frac{x_{2}}{\sqrt{x_{2}^{2} + {(x_{1} - σ_{1})}^{2}}} q_{0} (σ_{1}) d σ_{1} . \end{array}$ For a sufficiently large R and any natural numbers i and j (see [23]) we have $\begin{array}{l} \frac{\partial^{i + j} u (x_{1}, x_{2})}{\partial x_{1}^{i} \partial x_{2}^{j}} = & \sum_{n^{0}, n^{1}} \int_{- 1}^{1} {(x_{1} - σ_{1})}^{α_{n^{0}}} x_{2}^{β_{n^{0}}} K_{n^{0}} (\frac{\tilde{k} (0)}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) \\ \times P_{n^{0}} (\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{0} (σ_{1}) d σ_{1} \\ + \int_{- 1}^{1} {(x_{1} - σ_{1})}^{α_{n^{1}}} x_{2}^{β_{n^{1}}} K_{n^{1}} (\frac{\tilde{k} (0)}{2} \sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) \\ (29) & \times P_{n^{1}} (\sqrt{{(x_{1} - σ_{1})}^{2} + x_{2}^{2}}) q_{1} (σ_{1}) d σ_{1}, \end{array}$ where $α_{n^{0}}$ , $α_{n^{1}}$ , $β_{n^{0}}$ , $β_{n^{1}}$ , $n^{0}$ , $n^{1}$ are non-negative integers, $K_{n}$ are the Macdonald functions, and $P_{n}$ are rational functions.

If $σ_{1} \in [- 1; 1]$ , then the following estimates hold: $\begin{matrix} {(x_{1} - σ_{1})}^{2} + x_{2}^{2} ⩾ \{\begin{matrix} {(x_{1} + 1)}^{2} + x_{2}^{2}, & x_{1} < - 1, \\ x_{2}^{2}, & - 1 ⩽ x_{1} ⩽ 1, \\ {(x_{1} - 1)}^{2} + x_{2}^{2}, & x_{1} > 1 . \end{matrix} \end{matrix}$

For any $x > 1$ and any n, one has $| K_{n} (x) | ⩽ C \frac{e^{- x}}{\sqrt{x}}$ (see [23]).

From the two last estimates and equality (29), we get that $u \in W_{2}^{m} (R^{2} ∖ {\bar{B}}_{R} (0))$ . Since $u \in C^{\infty} (R^{2} ∖ l)$ , then $u \in W_{2}^{m} (Ω)$ . □
Remark 3.
Let u be a solution to problem (23)–(25). Then the function $f (x_{1}, x_{2}) = 0, 25 ({\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)) u (x_{1}, x_{2})$ belongs to the space $L_{2} (Ω)$ , where Ω is any bounded domain in $R^{2}$ containing l. Indeed, in Statement 2 it was shown that the function u is bounded in the domain Ω, whence the result.
Lemma 2.
Let u be a solution to problem ( 23 )–( 25 ). Then the function $f (x_{1}, x_{2}) = 0, 25 ({\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)) u (x_{1}, x_{2})$ belongs to $W_{2}^{1} (Ω)$ , where Ω is any bounded domain in $R^{2}$ containing l.
Proof.
From Remark 3 it follows that $f \in L_{2} (Ω)$ . Let us show that $\frac{\partial f}{\partial x_{1}} \in L_{2} (Ω)$ , the statement $\frac{\partial f}{\partial x_{2}} \in L_{2} (Ω)$ being proved in a similar way. From Statement 3, we have the following representation in $Ω ∖ l$ : $\begin{array}{l} \frac{\partial f (x_{1}, x_{2})}{\partial x_{1}} = & \frac{{\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)}{4} (\frac{- x_{2} q_{0} (1)}{2 π [{(1 - x_{1})}^{2} + x_{2}^{2}]} + \frac{x_{2} q_{0} (- 1)}{2 π [{(1 + x_{1})}^{2} + x_{2}^{2}]}) \\ + \frac{{\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)}{4} (- \frac{q_{1} (1) ln [{(1 - x_{1})}^{2} + x_{2}^{2}]}{4 π} + \frac{q_{1} (- 1) ln [{(1 + x_{1})}^{2} + x_{2}^{2}]}{4 π}) \\ + R (x_{1}, x_{2}), \end{array}$ where R is a bounded function in the domain Ω.

Denote by $B_{δ} (a; b)$ the sphere with a radius δ and centered at the point $(a, b) \in R^{2}$ .

Since in Ω, ${\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0) = x_{2} {({\tilde{k}}^{2} (ξ))}^{'}$ where ξ lies between 0 and $x_{2}$ , for the proof of the lemma it is sufficient to note that the function $x_{2}^{2} {({(1 - x_{1})}^{2} + x_{2}^{2})}^{- 1}$ belongs to $L_{2} (B_{δ} (1; 0))$ , the function $x_{2}^{2} {({(1 + x_{1})}^{2} + x_{2}^{2})}^{- 1}$ belongs to $L_{2} (B_{δ} (- 1; 0))$ , the function $x_{2} ln [{(1 - x_{1})}^{2} + x_{2}^{2}]$ belongs to $L_{2} (B_{δ} (1; 0))$ and the function $x_{2} ln [{(1 + x_{1})}^{2} + x_{2}^{2}]$ belongs to $L_{2} (B_{δ} (- 1; 0))$ .

Consider in $R^{2}$ the partial differential equation $\begin{matrix} (30) & Δ W (x_{1}, x_{2}) - 0, 25 {\tilde{k}}^{2} (x_{2}) W (x_{1}, x_{2}) = f (x_{1}, x_{2}), \end{matrix}$ where $f (x_{1}, x_{2}) = 0, 25 ({\tilde{k}}^{2} (x_{2}) - {\tilde{k}}^{2} (0)) u (x_{1}, x_{2})$ .

From Lemma 1 and Remark 3 it follows that f is in $L_{2} (R^{2})$ .

The solution to the generalized problem (30) in $R^{2}$ is a function $W \in W_{2}^{1} (R^{2})$ which for any $Φ \in W_{2}^{1} (R^{2})$ , satisfies the integral identity $\begin{matrix} (31) & \int_{R^{2}} (\frac{\partial W}{\partial x_{1}} \frac{\partial Φ}{\partial x_{1}} + \frac{\partial W}{\partial x_{2}} \frac{\partial Φ}{\partial x_{2}} + \frac{{\tilde{k}}^{2} (x_{2})}{4} W Φ) d x = - \int_{R^{2}} f Φ d x . \end{matrix}$

In a standard way by ${(\cdot; \cdot)}_{W_{2}^{1} (R^{2})}$ we denote a scalar product in $W_{2}^{1} (R^{2})$ , and by $(\cdot; \cdot)$ a scalar product in $L_{2} (R^{2})$ .

For a set of functions summable in $L_{2} (R^{2})$ together with their first-order derivatives, we introduce the scalar product ${(\cdot; \cdot)}_{{\tilde{W}}_{2}^{1} (R^{2})}$ , $\begin{matrix} (32) & {(W; Φ)}_{{\tilde{W}}_{2}^{1} (R^{2})} = \int_{R^{2}} (\frac{\partial W}{\partial x_{1}} \frac{\partial Φ}{\partial x_{1}} + \frac{\partial W}{\partial x_{2}} \frac{\partial Φ}{\partial x_{2}} + \frac{{\tilde{k}}^{2} (x_{2})}{4} W Φ) d x . \end{matrix}$

Note that due to the properties of the function k, the scalar product (32) is equivalent to the standard scalar product in $W_{2}^{1} (R^{2})$ . We denote the obtained space by ${\tilde{W}}_{2}^{1} (R^{2})$ and the scalar product in this space by ${(\cdot; \cdot)}_{{\tilde{W}}_{2}^{1} (R^{2})}$ . □
Theorem 1.
The generalized problem ( 30 ) in $R^{2}$ has a unique solution in $W_{2}^{1} (R^{2})$ .
Proof.
From the equivalency of the scalar products ${(\cdot; \cdot)}_{W_{2}^{1} (R^{2})}$ and ${(\cdot; \cdot)}_{{\tilde{W}}_{2}^{1} (R^{2})}$ , the Cauchy–Bunyakovsky inequality and the fact that $W_{2}^{l} (R^{2})$ forms a scale of spaces, it follows that $- (f; Φ)$ is a continuous functional in ${\tilde{W}}_{2}^{1} (R^{2})$ . Then from the Riesz theorem and the equivalence of the scalar products ${(\cdot; \cdot)}_{W_{2}^{1} (R^{2})}$ and ${(\cdot; \cdot)}_{{\tilde{W}}_{2}^{1} (R^{2})}$ , the statement of the theorem follows easily. □

Consider in $R^{2}$ a bounded domain Ω with a sufficiently smooth boundary. By $\tilde{\tilde{k}}$ , $\tilde{f}$ , $\tilde{W}$ we denote, respectively, the restriction of the functions $\frac{{\tilde{k}}^{2}}{4}$ , f and $W (x_{1}, x_{2})$ to Ω, by φ the trace of the solution to problem (30) in $R^{2}$ , i.e. the trace of the function W on $\partial Ω$ . From the trace theorem (see [15]) it follows that $φ \in L_{2} (\partial Ω)$ .

Consider in the domain Ω the auxiliary problem $\begin{array}{l} (33) & Δ \tilde{W} (x_{1}, x_{2}) - \tilde{\tilde{k}} (x_{2}) \tilde{W} (x_{1}, x_{2}) = \tilde{f} (x_{1}, x_{2}), \\ (34) & \tilde{W} (x_{1}, x_{2}) |_{\partial Ω} = φ (x_{1}, x_{2}) . \end{array}$
Definition 3.
The function $\tilde{W} (x_{1}, x_{2}) \in W_{2}^{1} (Ω)$ is called a solution to the generalized problem for problem (33)–(34), if it satisfies the integral identity $\begin{matrix} (35) & \int_{Ω} (\frac{\partial \tilde{W}}{\partial x_{1}} \frac{\partial \tilde{Φ}}{\partial x_{1}} + \frac{\partial \tilde{W}}{\partial x_{2}} \frac{\partial \tilde{Φ}}{\partial x_{2}} + \tilde{\tilde{k}} (x_{2}) \tilde{W} \tilde{Φ}) d x = - \int_{Ω} \tilde{f} \tilde{Φ} d x \end{matrix}$ for all $\tilde{Φ} \in {\overset{\circ}{W}}_{2}^{1} (Ω)$ , and if the trace of this function in $L_{2} (\partial Ω)$ coincides with φ.

The restriction of the solution to problem (30) in $R^{2}$ on the domain Ω will be a solution to the generalized problem (33)–(34) in the domain Ω.

Let $\tilde{Ω}$ be a domain in $R^{n}$ , and F a finite function in $\tilde{Ω}$ belonging to $L_{2} (\tilde{Ω})$ . We will extend it by zero outside of $\tilde{Ω}$ and consider for $h \neq 0$ the difference quotient $\begin{matrix} δ_{h}^{j} F (x) = \frac{F (x_{1}, x_{2}, \dots, x_{j - 1}, x_{j} + h, x_{j + 1}, \dots, x_{n}) - F (x_{1}, x_{2}, \dots, x_{j - 1}, x_{j}, x_{j + 1}, \dots, x_{n})}{h} . \end{matrix}$

Concerning difference quotients the following theorem holds (see [15]):
Theorem 2.
For F as above, we have the following assertions:

(1) If there exists a generalized derivative $F_{x_{j}}$ for some $j = 1, \dots, n$ , then for all sufficiently small modulus $h \neq 0$ , we have: $‖ δ_{h}^{j} F ‖_{L_{2} (\tilde{Ω})} ⩽ ‖ F_{x_{j}} ‖_{L_{2} (\tilde{Ω})}$ and $‖ δ_{h}^{j} F - F_{x_{j}} ‖_{L_{2} (\tilde{Ω})} \to 0$ as $h \to 0$ .

(2) If there exists a constant $c > 0$ such that for all sufficiently small modulus $h \neq 0$ , $‖ δ_{h}^{j} F ‖_{L_{2} (\tilde{Ω})} ⩽ c$ , then there exists a generalized derivative $F_{x_{j}}$ in $\tilde{Ω}$ of the function F and $‖ F_{x_{j}} ‖_{L_{2} (\tilde{Ω})} ⩽ c$ and $‖ δ_{h}^{j} F - F_{x_{j}} ‖_{L_{2} (\tilde{Ω})} \to 0$ for $h \to 0$ .

(3) If $g \in L_{2} (\tilde{Ω})$ , then for h smaller in absolute value than the distance between the boundary of the domain $\tilde{Ω}$ and the boundary of the domain outside which $F (x) \equiv 0$ , the following equality holds: ${(δ_{h}^{j} F; g)}_{L_{2} (\tilde{Ω})} = - {(F; δ_{- h}^{j} g)}_{L_{2} (\tilde{Ω})}$ .

Now we state and prove a theorem analogous to the corresponding statement from [15] for the case when $\tilde{\tilde{k}} (x_{2}) \equiv 0$ .

Let $δ > 0$ be sufficiently small, and $Ω_{r}$ some bounded domain in $R^{2}$ . Let $Ω_{r, δ}$ denote the points in the domain $Ω_{r}$ away from its boundary at a distance greater than δ. Theorem 3.
Let $\tilde{f} \in L_{2} (Ω) \cap W_{2, loc}^{i} (Ω)$ , $i = 0, 1, 2, \dots$ , $\tilde{\tilde{k}} \in C^{i} (Ω)$ . Let $\tilde{W} \in W_{2}^{1} (Ω)$ satisfying the integral identity ( 35 ) for all $\tilde{Φ} \in {\overset{\circ}{W}}_{2}^{1} (Ω)$ . Then $\tilde{W} \in W_{2, loc}^{i + 2} (Ω)$ and for any pair of subdomains $Ω_{1}$ and $Ω_{2}$ of the domain Ω such that $Ω_{1} \subset Ω_{2} \subset Ω$ (all embeddings of the domains are strict), the following inequality holds: $‖ \tilde{W} ‖_{W_{2}^{i + 2} (Ω_{1})} ⩽ c (‖ \tilde{f} ‖_{W_{2}^{i} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω)})$ with a positive constant $c = c (i, Ω_{1}, Ω_{2})$ .
Proof.
Denote by δ the distance between the boundaries $\partial Ω_{1}$ and $\partial Ω_{2}$ . Consider the function ξ, having the following properties: $ξ \in C^{\infty} (R^{2})$ , $ξ \equiv 1$ in $Ω_{2, δ}$ and $ξ \equiv 0$ outside of $Ω_{2, 2 δ / 3}$ . Put in (35) in place of $\tilde{Φ}$ , the function $ξ V_{0}$ , where $V_{0}$ is an arbitrary function from $W_{2}^{1} (Ω_{2})$ , continued by zero outside of $Ω_{2}$ . Since $\begin{matrix} \nabla \tilde{W} \cdot \nabla \tilde{Φ} = \nabla (ξ \tilde{W}) \cdot \nabla V_{0} + \nabla \tilde{W} \cdot \nabla ξ \cdot V_{0} - \tilde{W} \nabla ξ \cdot \nabla V_{0}, \end{matrix}$ identity (35) takes the form $\begin{array}{l} \int_{Ω_{2}} \nabla (ξ \tilde{W}) \cdot \nabla V_{0} d x + \int_{Ω_{2}} \nabla \tilde{W} \cdot \nabla ξ \cdot V_{0} d x - \int_{Ω_{2}} \tilde{W} \nabla ξ \cdot \nabla V_{0} d x + \int_{Ω_{2}} \tilde{\tilde{k}} (x_{2}) \tilde{W} ξ V_{0} d x \\ = - \int_{Ω_{2}} \tilde{f} ξ V_{0} d x, \end{array}$ where $x = (x_{1}, x_{2})$ .

We will rewrite the latter equality in the form $\begin{matrix} (36) & \int_{Ω_{2}} \nabla Θ \cdot \nabla V_{0} d x = \int_{Ω_{2}} F V_{0} d x + \int_{Ω_{2}} \tilde{W} \nabla ξ \cdot \nabla V_{0} d x, \end{matrix}$ where $Θ = ξ \tilde{W}$ and $F = - \tilde{f} ξ - \nabla \tilde{W} \cdot \nabla ξ - \tilde{\tilde{k}} (x_{2}) \tilde{W} ξ$ .

Take an arbitrary function $V_{1}$ , belonging to $W_{2}^{1} (Ω_{2})$ extended by zero outside of $Ω_{2}$ . For any $j = 1, 2$ and an arbitrary $h : 0 < | h | < \frac{δ}{2}$ consider a finite-difference quotient $δ_{- h}^{j} V_{1}$ . Take $V_{0} = δ_{- h}^{j} V_{1}$ in equation (36), we get $\begin{matrix} \int_{Ω_{2}} \nabla Θ \cdot \nabla δ_{- h}^{j} V_{1} d x = \int_{Ω_{2}} F \cdot δ_{- h}^{j} V_{1} d x + \int_{Ω_{2}} \tilde{W} \nabla ξ \cdot \nabla δ_{- h}^{j} V_{1} d x . \end{matrix}$

Using item (3) of Theorem 2, the latter equation can be written as $\begin{matrix} (37) & \int_{Ω_{2}} \nabla δ_{h}^{j} Θ \cdot \nabla V_{1} d x = - \int_{Ω_{2}} F \cdot δ_{- h}^{j} V_{1} d x + \int_{Ω_{2}} δ_{h}^{j} (\tilde{W} \nabla ξ) \cdot \nabla V_{1} d x . \end{matrix}$

Let us prove Theorem 3 for $i = 0$ .

Since $\tilde{W} \in W_{2}^{1} (Ω_{2})$ , the function ξ is bounded in $R^{2}$ together with all its derivatives, then from the form of the function F we derive the following estimate: $\begin{matrix} (38) & ‖ F ‖_{L_{2} (Ω_{2})} ⩽ c (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + | | | \nabla \tilde{W} | | |_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{L_{2} (Ω_{2})}) ⩽ c_{1} (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) . \end{matrix}$

From (37) by using the Cauchy–Bunyakovsky inequality, Theorem 2 and estimate (38), we obtain the inequality $\begin{array}{l} | \int_{Ω_{2}} \nabla δ_{h}^{j} Θ \cdot \nabla V_{1} d x | & ⩽ c (‖ F ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) | | | \nabla V_{1} | | |_{L_{2} (Ω_{2})} \\ ⩽ c_{1} (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) | | | \nabla V_{1} | | |_{L_{2} (Ω_{2})} . \end{array}$ Setting $V_{1} (x_{1}, x_{2}) = δ_{h}^{j} Θ (x_{1}, x_{2})$ in the latter inequality, we get the estimate $\begin{matrix} | | | \nabla δ_{h}^{j} Θ | {| |}_{L_{2} (Ω_{2})} ⩽ c (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) for j = 1, 2, 0 < | h | < \frac{δ}{2} . \end{matrix}$ This estimate and Theorem 2 imply the inequality $\begin{matrix} ‖ Θ ‖_{W_{2}^{2} (Ω_{2})} ⩽ c (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) . \end{matrix}$ Since in the domain $Ω_{1}$ , $Θ = \tilde{W}$ , where u is a solution to (23)–(25), we have $\begin{matrix} (39) & ‖ \tilde{W} ‖_{W_{2}^{2} (Ω_{1})} ⩽ c (‖ \tilde{f} ‖_{L_{2} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) . \end{matrix}$

We now prove the theorem for an arbitrary i.

Let $\tilde{f} \in W_{2, loc}^{i + 1} (Ω)$ . Assume that the function $\tilde{W} \in W_{2, loc}^{i + 2} (Ω)$ and such that for any pair of subdomains $Ω_{1}$ and $Ω_{2}$ of the domain Ω with $Ω_{1} \subset Ω_{2} \subset Ω$ (all embeddings of the domains are strict), one has the inequality $\begin{matrix} (40) & ‖ \tilde{W} ‖_{W_{2}^{i + 2} (Ω_{1})} ⩽ c (‖ \tilde{f} ‖_{W_{2}^{i} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}), \end{matrix}$ and moreover, that for any $α : | α | ⩽ i$ , $j = 1, 2$ and $0 < | h | < \frac{δ}{2}$ the following identity holds: $\begin{matrix} (41) & \int_{Ω_{2}} \nabla δ_{h}^{j} D^{α} Θ \cdot \nabla V_{i + 1} d x = - \int_{Ω_{2}} D^{α} F \cdot δ_{- h}^{j} V_{i + 1} d x + \int_{Ω_{2}} δ_{h}^{j} (D^{α} (\tilde{W} \nabla ξ)) \cdot \nabla V_{i + 1} d x, \end{matrix}$ where $V_{i + 1}$ is an arbitrary function from $W_{2}^{1} (Ω_{2})$ .

For $i = 0$ the validity of (40) and (41) was already established. For $α, | α | ⩽ i$ , $D^{α} Θ \in W_{2}^{2} (Ω_{2})$ and $D^{α} F \in W_{2}^{1} (Ω_{2})$ .

By Theorem 2 we can pass to the limit in (41) as $h \to 0$ . As a result, for any α, $| α | = i$ , $j = 1, 2$ we obtain $\begin{matrix} \int_{Ω_{2}} \nabla D^{α} Θ_{x_{j}} \cdot \nabla V_{i + 1} d x = - \int_{Ω_{2}} D^{α} F \cdot {(V_{i + 1})}_{x_{j}} d x + \int_{Ω_{2}} {(D^{α} (\tilde{W} \nabla ξ))}_{x_{j}} \cdot \nabla V_{i + 1} d x \end{matrix}$ and then, since $D^{α} F$ is equal to zero outside of $Ω_{2, 2 δ / 3}$ , we have $\begin{matrix} \int_{Ω_{2}} \nabla D^{α} Θ_{x_{j}} \cdot \nabla V_{i + 1} d x = \int_{Ω_{2}} D^{α} F_{x_{j}} \cdot V_{i + 1} d x + \int_{Ω_{2}} {(D^{α} (\tilde{W} \nabla ξ))}_{x_{j}} \cdot \nabla V_{i + 1} d x \end{matrix}$ for all $V_{i + 1} (x_{1}, x_{2}) \in W_{2}^{1} (Ω_{2})$ . Take in place of $V_{i + 1}$ the function $δ_{- h}^{p} V_{i + 2} (x_{1}, x_{2})$ in the last equation, where $V_{i + 2}$ is an arbitrary function from $W_{2}^{1} (Ω_{2})$ , for $p = 1, 2$ , $h : 0 < | h | < \frac{δ}{2}$ . As a result we get $\begin{array}{l} \int_{Ω_{2}} \nabla δ_{h}^{p} D^{α} Θ_{x_{j}} \cdot \nabla V_{i + 2} d x = & - \int_{Ω_{2}} D^{α} F_{x_{j}} \cdot δ_{- h}^{p} V_{i + 2} d x \\ (42) & + \int_{Ω_{2}} δ_{h}^{p} (D^{α} {(\tilde{W} \nabla ξ)}_{x_{j}}) \cdot \nabla V_{i + 2} d x . \end{array}$

The form of the function F implies the estimate $\begin{matrix} ‖ F ‖_{W_{2}^{i + 1} (Ω_{2})} ⩽ c (‖ \tilde{f} ‖_{W_{2}^{i + 1} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) . \end{matrix}$ Setting $V_{i + 2} (x_{1}, x_{2}) = δ_{h}^{p} D^{α} Θ_{x_{j}} (x_{1}, x_{2})$ in (42), we obtain the estimate $\begin{matrix} ‖ \tilde{W} ‖_{W_{2}^{i + 3} (Ω_{1})} ⩽ c (‖ \tilde{f} ‖_{W_{2}^{i + 1} (Ω_{2})} + ‖ \tilde{W} ‖_{W_{2}^{1} (Ω_{2})}) . \end{matrix}$ □
Theorem 4.
Let $k \in C^{m + 2} (R)$ , where $m = 2, \dots$ . Then problem ( 1 )–( 3 ) admits a solution U in $C^{m} (R^{2} ∖ l)$ . The functions U, $\frac{\partial U}{\partial x_{1}}$ and $\frac{\partial U}{\partial x_{2}}$ have the same asymptotic representation in the vicinity of l as the functions $u (x_{1}, x_{2}) e^{- \frac{k (x_{2})}{2}}$ , $\frac{\partial u (x_{1}, x_{2})}{\partial x_{1}} e^{- \frac{k (x_{2})}{2}}$ and $\frac{\partial u (x_{1}, x_{2})}{\partial x_{2}} e^{- \frac{k (x_{2})}{2}}$ , respectively, where u is a solution to problem ( 23 )–( 25 ).
Proof.
Let W be a solution to the generalized problem (30) in $R^{2}$ , Ω-an arbitrary bounded domain in $R^{2}$ , containing l, and $Ω_{1}$ -an arbitrary bounded domain in $R^{2}$ , such that the distance from $Ω_{1}$ to l is larger than some number. From Lemma 2 and Theorem 3 it follows that $W \in W_{2}^{3} (Ω)$ . Then the embedding $W_{2}^{r} (Ω) \subset C^{m} (Ω)$ , where $r - m > n / 2$ ( $n = 2$ is the dimension of space), yields $W \in C^{1} (Ω)$ . Similarly, by Lemma 1 and Theorem 3, we get $W \in C^{m} (Ω_{1})$ .

The solution to problem (1)–(3) can be represented in the form $U (x_{1}, x_{2}) = u (x_{1}, x_{2}) e^{- \frac{k (x_{2})}{2}} + W (x_{1}, x_{2}) e^{- \frac{k (x_{2})}{2}}$ . The statement of the theorem follows from the smoothness of the function W in the vicinity of the crack l and outside the crack l. □

6. Conclusion

We have studied the boundary-value thermal problem (1)–(3), describing the distribution of the heat in a FGM plane with a crack. The thermal conductivity coefficient of the FGM was given by the function $G (x_{2}) = e^{k (x_{2})}$ . After simple arrangements this problem was reduced to another problem (20)–(22) with the relationship $U (x_{1}, x_{2}) = e^{- \frac{k (x_{2})}{2}} V (x_{1}, x_{2})$ between the solutions of these two problems. This “reduced problem” (20)–(22) was investigated in detail. Besides, we studied the auxiliary problem (23)–(25) with constant coefficients for which an explicit representation of the solution was constructed and its asymptotic behavior near the crack tips was investigated by the procedure developed in [8]. The main result in our paper is the statement that the singular parts of the asymptotic representations of the solution to problem (20)–(22) and to problem (23)–(25) coincide. Thus, the asymptotic representation of the solution to problem (20)–(22) was constructed and, consequently, that of the solution to problem (1)–(3). The asymptotic expansions (18)–(19) are examples of such asymptotics for heat fluxes.

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