GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 3, 1999, 233-250

THE BASIC MIXED PROBLEM FOR AN ANISOTROPIC ELASTIC BODY

M. BASHELEISHVILI AND SH. ZAZASHVILI

Abstract. The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.

In [1,2], the basic plane mixed boundary value problem for isotropic elastic body was studied by reducing it to a singular integral equation with discontinuous coefficients that contains, together with an unknown function, a complex conjugate outside the characteristic part. This problem for an anisotropic elastic body was solved in [3] by means of the problem of linear conjugation with displacements both for several unknown functions with discontinuous coefficients and for systems of singular integral equations.

In [4, 5], the mixed problem for isotropic and anisotropic bodies was treated by the method of the potential theory and a system of singular integral equations. However this way of solving the problem is connected with the construction of Green’s tensor.

In [6], again applying the method of the potential theory and systems of singular integral equations, but without resorting to Green’s tensor, the mixed problem was solved for an anisotropic body.

In the present work we solve the mixed boundary value problem in a general statement for an anisotropic elastic body. Using the potential theory and some transformations, we reduce the problem to a singular integral equation similar to that constructed by D. I. Sherman in [1].

233

1991 Mathematics Subject Classification. 73C35.

Key words and phrases. Theory of elasticity, potential method, mixed problem..

1072-947X/99/0500-0233$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

§ 1. Statement of the Problem

Let an elastic anisotropic medium occupy a finite simply connected twodimensional domain D^{+} in the plane x_{1}0x_{2} bounded by a closed curve S of Hölder-continuous curvature.

By D^{−} we denote an infinite domain that complements D^{+} ∪ S to the whole plane. Assume that on the curve S we have the arcs S′_{j} = a_{j}b_{j}, j = 1, 2, …, p, which have no common ends, whose positive directions coincide with the positive direction of S, thus leaving the domain D^{+} on the left, and which are arranged one after another in that direction. The set of arcs S′_{j} will be denoted by S′ and the set of arcs S′′_{j} = b_{j}a_{j+1}, (a_{p+1} ≡ a_{1}) by S′′. The direction of the outward normal with respect to D^{+} is chosen as positive.

A system of equations of statics of an anisotropic elastic body free from body forces is written in terms of displacement vector components as

\begin{aligned}A_{11} \frac{∂^{2}u_{1}}{∂x_{1}^{2}} & + 2A_{13} \frac{∂^{2}u_{1}}{∂x_{1}∂x_{2}} + A_{33} \frac{∂^{2}u_{1}}{∂x_{2}^{2}} + \\ & + A_{13} \frac{∂^{2}u_{2}}{∂x_{1}^{2}} + (A_{12} + A_{33})\frac{∂^{2}u_{2}}{∂x_{1}∂x_{2}} + A_{23} \frac{∂^{2}u_{2}}{∂x_{2}^{2}} = 0, \\ A_{13} \frac{∂^{2}u_{1}}{∂x_{1}^{2}} & + (A_{12} + A_{33})\frac{∂^{2}u_{1}}{∂x_{1}∂x_{2}} + A_{23} \frac{∂^{2}u_{1}}{∂x_{2}^{2}} + \\ & + A_{33} \frac{∂^{2}u_{2}}{∂x_{1}^{2}} + 2A_{23} \frac{∂^{2}u_{2}}{∂x_{1}∂x_{2}} + A_{22} \frac{∂^{2}u_{2}}{∂x_{2}^{2}} = 0, \\ & x = (x_{1}, x_{2}) ∈ D^{+} ⊂ R^{2}, \end{aligned} (1.1)

where u = (u_{1}, u_{2})^{T} is the displacement vector, T denotes transposition and A_{11}, A_{12}, …, A_{33} are the elastic constants satisfying the conditions

A_{11} > 0, A_{22} > 0, A_{33} > 0, A_{11}A_{22} − A_{12}^{2} > 0,
A_{11}A_{33} − A_{13}^{2} > 0, A_{22}A_{33} − A_{23}^{2} > 0,

Δ = |\begin{matrix}A_{11} & A_{12} & A_{13} \\ A_{12} & A_{22} & A_{23} \\ A_{13} & A_{23} & A_{33} \end{matrix}| > 0,

which are obtained from the positive definiteness of potential energy.

We introduce the matrix differential operator T (∂_{x}, n) = [T_{k_{j}}(∂_{x}, n)]_{2×2}, where

T_{11}(∂_{x}, n) = n_{1}(x)(A_{11} \frac{∂}{∂x_{1}} + A_{13} \frac{∂}{∂x_{2}} ) + n_{2}(x)(A_{13} \frac{∂}{∂x_{1}} + A_{33} \frac{∂}{∂x_{2}} ),

T_{12}(∂_{x}, n) = n_{1}(x)(A_{13} \frac{∂}{∂x_{1}} + A_{12} \frac{∂}{∂x_{2}} ) + n_{2}(x)(A_{33} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ),

T_{21}(∂_{x}, n) = n_{1}(x)(A_{13} \frac{∂}{∂x_{1}} + A_{33} \frac{∂}{∂x_{2}} ) + n_{2}(x)(A_{12} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ),

T_{22}(∂_{x}, n) = n_{1}(x)(A_{33} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ) + n_{2}(x)(A_{23} \frac{∂}{∂x_{1}} + A_{22} \frac{∂}{∂x_{2}} );

n_{1}(x) and n_{2}(x) are the components of some unit vector n at the point x. Now T (∂_{x}, n)u(x) will be the stress vector acting on an element of the arc passing through x, with the normal n.

Denote by

T^{κ}(∂_{x}, n)u(x) = T (∂_{x}, n)u(x) + κ [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] \frac{∂u(x)}{∂s(x)}  (1.2)

the generalized stress vector, where κ is an arbitrary real constant, \frac{∂}{∂s(x)} = n_{1}(x)\frac{∂}{∂x_{2}} − n_{2}(x)\frac{∂}{∂x_{1}} is a derivative with respect to the tangent [8].

We can see from (1.2) that for κ = 0 the generalized stress vector coincides with the physical stress vector.

Definition 1. A vector u defined in the domain D^{+} is said to be regular if:

(a) u ∈ C(\overline{D}^{+}) ∩ C^{2}(D^{+});

(b) the vector T^{κ}(∂_{x}, n)u is continuously extendable to all points of the boundary S except possibly the points a_{j}, b_{j}, j = 1, 2, …, p, and in the neighborhood of these points it has an integrable singularity.

Definition 2. A vector u defined in the domain D^{−} is said to be regular if it satisfies conditions (a) and (b) of Definition 1 in the domain D^{−}, and for sufficiently large |x| = \sqrt{x_{1}^{2} + x_{2}^{2}}

u(x) = O(1), T^{κ}(∂_{x}, n)u(x) = O(|x|^{−2}).  (1.3)

Now we shall consider the Basic Mixed Problem: Find in the domain D^{+} a regular solution of system (1.1) by the boundary condition

{T^{κ}(∂_{t}, n)u(t)}^{+} = F (t), t ∈ S′,  (1.4)
{u(t)}^{+} = f(t), t ∈ S′′,  (1.5)

where F = (F_{1}, F_{2})^{T} and f = (f_{1}, f_{2})^{T} are the given vectors; the symbol {·}^{+} ({·}^{−}) denotes the boundary value on S from D^{+} (D^{−}) of the vector contained in the curly brackets.

Suppose: (1) f belongs to the Hölder class H on S′′; (2) f′ and F belong to the class H^{∗} at the nodes a_{j}, b_{j}, j = 1, 2, …, p (for the definition of the classes H and H^{∗} see [9]).

Note that in stating the problem the parameter κ is temporarily assumed to be arbitrary. Below it will be specified under which values of κ the problem is solvable. As mentioned above, for κ = 0 this problem was studied in [6] by means of the potential method and the existence of a unique solution was proved.

§ 2. Some Auxiliary Statements

It is known [10] that the characteristic equation of system (1.1)

a_{11}α^{4} − 2a_{13}α^{3} + (2a_{12} + a_{33})α^{2} − 2a_{23}α + a_{22} = 0,  (2.1)

where

a_{11} = Δ^{−1}(A_{22}A_{33} − A_{23}^{2}), a_{12} = Δ^{−1}(A_{13}A_{23} − A_{12}A_{33}),
a_{13} = Δ^{−1}(A_{12}A_{23} − A_{13}A_{22}), a_{22} = Δ^{−1}(A_{11}A_{33} − A_{13}^{2}),
a_{23} = Δ^{−1}(A_{12}A_{13} − A_{11}A_{23}), a_{33} = Δ^{−1}(A_{11}A_{22} − A_{12}^{2}),

admits only the complex roots α_{k} =a^{(k)}+ib^{(k)} (b^{(k)} >0), \overline{α}_{k}, k =1, 2. The matrix of fundamental solutions of system (1.1) is of the form

Γ(z, t) = Im ∑_{k=1}^{2} [\begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix}] ln σ_{k},  (2.2)

where

\begin{matrix}A_{k} = −\frac{2}{Δa_{11}} (A_{22}α_{k}^{2} + 2A_{23}α_{k} + A_{33})d_{k}, \\ B_{k} = \frac{2}{Δa_{11}} (A_{23}α_{k}^{2} + (A_{12} + A_{33})α_{k} + A_{13})d_{k}, \\ C_{k} = −\frac{2}{Δa_{11}} (A_{33}α_{k}^{2} + 2A_{13}α_{k} + A_{11})d_{k}, \\ d_{k}^{−1} = (−1)^{k}(α_{k} − \overline{α}_{1})(α_{2} − α_{1})(α_{k} − \overline{α}_{2}), k = 1, 2; \end{matrix}  (2.3)

here z = x_{1} + ix_{2} and t = y_{1} + iy_{2} are arbitrary points of the plane, and σ_{k} = z_{k} − t_{k}, z_{k} = x_{1} + α_{k}x_{2}, t_{k} = y_{1} + α_{k}y_{2}.

Note that in the sequel we shall use the notation Ψ(z) for for the function Ψ(x_{1}, x_{2}) of the variables x_{1} and x_{2}.

By equalities (2.3) the coefficients A_{k}, B_{k} and C_{k} satisfy the equation

A_{k}C_{k} − B_{k}^{2} = 0, k = 1, 2,  (2.4)

which differs from the characteristic equation (2.1) in a constant multiplier.

Note further that Γ(z, t) is a one-valued matrix whose every column is a solution of system (1.1) at every point of the plane except the point z = t at which it has a logarithmic singularity.

Introduce the notation [7]

C = 2i ∑_{k=1}^{2} d_{k}, A = 2i ∑_{k=1}^{2} α_{k}d_{k}, B = 2i ∑_{k=1}^{2} α_{k}^{2}d_{k}.  (2.5)

It can be easily verified that the real constants A, B, C satisfy the conditions

B > 0, C > 0, BC − A^{2} > 0.  (2.6)

From (2.5) we find that

\begin{matrix}d_{k} = −\frac{i}{2}(Cs_{k} − Ar_{k}), α_{k}d_{k} = −\frac{i}{2}(Ap_{k} − Cq_{k}), \\ α_{k}^{2}d_{k} = −\frac{i}{2}(Bp_{k} − Aq_{k}), Ap_{k} − Cq_{k} = As_{k} − Br_{k}, \end{matrix}  (2.7)

where

r_{k} = \frac{(−1)^{k}}{α_{1} − α_{2}} , s_{k} = \frac{(−1)^{k}α_{1}α_{2}}{α_{k}(α_{1} − α_{2})}, p_{k} = −α_{k}r_{k}, q_{k} = −α_{k}s_{k}. (2.8)

By virtue of (2.7) and (2.8) we readily obtain

[\begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix}] = \frac{2id_{k}}{BC − A^{2}} [ \begin{matrix}α_{k}^{2} & −α_{k} \\ −α_{k} & 1 \end{matrix} ] [\begin{matrix}C & A \\ A & B \end{matrix}] ,  (2.9)

∑_{k=1}^{2} [\begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix}] = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] ≡ E.  (2.10)

After applying the operator T^{κ}(∂_{t}, n) to the matrix Γ(z, t) given by (2.2) and taking into account (2.4), we have

[T^{κ}(∂_{t}, n)Γ(z, t)]^{T} = Im ∑_{k=1}^{2} [\begin{matrix}P_{k} & Λ_{k} \\ Q_{k} & S_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} ,  (2.11)

where

[\begin{matrix}P_{k} & Λ_{k} \\ Q_{k} & S_{k} \end{matrix}] = [\begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix}] + κ [\begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix}] [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] ,

M_{k} = −\frac{2}{a_{11}} (a_{11}d_{k}^{2} − a_{13}α_{k} + a_{12})d_{k}, N_{k} = −α_{k}M_{k},

R_{k} = −\frac{2}{a_{11}α_{k}} (a_{12}α_{k}^{2} − a_{23}α_{k} + a_{22})d_{k}, L_{k} = −α_{k}R_{k}.

Each column of matrix (2.11) treated as a vector with respect to the point z satisfies (1.1) everywhere except the point z = t, for any value of the parameter κ. In the neighborhood of the point z = t the terms of matrix (2.11) possess singularities (of order not higher than one).

Applying Viet’s theorem to equation (2.1), we can represent the coefficients N_{k}, L_{k}, M_{k}, R_{k} as

[\begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix}] = [\begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix}] + 2ωd_{k} [ \begin{matrix}α_{k} & α_{k}^{2} \\ −1 & −α_{k} \end{matrix} ] ,  (2.12)

where ω = a_{12}/a_{11} − Re(α_{1}α_{2}) ≡ a_{12}/a_{11} + b^{(1)}b^{(2)} − a^{(1)}a^{(2)}.

By (2.7) we have

[\begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix}] = [\begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix}] [ \begin{matrix}1 − iωA & −iωB \\ iωC & 1 + iωA \end{matrix} ] ,  (2.13)

or else by (2.9)

[\begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix}]=2d_{k}[ \begin{matrix}α_{k}^{2} & −α_{k} \\ −α_{k} & 1 \end{matrix} ] \{ω[ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]+ \frac{i}{BC−A^{2}} [\begin{matrix}C & A \\ A & B \end{matrix}]\} . (2.14)

Since α_{k} is the root of equation (2.1), the following relationships are established between the coefficients A_{k}, B_{k}, C_{k} and M_{k}, R_{k}:

A_{k} = \frac{a_{11}}{2d_{k}} M_{k}^{2}, B_{k} = \frac{a_{11}}{2d_{k}} M_{k}R_{k}, C_{k} = \frac{a_{11}}{2d_{k}} R_{k}^{2}.  (2.15)

By virtue of (2.14) and (2.12) relations (2.15) imply

[\begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix}]=[\begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix}] \{ωa_{11} [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]+ \frac{ia_{11}}{BC − A^{2}} [\begin{matrix}C & A \\ A & B \end{matrix}]\} . (2.16)

By (2.13) it follows from (2.16) that

∑_{k=1}^{2} [\begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix}] = \frac{im}{BC − A^{2}} [\begin{matrix}C & A \\ A & B \end{matrix}] ,  (2.17)

where m = a_{11}[1 − ω^{2}(BC − A^{2})]. Note that ω > 0 and m > 0 [7]. It is evident that for κ = 0 from (2.11) we obtain the matrix

[T^{0} (∂_{t}, n)Γ(z, t)]^{T} = [T (∂_{t}, n)Γ(z, t)]^{T} = Im ∑_{k=1}^{2} [\begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)}

whose every element by (2.13) and (2.10) will have a singularity of the type |z − t|^{−1} at the point z = t.

Matrix (2.11) for κ = κ_{N}, where

κ_{N} = \frac{ω(BC − A^{2})}{m},  (2.18)

takes the form either

[ T^{κ_{N}}(∂_{t}, n)Γ(z, t)]^{T} =[N(∂_{t}, n)Γ(z, t)]^{T} =Im ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} , (2.19)

where N(∂_{t}, n) is the so-called pseudostress operator [7],

[\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] = \frac{a_{11}}{m} [\begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix}] [ \begin{matrix}1 + iωA & iωC \\ iωB & 1 − iωA \end{matrix} ]  (2.20)

or

[\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] = −\frac{i}{m} [\begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix}] [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] .  (2.21)

We can see from (2.18) that κ_{N} > 0, since ω > 0, m > 0, and it follows from (2.6) that BC − A^{2} > 0.

By (2.13) or (2.17) it follows from (2.20) or (2.21) that

∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] = E.

Using the latter condition, we can show that when z and t are points of the Lyapunov curve, for z = t the elements of matrix (2.19) contain only singularities integrable in an ordinary sense.

We shall say that u = (u_{1}, u_{2})^{T} and v = (v_{1}, v_{2})^{T} are the conjugate vectors or satisfy the generalized Cauchy–Riemann conditions [11] if the following conditions are fulfilled:

\begin{matrix}A_{11} \frac{∂u_{1}}{∂x_{1}} + A_{12} \frac{∂u_{2}}{∂x_{2}} + A_{13}(\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) + κ_{N} \frac{∂u_{2}}{∂x_{2}} = \\ = \frac{1}{m}(B \frac{∂v_{1}}{∂x_{2}} − A\frac{∂v_{2}}{∂x_{2}} ), \\ A_{12} \frac{∂u_{1}}{∂x_{1}} + A_{22} \frac{∂u_{2}}{∂x_{2}} + A_{23}(\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) + κ_{N} \frac{∂u_{1}}{∂x_{1}} = \\ = \frac{1}{m}(A\frac{∂v_{1}}{∂x_{1}} − C \frac{∂v_{2}}{∂x_{1}} ), \\ A_{13} \frac{∂u_{1}}{∂x_{1}} + A_{23} \frac{∂u_{2}}{∂x_{2}} + A_{33}(\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) − κ_{N} \frac{∂u_{2}}{∂x_{1}} = \\ = −\frac{1}{m}(B \frac{∂v_{1}}{∂x_{1}} − A\frac{∂v_{2}}{∂x_{1}} ), \\ A_{13} \frac{∂u_{1}}{∂x_{1}} + A_{23} \frac{∂u_{2}}{∂x_{2}} + A_{33}(\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) − κ_{N} \frac{∂u_{1}}{∂x_{2}} = \\ = −\frac{1}{m}(A\frac{∂v_{1}}{∂x_{2}} − C \frac{∂v_{2}}{∂x_{2}} ). \end{matrix}  (2.22)

By virtue of equalities (2.22) we can readily show that the vector u is a solution of system (1.1).

By transforming equalities (2.22) it is proved in [11] that the equalities obtained formally from (2.22) by replacing u by v and v by (−u) are also valid. Hence the vector v is also a solution of system (1.1).

After multiplying the first equality from (2.22) by n_{1}(x) and the third equality by n_{2}(x) and summing the obtained equations, we obtain

[N(∂_{x}, n)u(x)]_{1} = \frac{1}{m}(B \frac{∂v_{1}(x)}{∂s(x)} − A\frac{∂v_{2}(x)}{∂s(x)}).

Similarly, having multiplied the second equality from formula (2.22) by n_{2}(x) and the fourth equality by n_{1}(x) and summing the obtained equations, we have

[N(∂_{x}, n)u(x)]_{2} = −\frac{1}{m}(A\frac{∂v_{1}(x)}{∂s(x)} − C \frac{∂v_{2}(x)}{∂s(x)}).

The latter two formulas finally yield

N(∂_{x}, n)u(x) = \frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] \frac{∂v(x)}{∂s(x)}.  (2.23)

The above arguments show that the equality

N(∂_{x}, n)v(x) = −\frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] \frac{∂u(x)}{∂s(x)}  (2.24)

is also valid.

Thus the conjugate vectors u and v satisfy conditions (2.23) and (2.24). We can easily show that if u and v are conjugate, then on account of

(2.23) and (2.24) we shall have

T^{κ}(∂_{x}, n)u(x)= \frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ]\frac{∂v(x)}{∂s(x)} +(κ − κ_{N}) [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]\frac{∂u(x)}{∂s(x)}. (2.25)

If we introduce the notation w = u + iv, then (2.23) and (2.24) can be rewritten as a single equality

N(∂_{x}, n)w(x) = −\frac{i}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] \frac{∂w(x)}{∂s(x)}.  (2.26)

As is known, any solution v of system (1.1) of the class C^{1}(\overline{D}^{+})∩C^{2}(D^{+}) can be written as [12, Ch. 1]

v(z)= \frac{1}{2π} ∫_{S} ([N(∂_{t}, n)Γ(z, t)]^{T} \{v(t)\}^{+}−Γ(z, t)\{N(∂_{t}, n)v(t)\}^{+})ds, z ∈ D^{+},

which after integrating by parts the second summand under the integral sign and taking into account (2.24) and (2.21), eventually gives

v(z) = \frac{1}{2π} ∫_{S} ( Im ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} \{v(t)\}^{+} −

− Re ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} \{u(t)\}^{+})ds, z ∈ D^{+},  (2.27)

where u is the conjugate to the vector v of the class C^{1}(\overline{D}^{+}) ∩ C^{2}(D^{+}).

Analogously, if u is any solution of system (1.1) of the class C^{1}(\overline{D}^{+}) ∩ C^{2}(D^{+}) and v is its conjugate vector of the same class, then

u(z) = \frac{1}{2π} ∫_{S} ( Im ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} \{u(t)\}^{+} +

+ Re ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} \{v(t)\}^{+})ds, z ∈ D^{+}.  (2.28)

Thus from (2.27) and (2.28) we obtain

w(z) = \frac{1}{2πi} ∫_{S} ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} \{w(t)\}^{+}ds, z ∈ D^{+}, (2.29)

which is a generalization of the Cauchy integral formula.

Let us consider the potential

w̃(g)(z) = \frac{1}{π} ∫_{S} [N(∂_{t}, n)Γ(z, t)]^{T} g(t)ds,  (2.30)

where the density g = (g_{1}, g_{2})^{T} is an arbitrary real vector. Obviously, potential (2.30) is a solution of system (1.1) both in D^{+} and in D^{−} for an arbitrary integrable density g.

Theorem 1. If S ∈ C^{2+α}, 0 < α < 1, and 0 < β < α, then:

(a) w̃(g) ∈ C^{k+β}(\overline{D}^{±}), if g ∈ C^{k+β}(S) for k = 0, 1, 2, and for arbitrary g ∈ C^{β}(S) and t_{0} ∈ S, we have the equality

\{w̃(g)(t_{0})\}^{±} = ±g(t_{0}) + \frac{1}{π} ∫_{S} [N(∂_{t}, n)Γ(t_{0}, t)]^{T} g(t)ds; (2.31)

(b) for arbitrary g ∈ C^{1+β}(S) the vector N(∂_{z}, n)w̃(g)(z) is continuously extendable to S from D^{±}, and for arbitrary t_{0} ∈ S we have the equality

\{N(∂_{t_{0}}, n)w̃(g)(t_{0})\}^{±} = −\frac{1}{π} Im ∑_{k=1}^{2} E_{(k)} ∫_{S} \frac{∂ ln σ_{k}}{∂s(t_{0})} \frac{∂g(t)}{∂s(t)}ds, (2.32)

where

E_{(k)} = \frac{2d_{k}}{a_{11}} [ \begin{matrix}α_{k}^{2} & −α_{k} \\ −α_{k} & 1 \end{matrix} ]+κ_{N}[ \begin{matrix}2L_{k} & R_{k} − N_{k} \\ R_{k} − N_{k} & −2M_{k} \end{matrix} ]+κ_{N}^{2}[ \begin{matrix}C_{k} & −B_{k} \\ −B_{k} & A_{k} \end{matrix} ] .

Finally, let us quote Green’s formula. If u ∈ C^{1}(\overline{D}^{+}) ∩ C^{2}(D^{+}) is a solution of system (1.1) in D^{+} and κ is an arbitrary constant, then the formula

∫_{D^{+}} T^{κ}(u, u)dσ = ∫_{S} {u}^{+}{T (∂_{t}, n)u}^{+}ds  (2.33)

is valid [7], where

T^{κ}(u, u) =A_{11}(\frac{∂u_{1}}{∂x_{1}} )^{2} + A_{22}(\frac{∂u_{2}}{∂x_{2}} )^{2} + 2A_{13} \frac{∂u_{1}}{∂x_{1}} (\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) +

+ 2A_{23} \frac{∂u_{2}}{∂x_{2}} (\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} ) + 2(A_{12} + κ)\frac{∂u_{1}}{∂x_{1}} \frac{∂u_{2}}{∂x_{2}} +

+ (A_{33} − \frac{κ}{2} )(\frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} )^{2} + \frac{κ}{2} (\frac{∂u_{2}}{∂x_{1}} − \frac{∂u_{1}}{∂x_{2}} )^{2}.

It is proved [12, Ch. 1] that the quadratic form T^{κ}(u, u) is positive definite for 0 ≤ κ < α^{∗}, where α^{∗} is the least positive root of the equation

α^{3} + 2(A_{12} − A_{33})α^{2} + (A_{12}^{2} − A_{11}A_{22} + 4A_{13}A_{33} − 4A_{12}A_{33})α + 2Δ = 0.

Note that κ_{N} < α^{∗}, and hence the quadratic form T^{κ}(u, u) is positive definite for κ = 0 and κ = κ_{N}.

The equation T^{κ}(u, u) = 0 admits, for κ = 0, a solution u(z) = (e_{1} − e_{3}x_{2}, e_{2} + e_{3}x_{1})^{T}, and for 0 < κ < α^{∗}, a solution u(z) = (e_{1}, e_{2})^{T}, where e_{1}, e_{2}, e_{3} are arbitrary real constants.

For the domain D^{−}, formula (2.33) (with the minus before the integral sign on the right-hand side) also holds if the vector u satisfies additionally conditions (1.3) at infinity.

Theorem 2. The homogeneous basic mixed problem (F = 0, f = 0) in the class of regular vector functions (in the sense of Definition 1) has, for 0 ≤ κ < α^{∗}, only a trivial solution.

Proof. Note first that formula (2.33) can also be applied to the regular solution u of system (1.1) in D^{+}. Indeed, denote by D_{∗}^{+} the domain which is obtained from D^{+} by removing infinitesimal circumferences of infinitesimal radii with centers at the points a_{j}, b_{j}, j = 1, 2, …, p, of the contour S. Obviously, u ∈ C^{1}(\overline{D}_{∗}^{+}) and therefore we can apply (2.33) to the domain D_{∗}^{+} and then pass to the limit as the radii of the above-mentioned circumferences tend to zero. Taking into account that the vector u is continuously

extendable at all points of the contour S, and T^{κ}(∂_{x}, n)u is continuously extendable at all points of the contour S, except possibly the points a_{j}, b_{j}, j = 1, 2, …, p, in whose neighborhood we have singularities less tthan 1, we find that the integrals on the right side extended to the arcs of these circumferences contained in D^{+} will tend to zero, and formula (2.33) will appear to be valid for a regular solution of system (1.1) in D^{+}.

Let now u_{0} be a regular solution of the homogeneous basic mixed problem (F = 0, f = 0). In that case, if we apply (2.33) to the vector u_{0}, then the integral on the right-hand side will be equal to zero, and, since the

quadratic form T^{κ}(u_{0}, u_{0}) is positive definite, for 0 ≤ κ < α^{∗} we shall have

T^{κ}(u_{0}, u_{0}) = 0. These arguments and the condition that on some part of S′′ the boundary value of the vector u_{0} is equal to zero imply that u_{0}(z) = 0, z ∈ D^{+}, which was to be proved.

§ 3. Reducting the Basic Mixed Problem to Integral Equations

A solution of the problem is sought in the form u(z) = Re w(z), where w(z) is represented in terms of a potential of type (2.29),

w(z) = \frac{1}{πi} ∫_{S} ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} g(t)ds  (3.1)

whose density g = (g_{1}, g_{2})^{T} is an unknown Hölder class real vector.

Then on the basis of (2.25) and (2.26) the generalized stress vector will be of the form

T^{κ}(∂_{z}, n)u(z)= \frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] \frac{∂v(z)}{∂s(z)} +(κ−κ_{N}) [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] \frac{∂u(z)}{∂s(z)}, (3.2)

where this time

v(z) = Im w(z) = Im \frac{1}{πi} ∫_{S} ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} g(t)ds.  (3.3)

By (3.2), the generalized principal stress vector acting from the side of the positive normal on an arbitrary arc l confined within D^{+}, connecting the points z_{0} and z of D^{+} (to within arbitrary constants of additive vectors) has the form

\frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] v(z) + (κ − κ_{N}) [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] u(z) =

= ∫_{z_{0}}^{z} T^{κ}(∂_{t}, n)u(t)ds + const .  (3.4)

By equalities (3.1), (3.3), (3.4), (2.31) and Theorem 1 the boundary conditions of problem (1.4) and (1.5) give, for the unknown vector g, a system of singular integral equations with discontinuous coefficients of the form

A(t_{0})g(t_{0}) + \frac{B(t_{0})}{π} ∫_{S} \frac{g(t)dt}{t − t_{0}} + \frac{1}{π} ∫_{S} K(t_{0}, t)g(t)ds =

= Ω(t_{0}) + D(t_{0}), t_{0} ∈ S,  (3.5)

where

A(t_{0}) = \begin{cases}(κ − κ_{N}) [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] , & t_{0} ∈ S′, \\ E, & t_{0} ∈ S′′, \end{cases}

B(t_{0}) = \begin{cases}−\frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] , & t_{0} ∈ S′, \\ 0, & t_{0} ∈ S′′, \end{cases}

K(t_{0}, t)= \frac{1}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] Q(t_{0}, t)+(κ−κ_{N}) [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] R(t_{0}, t),

t_{0} ∈ S′, t ∈ S,

K(t_{0}, t) = R(t_{0}, t), t_{0} ∈ S′′, t ∈ S,

Q(t_{0}, t) = i\frac{∂θ}{∂s(t)}E − Re ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂}{∂s(t)} ln (1 + ν_{k} \frac{\overline{t} − \overline{t_{0}}}{t − t_{0}} ),

R(t_{0}, t) = \frac{∂θ}{∂s(t)}E + Im ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂}{∂s(t)} ln (1 + ν_{k} \frac{\overline{t} − \overline{t_{0}}}{t − t_{0}} ),

ν_{k} = \frac{1 + iα_{k}}{1 − iα_{k}} , θ = arg(t_{0} − t),

Ω(t_{0}) = \begin{cases}∫_{a_{j}}^{t_{0}} F (t)ds, t_{0} ∈ S′_{j}, \\ f(t_{0}), t_{0} ∈ S′′, \end{cases} D(t_{0}) = \begin{cases}D^{(j)}, t_{0} ∈ S′_{j}, \\ 0, t_{0} ∈ S′′; \end{cases}

D^{(j)} = (D_{1}^{(j)}, D_{2}^{(j)})^{T} is an arbitrary real constant vector.

Multiplying the first equation of system (3.5) by an arbitrary constant M for t_{0} ∈ S′ and adding the obtained equaion to (3.5) we obtain

(κ − κ_{N})(g_{1}(t_{0})−Mg_{2}(t_{0}))+ \frac{1}{πm} ∫_{S} \frac{(A−MB)g_{1}(t)+(AM −C)g_{2}(t)}{t − t_{0}}dt+

+\frac{1}{π} ∫_{S} \{[K_{11}(t_{0}, t)M +K_{21}(t_{0}, t)]g_{1}(t)+[MK_{12}(t_{0}, t)+K_{22}(t_{0}, t)]g_{2}(t)\}ds=

= ∫_{a_{j}}^{t_{0}}(F_{2}(t) + MF_{1}(t))ds + D_{2}^{(j)} + MD_{1}^{(j)}, t_{0} ∈ S′_{j},  (3.6)

where K_{rq}(t_{0}, t), r = 1, 2, q = 1, 2, are the components of the matrix K(t_{0}, t).

Choose M by the condition that C − AM = M(A − MB). This implies that

M_{1,2} = \frac{A ± i\sqrt{BC − A^{2}}}{B} .

Let M = (A + i\sqrt{BC − A^{2}})B^{−1} and substitute it into (3.6). Then we obtain

(κ − κ_{N})ω(t_{0}) − \frac{i\sqrt{BC − A^{2}}}{πm} ∫_{S} \frac{ω(t)dt}{t − t_{0}} + \frac{1}{π} ∫_{S} K_{1}(t_{0}, t)ω(t)ds +

+\frac{1}{π} ∫_{S} \overline{K_{2}(t_{0}, t)}\overline{ω(t)}ds = ∫_{a_{j}}^{t_{0}}(F_{2}(t) + MF_{1}(t))ds + D_{2}^{(j)} + MD_{1}^{(j)}, (3.7)

where ω(t) = g_{1}(t) − Mg_{2}(t),

K_{1}(t_{0}, t) = \frac{1}{\overline{M} − M} \{\overline{M}[K_{21}(t_{0}, t) + MK_{11}(t_{0}, t)] +
+ MK_{12}(t_{0}, t) + K_{22}(t_{0}, t)},  (3.8)
\overline{K_{2}(t_{0}, t)} = \frac{1}{M − \overline{M}} \{M[K_{21}(t_{0}, t) + MK_{11}(t_{0}, t)] +
+ MK_{12}(t_{0}, t) + K_{22}(t_{0}, t)}, t_{0} ∈ S′, t ∈ S.

If now we multiply the second equation of system (3.5) by M for t_{0} ∈ S′′ and add the obtained equation to (3.5), we obtain

ω(t_{0})+ \frac{1}{π} ∫_{S} K_{1}(t_{0}, t)ω(t)ds+ \frac{1}{π} ∫_{S} \overline{K_{2}(t_{0}, t)}\overline{ω(t)}ds=f_{1}(t_{0})−Mf_{2}(t_{0}), (3.9)

where

\begin{aligned}K_{1}(t_{0}, t) & = \frac{1}{\overline{M} − M} \{\overline{M}[K_{11}(t_{0}, t) − MK_{21}(t_{0}, t)] + \\ & + K_{12}(t_{0}, t) − MK_{22}(t_{0}, t)\}, \\ \overline{K_{22}(t_{0}, t)} & = \frac{1}{M − \overline{M}} \{M[K_{11}(t_{0}, t) − MK_{21}(t_{0}, t)] + \\ & + K_{12}(t_{0}, t) − MK_{22}(t_{0}, t)\}, t_{0} ∈ S′′, t ∈ S. \end{aligned} (3.10)

It is obvious that by combining equations (3.7) and (3.9) we shall have a singular integral equation with discontinuous coefficients that contains, along with the unknown function, its conjugate lying outside the characteristic part and is of the form

a(t_{0})ω(t_{0}) + \frac{b(t_{0})}{πi} ∫_{S} \frac{ω(t)dt}{t − t_{0}} + \frac{1}{π} ∫_{S} K_{1}(t_{0}, t)ω(t)ds +

+\frac{1}{π} ∫_{S} \overline{K_{2}(t_{0}, t)}\overline{ω(t)}ds = φ(t_{0}) + D_{0}(t_{0}), t_{0} ∈ S,  (3.11)

where

a(t_{0}) = \begin{cases}κ − κ_{N}, t_{0} ∈ S′, \\ 1, t_{0} ∈ S′′, \end{cases} b(t_{0}) = \begin{cases}m^{−1}\sqrt{BC − A^{2}}, & t_{0} ∈ S′, \\ 0, & t_{0} ∈ S′′, \end{cases}

φ(t_{0}) = \begin{cases}∫_{a_{j}}^{t_{0}}(F_{2}(t) + MF_{1}(t))ds, & t_{0} ∈ S′_{j}, \\ f_{1}(t_{0}) − Mf_{2}(t_{0}), & t_{0} ∈ S′′, \end{cases}

D_{0}(t_{0}) = \begin{cases}D_{2}^{(j)} + MD_{1}^{(j)}, & t_{0} ∈ S′_{j}, \\ 0, & t_{0} ∈ S′′, \end{cases}

the functions K_{1}(t_{0}, t) and K_{2}(t_{0}, t) are defined by equalities (3.8) and (3.10), respectively.

The characteristic homogeneous equation corresponding to equation (3.11) is

a(t_{0})ω(t_{0}) + \frac{b(t_{0})}{πi} ∫_{S} \frac{ω(t)dt}{t − t_{0}} = 0.

The homogeneous problem of conjugation corresponding to the above equation is written as

Φ^{+}(t) = G(t)Φ^{−}(t), t ∈ S,  (3.12)

where

Φ(z) = \frac{1}{2πi} ∫_{S} \frac{ω(t)dt}{t − z},

G(t) = \frac{a(t) − b(t)}{a(t) + b(t)} = \begin{cases}(κ − δ_{0})(κ + δ_{1})^{−1}, & t_{0} ∈ S′, \\ 1, & t_{0} ∈ S′′, \end{cases}

δ_{0} and δ_{1} are the positive numbers [8]

δ_{0} = \frac{\sqrt{BC − A^{2}}}{a_{11}(1 − ω\sqrt{BC − A^{2}})}, δ_{1} = \frac{\sqrt{BC − A^{2}}}{a_{11}(1 + ω\sqrt{BC − A^{2}})}.

Since α^{∗} < δ_{0} [12, Ch. 1], for 0 ≤ κ < α^{∗} < δ_{0} we have (κ − δ_{0})(κ + δ_{1})^{−1} < 0. By the general theory [9], all nodes a_{j}, b_{j}, j = 1, 2, …, p, are nonsingular, and the canonical solution X(z) of the class of functions bounded at all nodes, i.e., of the class h_{2p}, of the problem of conjugation (3.12) is given by the formula

X(z) = ∏_{j=1}^{p}(z − a_{j})^{\frac{1}{2} +iβ}(z − b_{j})^{\frac{1}{2} −iβ}, β = \frac{1}{2π} ln \frac{δ_{0} − κ}{δ_{1} + κ} . (3.13)

Under X(z) is meant a branch which is holomorphic on the plane cut along S′ and satisfying the condition lim_{z→∞} z^{−p}X(z) = 1.

It follows from (3.13) that the index of the class h_{2p} of the problem of conjugation and hence of equation (3.11) is equal to −p, since the order X(z) is equal to p at infinity.

A solution ω(t) of equation (3.11) will be sought in the class h_{2p}.

Note that equation (3.11) does not actually differ from the equation obtained for the basec mixed problem in the isotropic case. Therefore if we take into account the restrictions imposed on the contour S and the boundary values f and F, then on the entire contour S, arguing as in [2], we can show that ω(t) belongs to the Hölder class, while the derivative ω′(t) to the class H^{∗}.

Since the index of the class h_{2p} of equation (3.11) is equal to −p, by the general theorem [2] we have ν − ν′ = −2p, where ν is the number of linearly independent solutions of the class h_{2p} of the homogeneous equation corresponding to (3.11), and ν′ is the number of linearly independent solutions of the class h′_{2p} of the adjoint homogeneous equation adjoint to h_{2p}.

Let us prove that ν = 0 and hence ν′ = 2p. Indeed, let the homogeneous equation corresponding to equation (3.11) have a nontrivial solution ω_{0}(t) in the class h_{2p}. We construct the

w_{0}(z) = \frac{1}{πi} ∫_{S} ∑_{k=1}^{2} [\begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix}] \frac{∂ ln σ_{k}}{∂s(t)} g_{0}(t)ds,  (3.14)

where

g_{0}(t) = (\frac{\overline{M}ω_{0}(t) − M\overline{ω_{0}(t)}}{\overline{M} − M}, \frac{ω_{0}(t) − \overline{ω_{0}(t)}}{\overline{M} − M})^{T} .

Then u_{0}(z) = Re w_{0}(z) will satisfy the homogeneous boundary conditions

{u_{0}(t)}^{+} = 0 for t_{0} ∈ S′′ and {T^{κ}(∂_{t}, n)u_{0}(t)}^{+} = 0 for t_{0} ∈ S′, except

possibly the points a_{j}, b_{j}, j = 1, 2, …, p, at which the the generalized stress vector may have only an integrable singularity.

Assume that 0 ≤ κ < α^{∗}. Then by Theorem 2 we conclude that u_{0}(z) = 0, z ∈ D^{+}. Therefore N(∂_{z}, n)u_{0}(z) = 0, and by equality (2.32) and Theorem 1 we have {N(∂_{t}, n)u_{0}(t)}^{+} = {N(∂_{t}, n)u_{0}(t)}^{−}, which holds at all points of the contour S, except possibly the points a_{j}, b_{j}, j = 1, 2, …, p. Thus u_{0}(z) satisfies system (1.1) in the domain D^{−}, the boundary condition {N(∂_{t}, n)u_{0}(t)}^{−} = 0 on S and the decreasing conditions (1.3) at infinity. Since κ_{N} < α^{∗}, we can easily show by formula (2.33) that u_{0}(z) = 0, z ∈ D^{−}. Finally, from the equality {u_{0}(t)}^{+} − {u_{0}(t)}^{−} = 2g_{0}(t) it follows that g_{0}(t) = 0, t ∈ S, which contradicts our assumption.

Thus the homogeneous equation corresponding to (3.11) has only a trivial solution in the class h_{2p} and hence, as mentioned above, ν′ = 2p. By the general theory [2], the condition of solvability of equation (3.11) in the class h_{2p} has the form

Re ∫_{S} (φ(t) + D_{0}(t))σ_{j}(t)dt = 0, j = 1, 2, …, 2p,  (3.15)

where σ_{j}, j = 1, 2, …, 2p, is a complete system of linearly independent solutions of the class h′_{2n} of the adjoint homogeneous equation. Conditions (3.15) can be rewritten as

∑_{r=1}^{p}(γ_{jr}D_{1}^{(r)} + δ_{jr}D_{2}^{(r)}) = η_{j}, j = 1, 2, …, 2p,  (3.16)

where

γ_{jr} = Re M ∫_{S′_{r}} σ_{j}(t)dt, δ_{jr} = Re ∫_{S′_{r}} σ_{j}(t)dt, η_{j} = − Re ∫_{S} φ(t)σ_{j}(t)dt.

Let us prove that with respect to the unknowns D_{1}^{(r)} and D_{2}^{(r)} the determinant of the system of algebraic equations (3.16) is different from zero.

Indeed, let φ = 0. Then in system (3.16) all η_{j} = 0 and it becomes the homogeneous system. If the determinant of the system is equal to zero, then the homogeneous system admits a solution different from zero. Let D_{10}^{(r)} and D_{20}^{(r)}, r = 1, 2, …, p, be such a solution. Then equation (3.11) will be solvable in the class h_{2p} for φ = 0 and D_{0}(t) = D_{20}^{(r)} + MD_{10}^{(r)}, t ∈ S′_{r}, r = 1, 2, …, p, and D_{0}(t) = 0, t ∈ S′′. Let ω_{0}(t) be a solution of (3.11). Again constructing potential (3.14) and repeating the above arguments, we obtain g_{0}(t) = 0, and hence D_{10}^{(r)} = D_{20}^{(r)} = 0, r = 1, 2, …, p, which contradicts our assumption.

Consequently, the determinant of system (3.16) is different from zero, system (3.16) is always uniquely solvable in the class h_{2p} and its solution leads to the solution of the considered problem.

Thus the following theorem is valid.

Theorem 3. Under conditions (1) and (2), the basic mixed problem (1.4), (1.5) has, for 0 ≤ κ < α^{∗}, a unique solution which is representable in the form u(z) = Re w(z), where w(z) is given by formula (3.1), and the vector g is defined uniquely by a solution of the class h_{2p} of equation (3.11).

References

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(Received 04.11.1996)

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

Georgia