Georgian Mathematical Journal Volume 10 (2003), Number 3, 427–465

SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS OF CRACK-TYPE BOUNDARY-CONTACT PROBLEMS OF THE

COUPLE-STRESS ELASTICITY

O. CHKADUA

Abstract. Spatial boundary value problems of statics of couple-stress elasticity for anisotropic homogeneous media (with contact on a part of the boundary) with an open crack are studied supposing that one medium has a smooth boundary and the other one has an open crack.

Using the method of the potential theory and the theory of pseudodifferential equations on manifolds with boundary, the existence and uniqueness theorems are proved in Besov and Bessel-potential spaces. The smoothness and a complete asymptotics of solutions near the contact boundaries and near crack edge are studied.

Properties of exponents of the first terms of the asymptotic expansion of solutions are established. Classes of isotropic, transversally-isotropic and anisotropic bodies are found, where oscillation vanishes.

2000 Mathematics Subject Classification: 74B05, 74G70, 35B40, 47G30. Key words and phrases: Couple-stress elasticity, anisotropic homogeneous medium, asymptotic expansion, strongly elliptic pseudodifferential equations, potentials.

Introduction

The paper is dedicated to the study of solvability and asymptotics of solutions of spatial crack type boundary-contact problems of statics of couple-stress elasticity for anisotropic homogeneous media with contact on a part of the boundary.

A vast number of works are devoted to the justification and axiomatization of elasticity and couple-stress elasticity. The fundamentals of the theory of couple-stress elasticity are included in the works by W. Voight [40], E. Cosserat, F. Cosserat [11], and developed later in the works by E. Aero and E. Kuvshinski [1], G. Grioli [21], R. Mindlin [31], W. Koiter [24], W. Nowacki [34], V. Kupradze, T. Gegelia, M. Basheleishvili, T. Burchuladze [26], T. Burchuladze and T. Gegelia [4] and others.

It is well-known that solutions of elliptic boundary value problems in domains with corners, edges and conical points have singularities regardless the smoothness properties of given data.

Among theoretical investigations the methods suggested and developed by V. Kondrat’ev [25], V, Maz’ya [27], V, Maz’ya and B. Plamenevsky [28]–[30], S. Nazarov and B. Plamenevsky [33], M. Dauge [13], P. Grisvard [22] and others

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

attracted attention of many scientists. They used the Mellin transform which allows them to reduce the problem to the investigation of spectral properties of ordinary differential operators depending on the parameter.

The method of the potential theory and the theory of pseudodifferential equations used in this paper makes it possible to obtain more precise asymptotic representations of solutions of the problems posed, which frequently have crucial importance in applications (e.g., in crack extension problems). For the development of this method see [18], [3], [10], [9] and other papers.

The method of the potential theory was successfully applied to the classical problems of elasticity and couple-stress elasticity theory by V. Kupradze and his disciples.

In the present paper we consider the contact of two media, one of which has a smooth boundary, while the other has a boundary containing a closed cuspidal edge (the corresponding dihedral angle is equal to 2π), i.e., an open crack.

Theorems on the existence and uniqueness of solutions of these boundarycontact problems are obtained using the potential theory and the general theory of pseudodifferential equations on a manifold with boundary.

Using the asymptotic expansion of solutions of strongly elliptic pseudodifferential equations obtained in [10] (see also [18], [3]) and also the asymptotic expansion of potential-type functions [9], we obtain a complete asymptotic expansion of solutions of boundary-contact problems near the contact boundaries and near the crack edge. Here it is worth noticing the effective formulae for calculating the exponent of the first terms of asymptotic expansion of solutions of these problems by means of the symbol of the corresponding boundary pseudodifferential equations.

The properties of exponents of the first terms of the asymptotic expansion of solutions are established. Important classes of isotropic, transversally-isotropic and anisotropic bodies are found, where oscillation vanishes.

These results are new even for the problems of elasticity.

1. Formulation of the Problems

Let D_{1} be a finite domain, D_{2} be a domain that can be both finite or infinite in the Euclidean space ℝ^{3} with compact boundaries ∂D_{1}, ∂D_{2} (∂D_{1} ∈ C^{∞}), and let there exist a surface S_{0} of the class C^{∞} of dimension two, which divides the domain D_{2} into two subdomains D_{2}^{(1)} and D_{2}^{(2)} with C^{∞} boundaries ∂D_{2}^{(1)} and ∂D_{2}^{(2)} (D_{2}^{(1)} ∩ D_{2}^{(2)} = ∅, \overline{D}_{2}^{(1)} ∩ \overline{D}_{2}^{(2)} = \overline{S}_{0}). Then ∂S_{0} is the boundary of the surface S_{0} (∂S_{0} ⊂ ∂D_{2}), representing one-dimensional closed cuspidal edge, where ∂S_{0} is the crack edge.

Let the domains D_{1} and D_{2} have the contact on the two-dimensional manifolds \overline{S} _{0}^{(1)} and \overline{S} _{0}^{(2)} of the class C^{∞}, i.e., ∂D_{1} ∩ ∂D_{2} = \overline{S} _{0}^{(1)} ∪ \overline{S} _{0}^{(2)}, D_{1} ∩ D_{2} = ∅, \overline{S} _{0}^{(1)} ∩ \overline{S} _{0}^{(2)} = ∅, and S_{1} = ∂D_{1} \ (\overline{S} _{0}^{(1)} ∪ \overline{S} _{0}^{(2)}). Then ∂D_{2}^{(1)} = S_{2}^{(1)} ∪ \overline{S} _{0}^{(1)} ∪ \overline{S}_{0}, ∂D_{2}^{(2)} = S_{2}^{(2)} ∪ \overline{S} _{0}^{(2)} ∪ \overline{S}_{0}.

Suppose that the domains D_{q}, q = 1, 2, are filled with anisotropic homogeneous elastic materials.

The basic static equations of couple-stress elasticity for anisotropic homogeneous media are written in terms of displacement and rotation components as (see [5], [19])

M^{(q)}(∂_{x})U^{(q)} + F^{(q)} = 0 in D_{q}, q = 1, 2,  (1.1)

where U^{(q)} = (u^{(q)}, ω^{(q)}), u^{(q)} = (u_{1}^{(q)}, u_{2}^{(q)}, u_{3}^{(q)}) is the displacement vector, ω^{(q)} = (ω_{1}^{(q)}, ω_{2}^{(q)}, ω_{3}^{(q)}) is the rotation vector, F^{(q)} = (F_{1}^{(q)}, …, F_{6}^{(q)}) is the mass force applied to D_{q}, and M^{(q)}(∂_{x}) is the matrix differential operator

M^{(q)}(∂_{x}) = ( \begin{matrix}M^{1}^{(q)}(∂_{x}) & M^{2}^{(q)}(∂_{x}) \\ M^{3}^{(q)}(∂_{x}) & M^{4}^{(q)}(∂_{x}) \end{matrix} )_{6×6} ,

M^{l} ^{(q)}(∂_{x}) = ∥ M^{l} _{jk}^{(q)}(∂_{x})∥_{3×3}, l = \overline{1, 4}, q = 1, 2,

\begin{matrix}M^{1} _{jk}^{(q)}(∂_{x}) = a_{ijlk}^{(q)}∂_{i}∂_{l}, M^{2} _{jk}^{(q)}(∂_{x}) = b_{ijlk}^{(q)}∂_{i}∂_{l} − ε_{lrk}a_{ijlr}^{(q)}∂_{i}, \\ M^{3} _{jk}^{(q)}(∂_{x}) = b_{lkij}^{(q)}∂_{i}∂_{l} + ε_{irj}a_{irlk}^{(q)}∂_{l}, \\ M^{4} _{jk}^{(q)}(∂_{x}) = c_{ijlk}^{(q)}∂_{i}∂_{l} − b_{lrij}^{(q)}ε_{lrk}∂_{i} + ε_{irj}b_{irlk}^{(q)}∂_{l} − ε_{ipj}ε_{lrk}a_{iplr}^{(q)}; \end{matrix}  (1.2)

ε_{ikj} is the Levi-Civita symbol, a_{ijlk}^{(q)}, b_{ijlk}^{(q)}, c_{ijlk}^{(q)} are the elastic constants satisfying the conditions

a_{ijlk}^{(q)} = a_{lkij}^{(q)}, c_{ijlk}^{(q)} = c_{lkij}^{(q)}, q = 1, 2.

In (1.2) and in what follows, under the repeated indices we understand the summation from 1 to 3.

It is assumed that the quadratic forms

a_{ijlk}^{(q)}ξ_{ij}ξ_{lk} + 2b_{ijlk}^{(q)}ξ_{ij}η_{lk} + c_{ijlk}^{(q)}η_{ij}η_{lk}, q = 1, 2,

with respect to variables ξ_{ij}, η_{ij} are positive-definite, i.e., ∃M > 0

a_{ijlk}^{(q)}ξ_{ij}ξ_{lk}+2b_{ijlk}^{(q)}ξ_{ij}η_{lk}+c_{ijlk}^{(q)}η_{ij}η_{lk} ≥M(ξ_{ij}ξ_{ij}+η_{lk}η_{lk}) for all ξ_{ij}, η_{lk}, q =1, 2. (1.3)

We introduce the differential stress operator

N^{(q)}(∂_{z}, n(z)) = ( \begin{matrix}N^{1} ^{(q)}(∂_{z}, n(z)) & N^{2} ^{(q)}(∂_{z}, n(z)) \\ N^{3} ^{(q)}(∂_{z}, n(z)) & N^{4} ^{(q)}(∂_{z}, n(z)) \end{matrix} )_{6×6} ,

N^{l} ^{(q)}(∂_{z}, n(z)) = ∥ N^{l} _{jk}^{(q)}(∂_{z}, n(z))∥_{3×3}, l = \overline{1, 4}, q = 1, 2,

N^{1} _{jk}^{(q)}(∂_{z}, n(z)) = a_{ijlk}^{(q)}n_{i}(z)∂_{l}, N^{2} _{jk}^{(q)}(∂_{z}, n(z)) = b_{ijlk}^{(q)}n_{i}(z)∂_{l} − a_{ijlk}^{(q)}ε_{lrk}n_{i}(z),

N^{3} _{jk}^{(q)}(∂_{z}, n(z)) = b_{lkij}^{(q)}n_{i}(z)∂_{l}, N^{4} _{jk}^{(q)}(∂_{z}, n(z)) = c_{ijlk}^{(q)}n_{i}(z)∂_{l} − b_{lrij}^{(q)}ε_{lrk}n_{i}(z),

where n(z) = (n_{1}(z), n_{2}(z), n_{3}(z)) is the unit normal of the manifold ∂D_{1} at a point z ∈ ∂D_{1} (external with respect to D_{1}) and a point z ∈ ∂D_{2} (internal with respect to D_{2}).

In what follows the stress operators are denoted by

N^{(q)} = N^{(q)}(∂_{z}, n(z)), q = 1, 2.

Let M^{(q)}(ξ) be the symbol of the differential operator M^{(q)}(∂_{x}) and M^{∘}^{(q)}(ξ) be the principal homogeneous symbol of the differential operator M^{(q)}(∂_{x}).

We introduce the following notation for Besov and Bessel potential spaces (see [39]):

𝔹_{p,r}^{s} = B_{p,r}^{s} × B_{p,r}^{s}, 𝔹̃_{p,r}^{s} = B̃_{p,r}^{s} × B̃_{p,r}^{s}, ℍ_{p}^{s} = H_{p}^{s} × H_{p}^{s}, ℍ̃_{p}^{s} = H̃_{p}^{s} × H̃_{p}^{s}.

From the symmetry of the coefficients a_{ijlk}^{(q)}, c_{ijlk}^{(q)} and the positive-definiteness of the quadratic forms (1.3) it follows (see [19]) that the operators M^{(q)}(∂_{x}), q = 1, 2, are strongly elliptic, formally self-adjoint differential operators and therefore for any real vector ξ ∈ ℝ^{3} and any complex vector η ∈ ℂ^{6} the relations

Re(M^{∘}^{(q)}(ξ)η, η) = (M^{∘}^{(q)}(ξ)η, η) ≥ P_{0}^{(q)}|ξ|^{2}|η|^{2}

are valid, where P_{0}^{(q)} = const > 0 depends only on the elastic constants. Thus the matrices M^{∘}^{(q)}(ξ) are positive-definite for ξ ∈ ℝ^{3} \ {0}.

Taking into account the property of the Levi-Civita symbol

ε_{ipj}ε_{lrk} = det ( \begin{matrix}δ_{il} & δ_{ir} & δ_{ik} \\ δ_{pl} & δ_{pr} & δ_{pk} \\ δ_{jl} & δ_{jr} & δ_{jk} \end{matrix} ) (δ_{pl} is the Kronecker symbol),

it is not difficult to observe that

(M^{(q)}(ξ)η, η) = (M^{∘}^{(q)}(ξ)η, η), q = 1, 2.

Then we obtain that the matrices M^{(q)}(ξ), q = 1, 2, are positive-definite for ξ ∈ ℝ^{3} \ {0}.

Since

detM^{(q)}(ξ) ≠ 0 for ξ ≠ 0.

Let U^{(1)} ∈ W_{p}^{1}(D_{1}), U^{(2)} ∈ W_{p,loc}^{1}(D_{2}). Then r_{1}U^{(2)} = r_{D_{2}^{(1)}}U^{(2)} ∈ W_{p}^{1}(D_{2}^{(1)})

and r_{2}U^{(2)} = r_{D_{2}^{(2)}}U^{(2)} ∈ W_{p,loc}^{1}(D_{2}^{(2)}), where r_{i} is the restriction operator on

D_{2}^{(i)}, i = 1, 2. From the theorem on traces (see [39]) it follows that the trace of the functions U^{(i)}, r_{i}U^{(2)}, i = 1, 2, exists on ∂D_{1}, ∂D_{2}^{(i)}, i = 1, 2, and {U^{(1)}}^{±} ∈ 𝔹_{p,p}^{1/p′}(∂D_{1}), {r_{i}U^{(2)}}^{±} ∈ 𝔹_{p,p}^{1/p′}(∂D_{2}^{(i)}), i = 1, 2, p′ = p/(p − 1). Let U^{(1)} ∈ W_{p}^{1}(D_{1}), U^{(2)} ∈ W_{p,loc}^{1}(D_{2}) be such that M^{(1)}(∂_{x})U^{(1)} ∈ L_{p}(D_{1}), M^{(2)}(∂_{x})U^{(2)} ∈ L_{p,comp}(D_{2}). Then {N^{(1)}U^{(1)}}^{±} and {N^{(2)}(r_{i}U^{(2)})}^{±}, i = 1, 2, are correctly defined by the equalities

∫_{D_{1}} [\overline{V}^{(1)}M^{(1)}(∂_{x})U^{(1)} + E^{(1)}(U^{(1)}, V^{(1)})]dx
= ±⟨{N^{(1)}U^{(1)}}^{±}, {V^{(1)}}^{±}⟩_{∂D_{1}} for all V^{(1)} ∈ W_{p′}^{1}(D_{1})  (1.4)

and

∫_{D_{2}^{(i)}} [\overline{V}_{2}^{(i)}M^{(2)}(∂_{x})(r_{i}U^{(2)}) + E^{(2)}(r_{i}U^{(2)}, V_{2}^{(i)})]dx
= ∓⟨{N^{(2)}(r_{i}U^{(2)})}^{±}, {V_{2}^{(i)}}^{±}⟩_{∂D_{2}^{(i)}}, i = 1, 2,  (1.5)
for all V_{2}^{(1)} ∈ W_{p′}^{1}(D_{2}^{(1)}) (V_{2}^{(2)} ∈ W_{p,comp}^{1}(D_{2}^{(2)})),

where

E^{(1)}(U^{(1)}, V^{(1)}) = a_{ijlk}^{(1)}ξ_{ij}(U^{(1)})ξ_{lk}(\overline{V}^{(1)}) + b_{ijlk}^{(1)}ξ_{ij}(U^{(1)})η_{lk}(\overline{V}^{(1)})
+ b_{ijlk}^{(1)}ξ_{ij}(\overline{V}^{(1)})η_{lk}(U^{(1)}) + c_{ijlk}^{(1)}η_{ij}(U^{(1)})η_{lk}(\overline{V}^{(1)});

here U^{(1)} = (u^{(1)}, ω^{(1)}),

ξ_{ij}(U^{(1)}) = ∂_{j}u_{i}^{(1)} − ε_{ijk}ω_{k} is the deformation component,
η_{ij}(U^{(1)}) = ∂_{j}ω_{i} is the bending torsion component.

The quadratic form E^{(2)}(r_{i}U^{(2)}, V_{2}^{(i)}), i = 1, 2, is defined analogously. In the case of the infinite domain D_{2}, for solving equation (1.1) the condition

U^{(2)}(x) = o(1) for |x| → ∞  (1.6)

is assumed to be fulfilled at infinity (see [4]).

We can prove (see [4]) that

∂^{μ}U^{(2)}(x) = O(|x|^{−1−|μ|}) for |x| → ∞

is valid for any solution of equation (1.1) satisfying (1.6) the relation. Let us consider the model problems M_{1} and M_{2}. We will study the solvability and asymptotics of solutions U^{(q)} ∈ W_{p}^{1}(D_{q}), q =

1, 2, (U^{(2)} ∈ W_{p,loc}^{1}(D_{2}) with condition (1.6) at infinity) the following boundarycontact problems of couple-stress elasticity:

Problem M_{1}:

M^{(q)}(∂_{x})U^{(q)} = 0  in D_{q}, q = 1, 2,
π_{S_{1}}{U^{(1)}}^{+} = φ_{1}  on S_{1},
π_{S_{2}^{(1)}}{N^{(2)}(r_{1}U^{(2)})}^{+} = φ_{2}  on S_{2}^{(1)},
π_{S_{2}^{(2)}}{N^{(2)}(r_{2}U^{(2)})}^{+} = φ_{3}  on S_{2}^{(2)},
π_{S_{0}^{(i)}}{U^{(1)}}^{+} − π_{S_{0}^{(i)}}{r_{i}U^{(2)}}^{+} = f_{i}  on S_{0}^{(i)},
π_{S_{0}^{(i)}}{N^{(1)}U^{(1)}}^{+} − π_{S_{0}^{(i)}}{N^{(2)}(r_{i}U^{(2)})}^{+} = h_{i} on S_{0}^{(i)}, i = 1, 2,

where

φ_{1} ∈ 𝔹_{p,p}^{1/p′}(S_{1}), φ_{2} ∈ 𝔹_{p,p}^{−1/p}(S_{2}^{(1)}), φ_{3} ∈ 𝔹_{p,p}^{−1/p}(S_{2}^{(2)}),

f_{i} ∈ 𝔹_{p,p}^{1/p′}(S_{0}^{(i)}), h_{i} ∈ 𝔹_{p,p}^{−1/p}(S_{0}^{(i)}), i = 1, 2, 1 < p < ∞, p′ = p/(p − 1).

If D_{2} is a finite domain, then we have the following wedge-type problem:

Wedge-type Problem M_{2}:

M^{(q)}(∂_{x})U^{(q)} = 0  in D_{q}, q = 1, 2,
π_{S_{1}}{N^{(1)}U^{(1)}}^{+} = φ_{1}  on S_{1},
π_{S_{2}^{(1)}}{N^{(2)}(r_{1}U^{(2)})}^{+} = φ_{2}  on S_{2}^{(1)},
π_{S_{2}^{(2)}}{N^{(2)}(r_{2}U^{(2)})}^{+} = φ_{3}  on S_{2}^{(2)},
π_{S_{0}^{(i)}}{U^{(1)}}^{+} − π_{S_{0}^{(i)}}{r_{i}U^{(2)}}^{+} = f_{i}  on S_{0}^{(i)},
π_{S_{0}^{(i)}}{N^{(1)}U^{(1)}}^{+} − π_{S_{0}^{(i)}}{N^{(2)}(r_{i}U^{(2)})}^{+} = h_{i} on S_{0}^{(i)}, i = 1, 2,

where

φ_{1} ∈ 𝔹_{p,p}^{1/p′}(S_{1}), φ_{2} ∈ 𝔹_{p,p}^{1/p′}(S_{2}^{(1)}), φ_{3} ∈ 𝔹_{p,p}^{1/p′}(S_{2}^{(2)}),

f_{i} ∈ 𝔹_{p,p}^{−1/p}(S_{0}^{(i)}), h_{i} ∈ 𝔹_{p,p}^{1/p′}(S_{0}^{(i)}), i = 1, 2, 1 < p < ∞.

2. Fundamental Solutions and Potentials

Consider the fundamental matrix-functions

H^{(q)}(x) = F_{ξ′→x′}^{−1}( ± ∫_{L_{±}} (M^{(q)}(iξ′, iτ))^{−1}e^{−iτx_{3}}dτ), q = 1, 2,

where the sign “−” refers to the case x_{3} > 0 and the sign “+” to the case x_{3} < 0; x = (x_{1}, x_{2}, x_{3}), x′ = (x_{1}, x_{2}), ξ′ = (ξ_{1}, ξ_{2}); ∫_{L_{±}} denotes integration over the contour L^{±}, where L_{+} (L_{−}) has the positive orientation and covers all roots of the polynomial detM^{(q)}(iξ′, iτ) with respect to τ in the upper (resp. lower) τ-half-plane. F^{−1} is the inverse Fourier transform.

The simple-layer potentials are of the form

𝕍^{(1)}(g_{1})(x) = ∫_{∂D_{1}} H^{(1)}(x − y)g_{1}(y)d_{y}S, x ∉ ∂D_{1},
𝕍^{(2)}(g_{2})(x) = ∫_{∂D_{2}^{(1)}} H^{(2)}(x − y)g_{2}(y)d_{y}S, x ∉ ∂D_{2}^{(1)},
𝕍^{(3)}(g_{3})(x) = ∫_{∂D_{2}^{(2)}} H^{(2)}(x − y)g_{3}(y)d_{y}S, x ∉ ∂D_{2}^{(2)}.

For these potentials the theorems below are valid.

Theorem 2.1. Let 1 < p < ∞, 1 ≤ r ≤ ∞. Then the operators 𝕍^{(i)}, i = 1, 2, 3, admit extensions to the operators which are continuous in the following spaces:

𝕍^{(1)} : 𝔹_{p,r}^{s}(∂D_{1}) → 𝔹_{p,r}^{s+1+1/p}(D_{1}) (𝔹_{p,p}^{s}(∂D_{1}) → ℍ_{p}^{s+1+1/p}(D_{1})),
𝕍^{(2)} : 𝔹_{p,r}^{s}(∂D_{2}^{(1)}) → 𝔹_{p,r}^{s+1+1/p}(D_{2}^{(1)}) (𝔹_{p,p}^{s}(∂D_{2}^{(1)}) → ℍ_{p}^{s+1+1/p}(D_{2}^{(1)})),
𝕍^{(3)} : 𝔹_{p,r}^{s}(∂D_{2}^{(2)}) → 𝔹_{p,r,loc}^{s+1+1/p}(D_{2}^{(2)}) (𝔹_{p,p}^{s}(∂D_{2}^{(2)}) → ℍ_{p,loc}^{s+1+1/p}(D_{2}^{(2)})).

SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS 433

Theorem 2.2. Let 1 < p < ∞, 1 ≤ r ≤ ∞, ε > 0, g_{1} ∈ 𝔹_{p,r}^{−1+ε}(∂D_{1}), g_{2} ∈ 𝔹_{p,r}^{−1+ε}(∂D_{2}^{(1)}), g_{3} ∈ 𝔹_{p,r}^{−1+ε}(∂D_{2}^{(2)}). Then

{𝕍^{(1)}(g_{1})(z)}^{±} = ∫_{∂D_{1}} H^{(1)}(z − y)g_{1}(y)d_{y}S, z ∈ ∂D_{1},

{𝕍^{(2)}(g_{2})(z)}^{±} = ∫_{∂D_{2}^{(1)}} H^{(2)}(z − y)g_{2}(y)d_{y}S, z ∈ ∂D_{2}^{(1)},
{𝕍^{(3)}(g_{3})(z)}^{±} = ∫_{∂D_{2}^{(2)}} H^{(2)}(z − y)g_{3}(y)d_{y}S, z ∈ ∂D_{2}^{(2)}.

Theorem 2.3. Let 1 < p < ∞, g_{1} ∈ 𝔹_{p,p}^{−1/p}(∂D_{1}), g_{2} ∈ 𝔹_{p,p}^{−1/p}(∂D_{2}^{(1)}), g_{3} ∈ 𝔹_{p,p}^{−1/p}(∂D_{2}^{(2)}). Then

\{N^{(1)}𝕍^{(1)}(g_{1})(z)\}^{±} = ∓\frac{1}{2} g_{1}(z)

+ ∫_{∂D_{1}} N^{(1)}(∂_{z}, n(z))H^{(1)}(z − y)g_{1}(y)d_{y}S, z ∈∂D_{1},

\{N^{(2)}𝕍^{(2)}(g_{2})(z)\}^{±} = ±\frac{1}{2} g_{2}(z)

+ ∫_{∂D_{2}^{(1)}} N^{(2)}(∂_{z}, n(z))H^{(2)}(z − y)g_{2}(y)d_{y}S, z ∈ ∂D_{2}^{(1)},

\{N^{(2)}𝕍^{(3)}(g_{3})(z)\}^{±} = ±\frac{1}{2} g_{3}(z)

+ ∫_{∂D_{2}^{(2)}} N^{(2)}(∂_{z}, n(z))H^{(2)}(z − y)g_{3}(y)d_{y}S, z ∈ ∂D_{2}^{(2)}.

Let us introduce the following notation: 𝕍_{−1}^{(1)}(g_{1})(z) = ∫_{∂D_{1}} H^{(1)}(z − y)g_{1}(y)d_{y}S, z ∈ ∂D_{1}, 𝕍_{−1}^{(2)}(g_{2})(z) = ∫_{∂D_{2}^{(1)}} H^{(2)}(z − y)g_{2}(y)d_{y}S, z ∈ ∂D_{2}^{(1)}, 𝕍_{−1}^{(3)}(g_{3})(z) = ∫_{∂D_{2}^{(2)}} H^{(2)}(z − y)g_{3}(y)d_{y}S, z ∈ ∂D_{2}^{(2)},

𝕍^{∗} _{0}^{(1)}(g_{1})(z) = ∫_{∂D_{1}} N^{(1)}(∂_{z}, n(z))H^{(1)}(z − y)g_{1}(y)d_{y}S, z ∈ ∂D_{1},

𝕍^{∗} _{0}^{(2)}(g_{2})(z) = ∫_{∂D_{2}^{(1)}} N^{(2)}(∂_{z}, n(z))H^{(2)}(z − y)g_{2}(y)d_{y}S, z ∈ ∂D_{2}^{(1)},

𝕍^{∗} _{0}^{(3)}(g_{3})(z) = ∫_{∂D_{2}^{(2)}} N^{(2)}(∂_{z}, n(z))H^{(2)}(z − y)g_{3}(y)d_{y}S, z ∈ ∂D_{2}^{(2)}.

Theorem 2.4. Let 1 < p < ∞, 1 ≤ r ≤ ∞. Then the operators 𝕍_{−1}^{(i)}, 𝕍^{∗} _{0}^{(i)}, i = 1, 2, 3, admit extensions to the operators which are continuous in the following spaces:

𝕍_{−1}^{(i)} : ℍ_{p}^{s}(∂Ω_{i}) → ℍ_{p}^{s+1}(∂Ω_{i})
(𝔹_{p,r}^{s}(∂Ω_{i}) → 𝔹_{p,r}^{s+1}(∂Ω_{i})), i = 1, 2, 3,
𝕍^{∗} _{0}^{(i)} : ℍ_{p}^{s}(∂Ω_{i}) → ℍ_{p}^{s}(∂Ω_{i})
(𝔹_{p,r}^{s}(∂Ω_{i}) → 𝔹_{p,r}^{s}(∂Ω_{i})), i = 1, 2, 3,

here Ω_{1} = D_{1}, Ω_{2} = D_{2}^{(1)}, Ω_{3} = D_{2}^{(2)}.

3. Uniqueness, Existence and Smoothness Theorems for Problem M_{1}

From the ellipticity of the differential operator M^{(2)}(∂_{x}) it follows that any generalized solution of the equation

M^{(2)}(∂_{x})U^{(2)} = 0 in D_{2}

is an analytic function in D_{2} (see [17]). Then we see that the equalities

\{ \begin{matrix}\{r_{1}U^{(2)}\}^{+} − \{r_{2}U^{(2)}\}^{+} = 0 & on S_{0}, \\ \{N^{(2)}(r_{1}U^{(2)})\}^{+} + \{N^{(2)}(r_{2}U^{(2)})\}^{+} = 0 & on S_{0} \end{matrix}  (3.1)

are valid on S_{0}.

Let us study the uniqueness of a solution of the boundary-contact problem M_{1} in the classes W_{2}^{1}(D_{q}), q = 1, 2 (W_{2,loc}^{1}(D_{2}) with condition (1.6) at infinity).

Lemma 3.1. A solution of the boundary-contact problem M_{1} is unique in the classes W_{2}^{1}(D_{q}), q = 1, 2 (W_{2,loc}^{1}(D_{2}) and satisfying condition (1.6) at infinity).

Proof. Let U^{(q)}, q = 1, 2, be a solution of the homogeneous problem M_{1}.

We write the Green formulae (see (1.4), (1.5)) in the domains D_{1}, D_{2}^{(1)}, D_{2}^{(2)} for the vector-functions U^{(1)}, r_{1}U^{(2)}, r_{2}U^{(2)} as

\begin{matrix}∫_{D_{1}} E^{(1)}(U^{(1)}, U^{(1)})dx = ⟨\{N^{(1)}U^{(1)}\}^{+}, \{U^{(1)}\}^{+}⟩_{∂D_{1}}, \\ ∫_{D_{2}^{(i)}} E^{(2)}(r_{i}U^{(2)}, r_{i}U^{(2)})dx= −⟨\{N^{(2)}(r_{i}U^{(2)})\}^{+}, \{r_{i}U^{(2)}\}^{+}⟩_{∂D_{2}^{(i)}}, i=1, 2. \end{matrix} (3.2)

Taking into account the boundary and boundary-contact conditions, the Green formulas (3.2) can be rewritten as

\begin{matrix}∫_{D_{1}} E^{(1)}(U^{(1)}, U^{(1)})dx = ⟨\{N^{(1)}U^{(1)}\}^{+}, \{U^{(1)}\}^{+}⟩_{S_{0}^{(1)}∪S_{0}^{(2)}}, \\ ∫_{D_{2}^{(i)}}E^{(2)}(r_{i}U^{(2)}, r_{i}U^{(2)})dx= −⟨\{N^{(2)}(r_{i}U^{(2)})\}^{+}, \{r_{i}U^{(2)}\}^{+}⟩_{S_{0}^{(i)}∪S_{0}}, i=1, 2. \end{matrix} (3.3)

Since equalities (3.1) are fulfilled for the function U^{(2)}, we have

⟨{N^{(2)}(r_{1}U^{(2)})}^{+}, {r_{1}U^{(2)}}^{+}⟩_{S_{0}} = −⟨{N^{(2)}(r_{2}U^{(2)})}^{+}, {r_{2}U^{(2)}}^{+}⟩_{S_{0}}. (3.4)

Summing now the Green formulas (3.3) and taking into account (3.4), we obtain

∫_{D_{1}} E^{(1)}(U^{(1)}, U^{(1)})dx + ∫_{D_{2}^{(1)}} E^{(2)}(r_{1}U^{(2)}, r_{1}U^{(2)})dx
+ ∫_{D_{2}^{(2)}} E^{(2)}(r_{2}U^{(2)}, r_{2}U^{(2)})dx = 0.  (3.5)

From equality (3.5) and the positive-definiteness of forms (1.3) we have

\{ \begin{matrix}∂_{j}u_{i}^{(q)} − ε_{ijk}ω_{k}^{(q)} = 0, \\ ∂_{j}ω_{i}^{(q)} = 0, q = 1, 2. \end{matrix}

Therefore

u^{(q)} = [a^{(1)} × x] + b^{(q)}, ω^{(q)} = a^{(q)}, q = 1, 2,

and

U^{(q)} = ([a^{(q)} × x] + b^{(q)}, a^{(q)}), q = 1, 2,

where a^{(q)} and b^{(q)}, q = 1, 2, are arbitrary three-dimensional constant vectors. By virtue of the contact conditions it is clear that

a^{(1)} = a^{(2)} and b^{(1)} = b^{(2)}.

Since {U^{(1)}}^{+} = 0 on S_{1}, we have

U^{(q)}(x) = 0, x ∈ D_{q}, q = 1, 2. □

Any extension Φ^{(1)} ∈ 𝔹_{p,p}^{1/p′}(∂D_{1}) of the function φ_{1} onto the whole boundary ∂D_{1} has the form

Φ^{(1)} = Φ_{0}^{(1)} + φ_{0}^{(1)} + ψ_{0}^{(1)},

where Φ_{0}^{(1)} is some fixed extension of φ_{1} and φ_{0}^{(1)} ∈ 𝔹̃_{p,p}^{1/p′}(S_{0}^{(1)}), ψ_{0}^{(1)} ∈ 𝔹̃_{p,p}^{1/p′}(S_{0}^{(2)}).

Any extension Φ^{(2)} ∈ 𝔹_{p,p}^{−1/p}(∂D_{2}^{(1)}) of the function φ_{2} onto the whole boundary ∂D_{2}^{(1)} has the form

Φ^{(2)} = Φ_{0}^{(2)} + φ_{0}^{(2)} + ψ_{0}^{(2)},

where Φ_{0}^{(2)} is some fixed extension of φ_{2} and φ_{0}^{(2)} ∈ 𝔹̃_{p,p}^{−1/p}(S_{0}^{(1)}), ψ_{0}^{(2)} ∈ 𝔹̃_{p,p}^{−1/p}(S_{0}).

Any extension Φ^{(3)} ∈ 𝔹_{p,p}^{−1/p}(∂D_{2}^{(2)}) of the function φ_{3} onto the whole boundary ∂D_{2}^{(2)} has the form

Φ^{(3)} = Φ_{0}^{(3)} + φ_{0}^{(3)} + ψ_{0}^{(3)},

where Φ_{0}^{(3)} is some fixed extension of φ_{3} and φ_{0}^{(3)} ∈ 𝔹̃_{p,p}^{−1/p}(S_{0}^{(2)}), ψ_{0}^{(3)} ∈ 𝔹̃_{p,p}^{−1/p}(S_{0}).

A solution of the boundary-contact problem M_{1} will be sought for in the form of the simple-layer potentials

U^{(1)} = 𝕍^{(1)}g_{1} in D_{1}, U^{(2)} = \{ \begin{matrix}𝕍^{(2)}g_{2} & in D_{2}^{(1)}, \\ 𝕍^{(3)}g_{3} & in D_{2}^{(2)}. \end{matrix}

Taking into account the boundary and boundary-contact conditions of problem M_{1} and equality (3.1) we obtain a system of equations with respect to (g_{1}, g_{2}, g_{3}, φ_{0}^{(1)}, ψ_{0}^{(1)}, φ_{0}^{(2)}, ψ_{0}^{(2)}, φ_{0}^{(3)}, ψ_{0}^{(3)}):

\{ \begin{matrix}𝕍_{−1}^{(1)}g_{1} − φ_{0}^{(1)} − ψ_{0}^{(1)} = Φ_{0}^{(1)} & on ∂D_{1}, \\ (\frac{1}{2}I+ 𝕍^{∗} _{0}^{(2)})g_{2} − φ_{0}^{(2)} − ψ_{0}^{(2)} = Φ_{0}^{(2)} & on ∂D_{2}^{(1)}, \\ (\frac{1}{2}I+ 𝕍^{∗} _{0}^{(3)})g_{3} − φ_{0}^{(3)} − ψ_{0}^{(3)} = Φ_{0}^{(3)} & on ∂D_{2}^{(2)}, \\ −π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}g_{2} + φ_{0}^{(1)} = f_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)} & on S_{0}^{(1)}, \\ π_{S_{0}^{(1)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})g_{1} − φ_{0}^{(2)} = h_{1} + π_{S_{0}^{(1)}}Φ_{0}^{(2)} & on S_{0}^{(1)}, \\ −π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}g_{3} + ψ_{0}^{(1)} = f_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)} & on S_{0}^{(2)}, \\ π_{S_{0}^{(2)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})g_{1} − φ_{0}^{(3)} = h_{2} + π_{S_{0}^{(2)}}Φ_{0}^{(3)} & on S_{0}^{(2)}, \\ π_{S_{0}}𝕍_{−1}^{(2)}g_{2} − π_{S_{0}}𝕍_{−1}^{(3)}g_{3} = 0 & on S_{0}, \\ ψ_{0}^{(2)} + ψ_{0}^{(3)} = −π_{S_{0}}Φ_{0}^{(2)} − π_{S_{0}}Φ_{0}^{(3)} & on S_{0}. \end{matrix}  (3.6)

It is almost obvious that system (3.6) has a solution if and only if the compatibility conditions on ∂S_{0}

∃Φ_{0}^{(2)} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(1)}), Φ_{0}^{(3)} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(2)}) : π_{S_{0}}Φ_{0}^{(2)}+π_{S_{0}}Φ_{0}^{(3)} ∈ 𝔹̃_{p,r}^{s−1}(S_{0}) (3.7)

hold for φ_{2} ∈ 𝔹_{p,r}^{s−1}(S_{2}^{(1)}), φ_{3} ∈ 𝔹_{p,r}^{s−1}(S_{2}^{(2)}), π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)} ∈ 𝔹_{p,r}^{s−1}(S_{0}), 1 ≤ r ≤ ∞, 1 < p < ∞, 1/p − 1/2 < s < 1/p + 1/2.

Note that these conditions hold automatically when 1/p − 1/2 < s < 1/p or 1/p < s < 1/p + 1/2 (see [39]).

Denote by A the operator corresponding to system (3.6) and acting in the spaces

A : H^{(1)} _{p}^{s} →H^{(2)} _{p}^{s} ( B^{(1)} _{p,r}^{s} →B^{(2)} _{p,r}^{s}),

where

H^{(1)} _{p}^{s} = ℍ_{p}^{s−1}(∂D_{1}) ⊕ ℍ_{p}^{s−1}(∂D_{2}^{(1)}) ⊕ ℍ_{p}^{s−1}(∂D_{2}^{(2)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(1)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(2)})
⊕ℍ̃_{p}^{s−1}(S_{0}^{(1)}) ⊕ ℍ̃_{p}^{s−1}(S_{0}) ⊕ ℍ̃_{p}^{s−1}(S_{0}^{(2)}) ⊕ ℍ̃_{p}^{s−1}(S_{0}),
H^{(2)} _{p}^{s} = ℍ_{p}^{s}(∂D_{1}) ⊕ ℍ_{p}^{s−1}(∂D_{2}^{(1)}) ⊕ ℍ_{p}^{s−1}(∂D_{2}^{(2)}) ⊕ ℍ_{p}^{s}(S_{0}^{(1)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(1)})
⊕ℍ_{p}^{s}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s}(S_{0}) ⊕ ℍ_{p}^{s−1}(S_{0}),
B^{(1)} _{p,r}^{s} = 𝔹_{p,r}^{s−1}(∂D_{1}) ⊕ 𝔹_{p,r}^{s−1}(∂D_{2}^{(1)}) ⊕ 𝔹_{p,r}^{s−1}(∂D_{2}^{(2)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(1)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(2)})
⊕𝔹̃_{p,r}^{s−1}(S_{0}^{(1)}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}^{(2)}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}),
B^{(2)} _{p,r}^{s} = 𝔹_{p,r}^{s}(∂D_{1}) ⊕ 𝔹_{p,r}^{s−1}(∂D_{2}^{(1)}) ⊕ 𝔹_{p,r}^{s−1}(∂D_{2}^{(2)}) ⊕ 𝔹_{p,r}^{s}(S_{0}^{(1)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(1)})
⊕𝔹_{p,r}^{s}(S_{0}^{(2)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(2)}) ⊕ 𝔹_{p,r}^{s}(S_{0}) ⊕ 𝔹_{p,r}^{s−1}(S_{0});

the symbol ⊕ denotes a direct sum of the spaces.

Consider the composition of the operators

D ∘ A,

where D is the invertible operator of the form

D = diag{I, −𝕍_{−1}^{(2)}, −𝕍_{−1}^{(3)}, I, …, I}_{54×54}.

Consider now the operator

A_{M} = T_{M} + D ∘ A, M = 2, 3, …,

where

T_{M} = diag{0, (−𝕍_{−1}^{(2)})^{M}, (−𝕍_{−1}^{(3)})^{M}, 0, …, 0}_{54×54}.

Since the operator A_{M} differs from the operator D ∘ A in a compact operator, it is sufficient to investigate the operator A_{M} acting in the spaces

A_{M} : H^{(1)} _{p}^{s} →H^{(3)} _{p}^{s} ( B^{(1)} _{p,r}^{s} →B^{(3)} _{p,r}^{s}),

where

H^{(3)} _{p}^{s} = ℍ_{p}^{s}(∂D_{1}) ⊕ ℍ_{p}^{s}(∂D_{2}^{(1)}) ⊕ ℍ_{p}^{s}(∂D_{2}^{(2)}) ⊕ ℍ_{p}^{s}(S_{0}^{(1)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(1)})
⊕ℍ_{p}^{s}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s}(S_{0}) ⊕ ℍ_{p}^{s−1}(S_{0}),
B^{(3)} _{p,r}^{s} = 𝔹_{p,r}^{s}(∂D_{1}) ⊕ 𝔹_{p,r}^{s}(∂D_{2}^{(1)}) ⊕ 𝔹_{p,r}^{s}(∂D_{2}^{(2)}) ⊕ 𝔹_{p,r}^{s}(S_{0}^{(1)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(1)})
⊕𝔹_{p,r}^{s}(S_{0}^{(2)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(2)}) ⊕ 𝔹_{p,r}^{s}(S_{0}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}).

Now consider the system of equations that corresponds to the operator A_{M} given by

\{ \begin{matrix}𝕍_{−1}^{(1)}g̃_{1} − φ̃_{0}^{(1)} − ψ̃_{0}^{(1)} = Ψ_{0}^{(1)} & on ∂D_{1}, \\ [(−𝕍_{−1}^{(2)})^{M} −𝕍_{−1}^{(2)}(\frac{1}{2}I+ 𝕍^{∗} _{0}^{(2)})]g̃_{2}+𝕍_{−1}^{(2)}φ̃_{0}^{(2)}+𝕍_{−1}^{(2)}ψ̃_{0}^{(2)} =Ψ_{0}^{(2)} & on ∂D_{2}^{(1)}, \\ (−𝕍_{−1}^{(3)})^{M} −𝕍_{−1}^{(3)}(\frac{1}{2}I+ 𝕍^{∗} _{0}^{(3)})g̃_{3}+𝕍_{−1}^{(3)}φ̃_{0}^{(3)}+𝕍_{−1}^{(3)}ψ̃_{0}^{(3)} =Ψ_{0}^{(3)} & on ∂D_{2}^{(2)}, \\ −π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}g̃_{2} + φ̃_{0}^{(1)} = G_{1} & on S_{0}^{(1)}, \\ π_{S_{0}^{(1)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})g̃_{1} − φ̃_{0}^{(2)} = G_{2} & on S_{0}^{(1)}, \\ −π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}g̃_{3} + ψ̃_{0}^{(1)} = F_{1} & on S_{0}^{(2)}, \\ π_{S_{0}^{(2)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})g̃_{1} − φ̃_{0}^{(3)} = F_{2} & on S_{0}^{(2)}, \\ π_{S_{0}}𝕍_{−1}^{(2)}g̃_{2} − π_{S_{0}}𝕍_{−1}^{(3)}g̃_{3} = E_{1} & on S_{0}, \\ ψ̃_{0}^{(2)} + ψ̃_{0}^{(3)} = E_{2} & on S_{0}, \end{matrix} (3.8)

where

Ψ_{0}^{(1)} ∈ ℍ_{p}^{s}(∂D_{1}) (𝔹_{p,r}^{s}(∂D_{1})), Ψ_{0}^{(2)} ∈ ℍ_{p}^{s}(∂D_{2}^{(1)}) (𝔹_{p,r}^{s}(∂D_{2}^{(1)})),
Ψ_{0}^{(3)} ∈ ℍ_{p}^{s}(∂D_{2}^{(2)}) (𝔹_{p,r}^{s}(∂D_{2}^{(2)})),

G_{1} ∈ ℍ_{p}^{s}(S_{0}^{(1)}) (𝔹_{p,r}^{s}(S_{0}^{(1)})), G_{2} ∈ ℍ_{p}^{s−1}(S_{0}^{(1)}) (𝔹_{p,r}^{s−1}(S_{0}^{(1)})),
F_{1} ∈ ℍ_{p}^{s}(S_{0}^{(2)}) (𝔹_{p,r}^{s}(S_{0}^{(2)})), F_{2} ∈ ℍ_{p}^{s−1}(S_{0}^{(2)}) (𝔹_{p,r}^{s−1}(S_{0}^{(2)})),
E_{1} ∈ ℍ_{p}^{s}(S_{0}) (𝔹_{p,r}^{s}(S_{0})), E_{2} ∈ ℍ_{p}^{s−1}(S_{0}) (𝔹_{p,r}^{s−1}(S_{0})).

The ΨDO −𝕍_{−1}^{(1)} is positive and the operators −𝕍_{−1}^{(i)}(\frac{1}{2}I+ 𝕍^{∗} _{0}^{(i)}), i = 2, 3, are nonnegative, i.e.,

⟨−𝕍_{−1}^{(1)}φ, φ⟩_{∂D_{1}} > 0 for all φ ∈ ℍ_{2}^{−1/2}(∂D_{1}), φ ≠ 0,

⟨ − 𝕍_{−1}^{(2)}(\frac{1}{2}I+ 𝕍^{∗} _{0}^{(2)})ψ, ψ⟩_{∂D_{2}^{(1)}} ≥ 0 for all ψ ∈ ℍ_{2}^{−1/2}(∂D_{2}^{(1)})

and

⟨ − 𝕍_{−1}^{(3)}(\frac{1}{2}I+ 𝕍^{∗} _{0}^{(3)})ψ, ψ⟩_{∂D_{2}^{(2)}} > 0 for all ψ ∈ ℍ_{2}^{−1/2}(∂D_{2}^{(2)}),

the equality being fulfilled only when ψ = ([a × x] + b, a), where a and b are arbitrary three-dimensional constant vectors.

The proof of these inequalities follows from the Green formulae (see [32]). Then the ΨDOs

B_{M}^{(i)} = (−𝕍_{−1}^{(i)})^{M} − 𝕍_{−1}^{(i)}(\frac{1}{2} I+ 𝕍^{∗} _{0}^{(i)}), i = 2, 3,

are positive operators, i.e.,

⟨B_{M}^{(2)}φ, φ⟩_{∂D_{2}^{(1)}} > 0 for all ψ ∈ ℍ_{2}^{−1/2}(∂D_{2}^{(1)}), φ ≠ 0,
⟨B_{M}^{(3)}ψ, ψ⟩_{∂D_{2}^{(2)}} > 0 for all ψ ∈ ℍ_{2}^{−1/2}(∂D_{2}^{(2)}), ψ ≠ 0.

Hence the ΨDOs 𝕍_{−1}^{(1)} and B_{M}^{(i)}, i = 2, 3, are invertible (which is proved as in [32], [7]). The first, second and third equations of system (3.8) imply

g̃_{1} = (𝕍_{−1}^{(1)})^{−1}φ̃_{0}^{(1)} + (𝕍_{−1}^{(1)})^{−1}ψ̃_{0}^{(1)} + (𝕍_{−1}^{(1)})^{−1}Ψ_{0}^{(1)},
g̃_{i} = −(B_{M}^{(i)})^{−1}𝕍_{−1}^{(i)}φ̃_{0}^{(i)} − (B_{M}^{(i)})^{−1}𝕍_{−1}^{(i)}ψ̃_{0}^{(i)} + (B_{M}^{(i)})^{−1}Ψ_{0}^{(i)}, i = 2, 3.

After substituting g̃_{1}, g̃_{2}, g̃_{3} into the remaining equations of system (3.8), we obtain a system of equations whose corresponding operator has the form

P = ( \begin{matrix}π_{S_{0}^{(1)}}A(x, D) & 0 & 0 \\ 0 & π_{S_{0}^{(2)}}B(x, D) & 0 \\ 0 & 0 & π_{S_{0}}C(x, D) \end{matrix} )_{36×36} + T_{−∞},

where

A(x, D) = ( \begin{matrix}π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}(B_{M}^{(2)})^{−1}𝕍_{−1}^{(2)} & I \\ −I & π_{S_{0}^{(1)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1} \end{matrix} ) ,

B(x, D) = ( \begin{matrix}π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}(B_{M}^{(3)})^{−1}𝕍_{−1}^{(3)} & I \\ −I & π_{S_{0}^{(2)}}(−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1} \end{matrix} ) ,

C(x, D) = ( \begin{matrix}−π_{S_{0}}𝕍_{−1}^{(2)}(B_{M}^{(2)})^{−1}𝕍_{−1}^{(2)} & π_{S_{0}}𝕍_{−1}^{(3)}(B_{M}^{(3)})^{−1}𝕍_{−1}^{(3)} \\ I & I \end{matrix} )

and T_{−∞} is the operator of order −∞.

Further, after the localization, the operators A(x, D) and B(x, D) are reduced by means of lifting to the strongly elliptic ΨDOs of order 1, while the operator C(x, D) is reduced to positive-definite ΨDO.

Indeed, let A(x′, D′) and B(x′, D′) be ΨDOs with the symbols σ_{A}(x′, ξ′) and σ_{B}(x′, ξ′) (ξ′ = (ξ_{1}, ξ_{2})) “frozen” at the points and written in terms of some local coordinate system of the manifolds S_{0}^{(1)} and S_{0}^{(2)}, respectively.

Denote

R(x′, D′) = ( \begin{matrix}L_{−} & 0 \\ 0 & I \end{matrix} ) ∘ A(x′, D′) ∘ ( \begin{matrix}L_{+} & 0 \\ 0 & I \end{matrix} )

and

Q(x′, D′) = ( \begin{matrix}L_{−} & 0 \\ 0 & I \end{matrix} ) ∘ B(x′, D′) ∘ ( \begin{matrix}L_{+} & 0 \\ 0 & I \end{matrix} ) ,

where L_{+} = diagΛ_{+}, L_{−} = diagπ_{+}Λ_{−}ℓ are 6 × 6 matrix operators, Λ_{±} is a ΨDO operator with the symbol Λ_{±}(ξ′) = ξ_{2} ± i ± i|ξ_{1}|, π_{+} denotes the operator of restriction onto ℝ_{+}^{2}, and ℓ is an extension operator.

The operators

( \begin{matrix}L_{±} & 0 \\ 0 & I \end{matrix} )

are invertible in the respective spaces (see [39]).

The principal homogeneous symbols of the ΨDOs R(x′, D′) and Q(x′, D′) are written as

σ_{R}(x′, ξ′)=( \begin{matrix}(ξ_{n−1}−i|ξ′′|)σ_{N_{2}}(x′, ξ′)(ξ_{n−1}+i|ξ′′|) & (ξ_{n−1}−i|ξ′′|)I \\ −(ξ_{n−1} + i|ξ′′|)I & σ_{N_{1}}(x′, ξ′) \end{matrix} ) , x′ ∈\overline{S}_{0}^{(1)},

σ_{Q}(x′, ξ′)=( \begin{matrix}(ξ_{n−1}−i|ξ′′|)σ_{N_{3}}(x′, ξ′)(ξ_{n−1}+i|ξ′′|) & (ξ_{n−1}−i|ξ′′|)I \\ −(ξ_{n−1} + i|ξ′′|)I & σ_{N_{1}}(x′, ξ′) \end{matrix} ) , x′ ∈\overline{S}_{0}^{(2)},

where σ_{N_{1}}(x′, ξ′), σ_{N_{2}}(x′, ξ′) and σ_{N_{3}}(x′, ξ′) are the principal homogeneous symbols of the ΨDOs

N_{1} = (−\frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1}, N_{2} = 𝕍_{−1}^{(2)}(B_{M}^{(2)})^{−1}𝕍_{−1}^{(2)}, N_{3} = 𝕍_{−1}^{(3)}(B_{M}^{(3)})^{−1}𝕍_{−1}^{(3)},

respectively, written in terms of a given local coordinate system, and I is the identity matrix.

Let λ_{R}^{(k)}, k = 1, …, 12, be the eigenvalues of the matrix

(σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1), x_{1} ∈ ∂S_{0}^{(1)},

where

σ_{R}(x_{1}, 0, −1) = ( \begin{matrix}σ_{N_{2}}(x_{1}, 0, −1) & −I \\ I & σ_{N_{1}}(x_{1}, 0, −1) \end{matrix} ) ,

σ_{R}(x_{1}, 0, +1) = ( \begin{matrix}σ_{N_{2}}(x_{1}, 0, +1) & I \\ −I & σ_{N_{1}}(x_{1}, 0, +1) \end{matrix} ) ,

and let λ_{Q}^{(k)}, k = 1, …, 12, be the eigenvalues of the matrix

(σ_{Q}(x_{1}, 0, +1))^{−1}σ_{Q}(x_{1}, 0, −1), x_{1} ∈ ∂S_{0}^{(2)},

where

σ_{Q}(x_{1}, 0, −1) = ( \begin{matrix}σ_{N_{3}}(x_{1}, 0, −1) & −I \\ I & σ_{N_{1}}(x_{1}, 0, −1) \end{matrix} ) ,

σ_{Q}(x_{1}, 0, +1) = ( \begin{matrix}σ_{N_{3}}(x_{1}, 0, +1) & I \\ −I & σ_{N_{1}}(x_{1}, 0, +1) \end{matrix} )

Introduce the notation

δ_{R} = sup_{\begin{matrix}1≤j≤12 \\ x_{1}∈∂S_{0}^{(1)} \end{matrix}} |\frac{1}{2π} arg λ_{R}^{(j)}(x_{1})|, δ_{Q} = sup_{\begin{matrix}1≤j≤12 \\ x_{1}∈∂S_{0}^{(2)} \end{matrix}} |\frac{1}{2π} arg λ_{Q}^{(j)}(x_{1})|,

δ = max(δ_{R}, δ_{Q}).

Using the general theory of pseudodifferential operators (ΨDOs) (see [35], [36], the following propositions are valid.

Lemma 3.2. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p − 1/2 + δ < s < 1/p + 1/2 − δ. Then the operators

R(x′, D′) : ℍ̃_{p}^{s}(ℝ_{+}^{2}) ⊕ ℍ̃_{p}^{s}(ℝ_{+}^{2}) → ℍ_{p}^{s−1}(ℝ_{+}^{2}) ⊕ ℍ_{p}^{s−1}(ℝ_{+}^{2})
(𝔹̃_{p,r}^{s}(ℝ_{+}^{2}) ⊕ 𝔹̃_{p,r}^{s}(ℝ_{+}^{2}) → 𝔹_{p,r}^{s−1}(ℝ_{+}^{2}) ⊕ 𝔹_{p,r}^{s−1}(ℝ_{+}^{2})),
Q(x′, D′) : ℍ̃_{p}^{s}(ℝ_{+}^{2}) ⊕ ℍ̃_{p}^{s}(ℝ_{+}^{2}) → ℍ_{p}^{s−1}(ℝ_{+}^{2}) ⊕ ℍ_{p}^{s−1}(ℝ_{+}^{2})
(𝔹̃_{p,r}^{s}(ℝ_{+}^{2}) ⊕ 𝔹̃_{p,r}^{s}(ℝ_{+}^{2}) → 𝔹_{p,r}^{s−1}(ℝ_{+}^{2}) ⊕ 𝔹_{p,r}^{s−1}(ℝ_{+}^{2}))

are Fredholm.

Note that the ΨDOs R(x′, D′) and Q(x′, D′) are Fredholm in the anisotropic Bessel potential spaces with weight

ℍ̃_{p}^{(μ,s),k}(ℝ_{+}^{2}) ⊕ ℍ̃_{p}^{(μ,s),k}(ℝ_{+}^{2}) → ℍ_{p}^{(μ,s−1),k}(ℝ_{+}^{2}) ⊕ ℍ_{p}^{(μ,s−1),k}(ℝ_{+}^{2})

for all μ ∈ ℝ and k = 0, 1, … (see [10]).

Since the operators π_{S_{0}^{(1)}}N_{1}, π_{S_{0}^{(1)}}N_{2}, π_{S_{0}^{(2)}}N_{1} and π_{S_{0}^{(2)}}N_{2} are positive-definite, we obtain a strong Gårding inequality for the operators A(x, D) and B(x, D) (see [6, Lemma 3.3]). Hence, using the results obtained in [2], [23], we have

Lemma 3.3. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p − 1/2 + δ < s < 1/p + 1/2 − δ. Then the operators

π_{S_{0}^{(1)}}A(x, D) : ℍ̃_{p}^{s−1}(S_{0}^{(1)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(1)}) → ℍ_{p}^{s}(S_{0}^{(1)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(1)})
(𝔹̃_{p,r}^{s−1}(S_{0}^{(1)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(1)}) → 𝔹_{p,r}^{s}(S_{0}^{(1)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(1)}))
π_{S_{0}^{(2)}}A(x, D) : ℍ̃_{p}^{s−1}(S_{0}^{(2)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(2)}) → ℍ_{p}^{s}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(2)})

SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS 441

(𝔹̃_{p,r}^{s−1}(S_{0}^{(2)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(2)}) → 𝔹_{p,r}^{s}(S_{0}^{(2)}) ⊕ 𝔹_{p,r}^{s−1}(S_{0}^{(2)}))

are invertible.

Let us consider the operator C(x, D). The corresponding system

π_{S_{0}}C(x, D) ( \begin{matrix}φ_{0}^{(2)} \\ φ_{0}^{(3)} \end{matrix} ) = ( Ẽ_{1} ; E_{2} )

is reduced to pseudodifferential equation on the open manifold S_{0}

π_{S_{0}}Nψ̃_{0}^{(3)} = Ẽ_{1}, ψ̃_{0}^{(2)} = −ψ̃_{0}^{(3)} + E_{2},

where N = N_{2} + N_{3}.

The ΨDO π_{S_{0}}N is positive-definite and the following proposition holds for it. Lemma 3.4. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p − 1/2 + δ < s < 1/p + 1/2 − δ.

Then the ΨDOs

π_{S_{0}}N : ℍ̃_{p}^{s−1}(S_{0}) → ℍ_{p}^{s}(S_{0})
(𝔹̃_{p,r}^{s−1}(S_{0}) → 𝔹_{p,r}^{s}(S_{0}))

and

π_{S_{0}}C(x, D) : ℍ̃_{p}^{s−1}(S_{0}) ⊕ ℍ̃_{p}^{s−1}(S_{0}) → ℍ_{p}^{s}(S_{0}) ⊕ ℍ_{p}^{s}(S_{0})
(𝔹̃_{p,r}^{s−1}(S_{0}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}) → 𝔹_{p,r}^{s}(S_{0}) ⊕ 𝔹_{p,r}^{s}(S_{0}))

are invertible.

Note that the ΨDO π_{S_{0}}N is invertible in anisotropic Bessel potential spaces with weight ℍ̃_{p}^{(μ,s−1),k} → ℍ_{p}^{(μ,s),k}(S_{0}) (see [10]).

Lemmas 3.3 and 3.4 imply the validity of the following proposition.

Lemma 3.5. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p − 1/2 + δ < s < 1/p + 1/2 − δ. Then the operator

P : \begin{matrix}ℍ̃_{p}^{s−1}(S_{0}^{(1)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(1)}) \\ ⊕ \\ ℍ̃_{p}^{s−1}(S_{0}^{(2)}) ⊕ ℍ̃_{p}^{s}(S_{0}^{(2)}) \\ ⊕ \\ ℍ̃_{p}^{s−1}(S_{0}) ⊕ ℍ̃_{p}^{s−1}(S_{0}) \end{matrix} → \begin{matrix}ℍ_{p}^{s}(S_{0}^{(1)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(1)}) \\ ⊕ \\ ℍ_{p}^{s}(S_{0}^{(2)}) ⊕ ℍ_{p}^{s−1}(S_{0}^{(2)}) \\ ⊕ \\ ℍ_{p}^{s}(S_{0}) ⊕ ℍ_{p}^{s}(S_{0}) \end{matrix}

( \begin{matrix}𝔹̃_{p,r}^{s−1}(S_{0}^{(1)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(1)}) \\ ⊕ \\ 𝔹̃_{p,r}^{s−1}(S_{0}^{(2)}) ⊕ 𝔹̃_{p,r}^{s}(S_{0}^{(2)}) \\ ⊕ \\ 𝔹̃_{p,r}^{s−1}(S_{0}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}) \end{matrix} → \begin{matrix}𝔹_{p,r}^{s}(S_{0}^{(1)}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}^{(1)}) \\ ⊕ \\ 𝔹_{p,r}^{s}(S_{0}^{(2)}) ⊕ 𝔹̃_{p,r}^{s−1}(S_{0}^{(2)}) \\ ⊕ \\ 𝔹_{p,r}^{s}(S_{0}) ⊕ 𝔹_{p,r}^{s}(S_{0}) \end{matrix} )

is invertible.

Theorem 3.6. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p−1/2+δ < s < 1/p+1/2−δ, M = 2, 3, . . .. Then the operator

A_{M} : H^{(1)} _{p}^{s} →H^{(3)} _{p}^{s} ( B^{(1)} _{p,r}^{s} →B^{(3)} _{p,r}^{s})

is invertible.

Lemma 2.1 and Theorem 3.6 imply that the following proposition is valid.

Theorem 3.7. Let 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p−1/2+δ < s < 1/p+1/2−δ. Then the operator

A : H^{(1)} _{p}^{s} →H^{(2)} _{p}^{s} ( B^{(1)} _{p,r}^{s} →B^{(2)} _{p,r}^{s} )

is invertible.

If we take s = 1/p′ in the condition for the operator A to be invertible (see Theorem 3.7), we conclude that p must satisfy the equality

\frac{4}{3 − 2δ} < p < \frac{4}{1 + 2δ} .

Theorem 3.7 and the above reasoning imply the validity of the solution existence and uniqueness for problem M_{1}.

Theorem 3.8. Let 4/(3−2δ) < p < 4/(1+2δ) and the compatibility condition (3.7) be fulfilled for s = 1 − 1/p. Then the boundary-contact problem M_{1} has a unique solution in the classes W_{p}^{1}(D_{q}), q = 1, 2, (W_{p,loc}^{1}(D_{2}) with condition (1.6) at infinity), which is given by the formulae

U^{(1)} = 𝕍^{(1)}g_{1} in D_{1}, U^{(2)} = \{ \begin{matrix}𝕍^{(2)}g_{2} & in D_{2}^{(1)}, \\ 𝕍^{(3)}g_{3} & in D_{2}^{(2)}, \\ π_{S_{0}}𝕍_{−1}^{(2)}g_{2} = π_{S_{0}}𝕍_{−1}^{(3)}g_{3} & on S_{0}, \end{matrix}

where g_{q}, q = 1, 2, 3, are obtained from system (3.6).

Theorems 2.1, 3.7 and the embedding theorems (see [39]) imply

Theorem 3.9. Let 4/(3 − 2δ) < p < 4/(1 + 2δ), 1 ≤ t ≤ ∞, 1 < r < ∞, 1/r−1/2+δ < s < 1/r+1/2−δ, the compatibility condition (3.7) with t instead of p be fulfilled, U^{(q)} ∈ W_{p}^{1}(D_{q}), q = 1, 2, (U^{(2)} ∈ W_{p,loc}^{1}(D_{2}) with condition (1.6) at infinity) be a solution of the boundary-contact problem M_{1}. Then:

if φ_{1} ∈ 𝔹_{r,r}^{s}(S_{1}), φ_{2} ∈ 𝔹_{r,r}^{s−1}(S_{2}^{(1)}), φ_{3} ∈ 𝔹_{r,r}^{s−1}(S_{2}^{(2)}), f_{i} ∈ 𝔹_{r,r}^{s}(S_{0}^{(i)}), h_{i} ∈ 𝔹_{r,r}^{s−1}(S_{0}^{(i)}), i = 1, 2, we have U^{(q)} ∈ ℍ_{r}^{s+1/r}(D_{q}), q = 1, 2, (U^{(2)} ∈ ℍ_{r,loc}^{s+1/r}(D_{2}));

if φ_{1} ∈ 𝔹_{r,t}^{s}(S_{1}), φ_{2} ∈ 𝔹_{r,t}^{s−1}(S_{2}^{(1)}), φ_{3} ∈ 𝔹_{r,t}^{s−1}(S_{2}^{(2)}), f_{i} ∈ 𝔹_{r,t}^{s}(S_{0}^{(i)}), h_{i} ∈ 𝔹_{r,t}^{s−1}(S_{0}^{(i)}), i = 1, 2, we have U^{(q)} ∈ 𝔹_{r,t}^{s+1/r}(D_{q}), q = 1, 2, (U^{(2)} ∈ 𝔹_{r,t,loc}^{s+1/r}(D_{2})).

4. Asymptotics of Solutions

Now we will write the asymptotics of solutions of the boundary-contact problem M_{1}. The boundary and contact data are assumed to be sufficiently smooth, i.e.,

φ_{1} ∈ ℍ_{r}^{(∞,s+2M+1),∞}(S_{1}), φ_{2} ∈ ℍ_{r}^{(∞,s+2M),∞}(S_{2}^{(1)}), φ_{3} ∈ ℍ_{r}^{(∞,s+2M),∞}(S_{2}^{(2)}),

f_{i} ∈ ℍ_{r}^{(∞,s+2M+1),∞}(S_{0}^{(i)}), h_{i} ∈ ℍ_{r}^{(∞,s+2M),∞}(S_{0}^{(i)}), i = 1, 2;

here the numbers r and s satisfy the conditions of Theorem 3.9.

Let m_{1}, …, m_{2ℓ} be algebraic multiplicities of the eigenvalues λ_{1}, …, λ_{2ℓ}, ∑_{j=1}^{2ℓ} m_{j} = 12, where λ_{j} = λ_{R}^{(j)}, j = 1, …, 2ℓ.

We introduce the notation

b_{R}(x_{1}) = (σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1).

Let

b_{0R}(x_{1}) = K^{−1}(x_{1})b_{R}(x_{1})K(x_{1}), x_{1} ∈ ∂S_{0}^{(1)},

be a quasi-diagonal form, where K is some nondegenerate matrix function detK(x_{1}) ≠ 0, and K ∈ C^{∞} (see [20]).

The asymptotics of solutions to a strongly elliptic pseudodifferential equation (see [10, Theorem 2.1]) implies the asymptotics of solutions of the pseudodifferential equation

R(x′, D′)χ = F, F ∈ ℍ_{r,comp}^{(∞,s+M),∞}(ℝ_{+}^{2}) × ℍ_{r,comp}^{(∞,s+M),∞}(ℝ_{+}^{2})

written in terms of some local coordinate system of the manifold S_{0}^{(1)}. Thus we obtain an asymptotic expansion of the solution χ = (χ_{1}, χ_{2})^{⊤}

χ(x_{1}, x_{2,+}) = K(x_{1})x_{2,+}^{1/2+Δ(x_{1})}𝔹_{a_{pr}}^{0}( − \frac{1}{2πi} log x_{2,+})K^{−1}(x_{1})c_{0}(x_{1})

+ ∑_{k=1}^{M} K(x_{1})x_{2,+}^{1/2+Δ(x_{1})+k}𝔹_{k}(x_{1}, log x_{2,+}) + χ_{M+1}(x_{1}, x_{2,+}),  (4.1)

for all sufficiently small x_{2,+} > 0, χ_{M+1} ∈ℍ̃_{r,comp}^{(∞,s+M+1),∞}(ℝ_{+}^{2})×ℍ̃_{r,comp}^{(∞,s+M+1),∞}(ℝ_{+}^{2}); 𝔹_{a_{pr}}^{0}(t) is defined in [10]; the 12 × 12 matrix function 𝔹_{k}(x_{1}, t) is a polynomial of order ν_{k} = k(2m_{0} − 1) + m_{0} − 1, m_{0} = max{m_{1}, …, m_{2ℓ}} with respect to the variable t with 12-dimensional vector coefficients which depend on the variable x_{1} and

Δ(x_{1}) = (Δ_{1}(x_{1}), Δ_{2}(x_{1}));

here

Δ_{j}(x_{1}) = (\underbrace{δ_{1}^{(j)}(x_{1}), …, δ_{1}^{(j)}(x_{1})}_{m_{1}-times}, …, \underbrace{δ_{ℓ}^{(j)}(x_{1}), …, δ_{ℓ}^{(j)}(x_{1})}_{m_{ℓ}-times}) , j = 1, 2,

δ_{k}^{(1)}(x_{1}) = \frac{1}{2π} arg λ_{k}(x_{1}) − \frac{i}{2π} |λ_{k}(x_{1})|,

δ_{k}^{(2)}(x_{1}) = −\frac{1}{2π} arg λ_{k}(x_{1}) − \frac{i}{2π} |λ_{k}(x_{1})|, k = 1, …, ℓ.

Without loss of generality suppose that the matrix 𝔹_{a_{pr}}^{0}(t) has the form

𝔹_{a_{pr}}^{0}(t) = diag{B_{a_{pr}}^{0}(t), B_{a_{pr}}^{0}(t)};

here B_{a_{pr}}^{0}(t) is the upper triangular block-diagonal 6×6-matrix-function defined in [9].

Hence for the functions χ_{1} and χ_{2} we can write an asymptotic expansion. Indeed, let

K(x_{1}) = ( \begin{matrix}K_{11}(x_{1}) & K_{12}(x_{1}) \\ K_{21}(x_{1}) & K_{22}(x_{1}) \end{matrix} )_{12×12}

and

K^{−1}(x_{1})c_{0}(x_{1}) = (c_{0}^{(1)}(x_{1}), c_{0}^{(2)}(x_{1}))^{⊤},  (4.2)

where K_{ij}(x_{1}), i, j = 1, 2, are 6 × 6-matrices, c_{0}^{(i)}, i = 1, 2, are six-dimensional vector functions. Then

χ_{i}(x_{1}, x_{2,+}) = ∑_{j=1}^{2} K_{ij}(x_{1})x_{2,+}^{1/2+Δ_{j}(x_{1})}B_{a_{pr}}^{0}( − \frac{1}{2πi} log x_{2,+})c_{0}^{(i)}(x_{1})

+ ∑_{j=1}^{2} ∑_{k=1}^{M} K_{ij}(x_{1})x_{2,+}^{1/2+Δ_{j}(x_{1})+k}B_{kj}^{(i)}(x_{1}, log x_{2,+}) + χ_{M+1}^{(i)}(x_{1}, x_{2,+}), i = 1, 2, (4.3)

where χ_{M+1}^{(i)} ∈ ℍ̃_{r,comp}^{(∞,s+M+1)}(ℝ_{+}^{2}) and B_{kj}^{(i)}(x_{1}, t) is a polynomial of order ν_{k} = k(2m_{0} − 1) + m_{0} − 1 with respect to the variable t with six-dimensional vector coefficients which depend on the variable x_{1}.

We can also obtain an analogous asymptotic expansion of the solution χ̃ = (χ̃_{1}, χ̃_{2}) of the strongly elliptic equation

Q(x′, D′)χ̃ = F̃, F̃ ∈ ℍ_{r,comp}^{(∞,s+M),∞}(ℝ_{+}^{2}) × ℍ_{r,comp}^{(∞,s+M),∞}(ℝ_{+}^{2})

in terms of some local coordinate system on the manifold S_{0}^{(2)}. Indeed, we have

χ̃_{i}(x_{1}, x_{2,+}) = ∑_{j=1}^{2} K_{ij}(x_{1})x_{2,+}^{1/2+Δ_{j}(x_{1})}B_{a_{pr}}^{0}( − \frac{1}{2πi} log x_{2,+})b_{0}^{(i)}(x_{1})

+ ∑_{j=1}^{2} ∑_{k=1}^{M} K_{ij}(x_{1})x_{2,+}^{1/2+Δ̃_{j}(x_{1})+k}B_{kj}^{(i)}(x_{1}, log x_{2,+}) + χ̃_{M+1}^{(i)}(x_{1}, x_{2,+}), i = 1, 2, (4.4)

where χ̃_{M+1}^{(i)} ∈ ℍ̃_{r,comp}^{(∞,s+M+1),∞}(ℝ_{+}^{2}), Δ̃_{j}, j = 1, 2, are defined as Δ_{j}, j = 1, 2, by means of the eigenvalues λ_{Q}^{(k)}, k = 1, …, 12, of the matrix b_{Q}.

Let us consider the pseudodifferential equation

π_{S_{0}}Nψ̃_{0}^{(3)} = Ẽ_{1} and ψ̃_{0}^{(2)} = −ψ̃_{0}^{(3)} + E_{2},

where

N = 𝕍_{−1}^{(2)}(B_{M}^{(2)})^{−1}𝕍_{−1}^{(2)} + 𝕍_{−1}^{(3)}(B_{M}^{(3)})^{−1}𝕍_{−1}^{(3)}.

The following equalities hold for the principal homogeneous symbols of the operators 𝕍_{−1}^{(i)} and 𝕍^{∗} _{0}^{(i)}, i = 2, 3:

\begin{matrix}σ_{𝕍_{−1}^{(2)}}(x′, ξ′) = σ_{𝕍_{−1}^{(3)}}(x′, ξ′) for x′ ∈ \overline{S}_{0}, \\ σ𝕍^{∗}_{{(2) \atop 0}}(x′, ξ′) = −σ𝕍^{∗}_{{(3) \atop 0}}(x′, ξ′) for x′ ∈ \overline{S}_{0}. \end{matrix}  (4.5)

In view of equality (4.5) we can write the symbol σ_{N}(x′, ξ′) of the ΨDO operator N as follows:

σ_{N}(x′, ξ′) = [(\frac{1}{2}I + σ𝕍^{∗}_{{(2) \atop 0}}(x′, ξ′))(σ_{−𝕍_{−1}^{(2)}}(x′, ξ′))^{−1}]^{−1}

+ [(\frac{1}{2}I − σ𝕍^{∗}_{{(2) \atop 0}}(x′, ξ′))(σ_{−𝕍_{−1}^{(2)}}(x′, ξ′))^{−1}]^{−1}.

Since the symbol σ𝕍^{∗}_{{(2) \atop 0}}(x′, ξ′) is an odd matrix function with respect to ξ′, while the symbol σ_{−𝕍_{−1}^{(2)}}(x′, ξ′) is an even matrix function, one can easily ascertain that the symbol σ_{N}(x′, ξ′) is even with respect to the variable ξ′, i.e.,

σ_{N}(x′, −ξ′) = σ_{N}(x′, ξ′), x′ ∈ \overline{S}_{0},

and all eigenvalues of the matrix

(σ_{N}(x_{1}, 0, +1))^{−1}σ_{N}(x_{1}, 0, −1) = I, x_{1} ∈ ∂S_{0},

are trivial λ_{N}^{(j)} = 1, j = 1, …, 6.

Applying the result on strongly elliptic pseudodifferential equations (see [10, Theorem 2.1]), we obtain, in terms of some local coordinate system, the following result on asymptotic expansion of the functions ψ_{0}^{(i)}, i = 2, 3:

ψ_{0}^{(i)}(x_{1}, x_{2,+}) = (−1)^{i+1}c_{0}(x_{1})x_{2,+}^{−1/2}+∑_{k=1}^{M} x_{2,+}^{−1/2+k}d_{k}^{(i)}(x_{1})+ψ_{M+1}^{(i)}(x_{1}, x_{2,+}), (4.6)

where c_{0}, d_{k}^{(i)} ∈ C_{0}^{∞}(ℝ), and the remainder ψ_{M+1}^{(i)} ∈ ℍ̃_{r,comp}^{(∞,s+M+1),∞}(ℝ_{+}^{2}), i = 2, 3, M ∈ ℕ. As we can see from (4.6), due to the properties of the symbol σ_{N}(x′, ξ′) (see [12]) there are no logarithms in the entire asymptotic expansion.

Let g = (g_{1}, g_{2}, g_{3}, φ_{0}^{(1)}, ψ_{0}^{(1)}φ_{0}^{(2)}, ψ_{0}^{(2)}, φ_{0}^{(3)}, ψ_{0}^{(3)}) be a solution of system (3.6), i.e.,

Ag = Φ,

where

Φ = (Φ_{0}^{(1)}, Φ_{0}^{(2)}, Φ_{0}^{(3)}, f_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)}, h_{1} + π_{S_{0}^{(1)}}Φ_{0}^{(2)}, f_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)},
h_{2} + π_{S_{0}^{(2)}}Φ_{0}^{(3)}, 0, −(π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)})).

Then

D ∘ Ag = Ψ;  (4.7)

here

Ψ = (Φ_{0}^{(1)}, −𝕍_{−1}^{(2)}Φ_{0}^{(2)}, −𝕍_{−1}^{(3)}Φ_{0}^{(3)}, f_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)}, h_{1} + π_{S_{0}^{(1)}}Φ_{0}^{(2)}, f_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)},

h_{2} + π_{S_{0}^{(2)}}Φ_{0}^{(3)}, 0, −(π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)})).

Now adding the expression

T_{2M+1}g = diag{0, −(𝕍_{−1}^{(2)})^{2M+1}, −(𝕍_{−1}^{(3)})^{2M+1}, 0, …, 0}g

to both parts of system (4.7), we obtain the equality

A_{2M+1}g = Ψ̃,  (4.8)

where Ψ̃ = (Φ_{0}^{(1)}, −𝕍_{−1}^{(2)}Φ_{0}^{(2)} − (𝕍_{−1}^{(2)})^{2M+1}g_{2}, −𝕍_{−1}^{(2)}Φ_{0}^{(3)} − (𝕍_{−1}^{(3)})^{2M+1}g_{3}, f_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)},

h_{1} + π_{S_{0}^{(1)}}Φ_{0}^{(2)}, f_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)}, h_{2} + π_{S_{0}^{(2)}}Φ_{0}^{(3)}, 0, −(π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)})).

Thus (−L_{+}^{−1}φ_{0}^{(2)}, φ_{0}^{(1)}) satisfies, in some local coordinate system of the manifold S_{0}^{(1)}, the pseudodifferential equation

R(x′, D′) ( \begin{matrix}χ_{1} \\ χ_{2} \end{matrix} ) = F,

and (−L_{+}^{−1}ψ_{0}^{(3)}, φ_{0}^{(1)}) satisfies, in some local coordinate system of the manifold S_{0}^{(2)}, the pseudodifferential equation

Q(x′, D′) ( χ̃_{1} ; χ̃_{2} ) = F̃,

where

F = (L_{−}F_{1}, F_{2}),

F_{1} = f_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)} − π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}Φ_{0}^{(2)}

−π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}ψ_{0}^{(2)} − π_{S_{0}^{(1)}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}(𝕍_{−1}^{(2)})^{2M+1}g_{2},

F_{2} = h_{1} + π_{S_{0}^{(1)}}Φ_{0}^{(2)} − π_{S_{0}^{(1)}}( − \frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1}Φ_{0}^{(1)}

−π_{S_{0}^{(1)}}( − \frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1}ψ_{0}^{(1)}

and

F̃ = (L_{−}F̃_{1}, F̃_{2}),

F̃_{1} = f_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)} + π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}(B_{2M+1}^{(3)})^{−1}𝕍_{−1}^{(3)}Φ_{0}^{(3)}

−π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}(B_{2M+1}^{(3)})^{−1}(𝕍_{−1}^{(3)})^{2M+1}ψ_{0}^{(3)} − π_{S_{0}^{(2)}}𝕍_{−1}^{(3)}(B_{2M+1}^{(3)})^{−1}(𝕍_{−1}^{(3)})^{2M+1}g_{3},

F̃_{2} = h_{2} + π_{S_{0}^{(2)}}Φ_{0}^{(3)} − π_{S_{0}^{(2)}}( − \frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1}Φ_{0}^{(1)}

−π_{S_{0}^{(2)}}( − \frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1}φ_{0}^{(1)};

here

F_{i} ∈ ℍ_{r,comp}^{(∞,s+2M),∞}(ℝ_{+}^{2}), F̃_{i} ∈ ℍ_{r,comp}^{(∞,s+2M),∞}(ℝ_{+}^{2}), i = 1, 2.

Further, we have that (ψ_{0}^{(2)}, ψ_{0}^{(3)}) is a solution of the system

π_{S_{0}}C(x, D) ( \begin{matrix}ψ_{0}^{(2)} \\ ψ_{0}^{(3)} \end{matrix} ) = ( \begin{matrix}0 \\ −(π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)}) \end{matrix} ) .

This system can be reduced to a pseudodifferential equation with the positivedefinite operator

π_{S_{0}}Nψ_{0}^{(3)} = E_{1}, ψ_{0}^{(2)} = −ψ_{0}^{(3)} − (π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)}),

where

N = 𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)} + 𝕍_{−1}^{(3)}(B_{2M+1}^{(3)})^{−1}𝕍_{−1}^{(3)}

and

E_{1} = −π_{S_{0}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}Φ_{0}^{(2)} − π_{S_{0}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}(𝕍_{−1}^{(2)})^{2M+1}g_{2}

+ π_{S_{0}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}φ_{0}^{(2)} − π_{S_{0}}𝕍_{−1}^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}(π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)});

here

E_{1} ∈ ℍ_{r,comp}^{(∞,s+2M),∞}(ℝ_{+}^{2}).

Hence we can obtain asymptotic expansions (4.3), (4.4) and (4.6) for the functions −L_{+}^{−1}φ_{0}^{(2)}, φ_{0}^{(2)}, −L_{+}^{−1}ψ_{0}^{(3)}, φ_{0}^{(1)} and ψ_{0}^{(3)}, ψ_{0}^{(2)}, respectively.

We now define g_{1}, g_{2} and g_{3} by the first three equations of system (4.8)

g_{1} = (𝕍_{−1}^{(1)})^{−1}φ_{0}^{(1)} + (𝕍_{−1}^{(1)})^{−1}ψ_{0}^{(1)} + (𝕍_{−1}^{(1)})^{−1}Φ_{0}^{(1)},  (4.9)
g_{i} = −(B_{2M+1}^{(i)})^{−1}𝕍_{−1}^{(i)}φ_{0}^{(i)} − (B_{2M+1}^{(i)})^{−1}𝕍_{−1}^{(i)}ψ_{0}^{(i)}
+ (B_{2M+1}^{(i)})^{−1}Φ_{0}^{(i)} + G_{i}, i = 2, 3,  (4.10)

where

G_{i} = (B_{2M+1}^{(i)})^{−1}(−𝕍_{−1}^{(i)})^{2M+1}g_{i}, i = 2, 3,
G_{2} ∈ ℍ_{r}^{(∞,s+2M),∞}(∂D_{2}^{(1)}), G_{3} ∈ ℍ_{r}^{(∞,s+2M),∞}(∂D_{2}^{(2)}).

Using expansions (4.9) and (4.10), we obtain the following representation, i.e., the solutions of the boundary-contact problem M_{1} are expressed by the potential-type functions

U^{(1)} = 𝕍^{(1)}(𝕍_{−1}^{(1)})^{−1}φ_{0}^{(1)} + 𝕍^{(1)}(𝕍_{−1}^{(1)})^{−1}ψ_{0}^{(1)} + R_{1},  (4.11)
r_{1}U^{(2)} = −𝕍^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}φ_{0}^{(2)} − 𝕍^{(2)}(B_{2M+1}^{(2)})^{−1}𝕍_{−1}^{(2)}ψ_{0}^{(2)} + R_{2}, (4.12)
r_{2}U^{(2)} = −𝕍^{(3)}(B_{2M+1}^{(3)})^{−1}𝕍_{−1}^{(3)}φ_{0}^{(3)} − 𝕍^{(3)}(B_{2M+1}^{(3)})^{−1}𝕍_{−1}^{(3)}ψ_{0}^{(3)} + R_{3}, (4.13)

where

R_{1} ∈ C^{M+1}(\overline{D}_{1}), R_{2} ∈ C^{M+1}(\overline{D}_{2}^{(1)}), R_{3} ∈ C^{M+1}(\overline{D}_{2}^{(2)}),
suppφ_{0}^{(1)} ⊂ \overline{S} _{0}^{(1)}, suppψ_{0}^{(1)} ⊂ \overline{S} _{0}^{(2)}, suppφ_{0}^{(2)} ⊂ \overline{S} _{0}^{(1)},
suppψ_{0}^{(2)} ⊂ \overline{S}_{0}, suppφ_{0}^{(3)} ⊂ \overline{S} _{0}^{(2)}, suppψ_{0}^{(3)} ⊂ \overline{S}_{0}.

Thus, taking into account (4.11), (4.12), (4.13), using the asymptotic expansions of the functions −L_{+}^{−1}φ_{0}^{(2)}, φ_{0}^{(1)}, −L_{+}^{−1}ψ_{0}^{(3)}, φ_{0}^{(1)} and ψ_{0}^{(3)}, ψ_{0}^{(2)} (see (4.3), (4.4), (4.6)), the asymptotic expansion of potential-type functions (see [9, Theorems 2.2 and 2.3]) we derive the following asymptotic expansions of the solutions of the considered boundary-contact problem M_{1} in terms of some local coordinate systems of curves ∂S_{0}^{(1)}, ∂S_{0}^{(2)}, ∂S_{0}:

a) the asymptotic expansion near the contact boundary ∂S_{0}^{(1)}:

U^{(q)}(x_{1}, x_{2}, x_{3}) = (u^{(q)}, ω^{(q)})(x_{1}, x_{2}, x_{3})

= ∑_{θ=±1} ∑_{j=1}^{2} ∑_{s=1}^{ℓ_{0}} Re{ ∑_{m=0}^{n_{s}−1} x_{3}^{m}[d_{sjm}^{(q)}(x_{1}, θ)(z_{s,θ}^{(q)})^{1/2+Δ_{j}(x_{1})−m}

×B_{a_{pr}}^{0}( − \frac{1}{2πi} log z_{s,θ}^{(q)})]c_{jm}^{(q)}(x_{1})

+ \begin{matrix}∑_{k,l=0}^{M+2} & ∑_{p+m=0}^{M+2−l} \\ k+l+p+m≠0 \end{matrix} x_{2}^{l}x_{3}^{m}d_{slmpj}^{(q)}(x_{1}, ϑ)(z_{s,ϑ}^{(q)})^{1/2+Δ_{j}(x_{1})+k+p}B_{skmpj}^{(q)}(x_{1}, log z_{s,ϑ}^{(q)})\}

+U_{M+1}^{(q)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s − 1], 0\}, q = 1, 2, (4.14)

with the coefficients d_{sjm}^{(q)}(·, ±1), c_{jm}^{(q)}, d_{slmpj}^{(q)}(·, ±1) ∈ C_{0}^{∞}(ℝ) and the remainder U_{M+1}^{(q)} ∈ C_{0}^{M+1}(\overline{ℝ_{±}^{3}}), q = 1, 2, where the signs “+” and “-” refer to the cases q = 1 and q = 2, respectively. Here

z_{s,+1}^{(q)} = (−1)^{q}[x_{2} + x_{3}τ_{s,+1}^{(q)}], z_{s,−1}^{(q)} = (−1)^{q+1}[x_{2} − x_{3}τ_{s,−1}^{(q)}],
−π < Argz_{s,±1} < π, τ_{s,±1} ∈ C_{0}^{∞}(ℝ),

{τ_{s,±1}^{(q)}}_{s=1}^{ℓ_{0}} are all different roots of the polynomial det M^{∘} ^{(q)}((J_{κ}^{⊤}(x_{1}, 0))^{−1} · (0, ±1, τ)) of multiplicity n_{s}, s = 1, …, ℓ_{0}, in the complex lower half-plane (n_{s} and ℓ_{0} depend on q). B_{skmpj}^{(q)}(x_{1}, t) is a polynomial of order ν_{kmp} = ν_{k} + p + m

(ν_{k} = k(2m_{0} − 1) + m_{0} − 1, m_{0} = max{m_{1}, …, m_{ℓ}}, ∑_{j=1}^{ℓ} m_{j} = 6) with respect

to the variable t with vector coefficients which depend on the variable x_{1}.

We write the following relation between the leading (first) coefficients of the asymptotic expansions (4.14) and (4.3) (see [9, Theorem 2.3]):

d_{sjm}^{(1)}(x_{1}, +1) = \frac{1}{2π} G_{κ}(x_{1}, 0) 𝕍^{1} _{−1,m}^{(s)}(x_{1}, 0, +1)σ_{𝕍_{−1}^{(1)}}^{−1}(x_{1}, 0, +1)K_{2j}(x_{1}),

d_{sjm}^{(1)}(x_{1}, −1) = −\frac{1}{2π} G_{κ}(x_{1}, 0) 𝕍^{1} _{−1,m}^{(s)}(x_{1}, 0, −1)σ_{𝕍_{−1}^{(1)}}^{−1}(x_{1}, 0, −1)

× K_{2j}(x_{1})e^{iπ(−1/2−Δ_{j}(x_{1}))},

d_{sjm}^{(2)}(x_{1}, +1) = \frac{(−1)^{m+1}}{2π} G_{κ}(x_{1}, 0) 𝕍^{2} _{−1,m}^{(s)}(x_{1}, 0, +1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(2)}}^{−1}(x_{1}, 0, +1)K_{1j}(x_{1}),

d_{sjm}^{(2)}(x_{1}, −1) = \frac{(−1)^{m+1}}{2π} G_{κ}(x_{1}, 0) 𝕍^{2} _{−1,m}^{(s)}(x_{1}, 0, −1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(2)}}^{−1}(x_{1}, 0, −1)

× K_{1j}(x_{1})e^{iπ(−1/2−Δ_{j}(x_{1}))},
j = 1, 2, s = 1, …, ℓ_{0}, m = 0, …, n_{s} − 1;

here G_{κ} is the square root from the Gram determinant of the diffeomorphism κ and

𝕍^{q} _{−1,m}^{(s)}(x_{1}, 0, ±1) = \frac{i^{m+1}}{m!(n_{s} − 1 − m)!} \frac{d^{n_{s}−1−m}}{dτ^{n_{s}−1−m}} (τ − τ_{s,±1}^{(q)})^{n_{s}}

×[ M^{∘}^{(q)}((J_{κ}^{⊤}(x_{1}, 0))^{−1}(0, ±1, τ))]^{−1}|_{τ=τ_{s,±1}^{(q)}}, q = 1, 2.

The coefficients c_{jm}^{(q)}(x_{1}) in (4.14) are defined as follows:

\begin{matrix}c_{jm}^{(1)}(x_{1}) = a_{jm}(x_{1})B_{a_{pr}}^{−}(\frac{1}{2} + Δ_{j}(x_{1}))c_{0}^{(2)}(x_{1}), \\ c_{jm}^{(2)}(x_{1}) = a_{jm}(x_{1})B_{a_{pr}}^{−}(\frac{1}{2} + Δ_{j}(x_{1}))c_{0}^{(1)}(x_{1}), \end{matrix} j = 1, 2, m = 0, …, n_{s} − 1,

where B_{a_{pr}}^{−}(t) = diag{B_{−}^{m_{1}}(t), …, B_{−}^{m_{l}})t)},

B_{−}^{m_{r}}(t) = B^{m_{r}}( − \frac{1}{2πi}∂t) (Γ(t + 1)e^{\frac{iπ(t+1}{2}} ) ,

B^{m_{r}}(t) = ∥b_{kp}^{m_{r}}(t)∥_{m_{r}×m_{r}},

b_{kp}^{m_{r}}(t) = \{ \begin{matrix}(\frac{1}{2πi})^{p−k} \frac{(−1)^{p+k}}{(p − k)!} \frac{d^{p−k}}{dt^{p−k}} (Γ(t + 1)e^{\frac{iπ(t+1)}{2}} ) , & k ≤ p, \\ 0, & k > p, \end{matrix}

p = 0, …, m_{r} − 1, r = 1, …, ℓ.

Further, a_{jm}(x_{1}) = diag{a^{m_{1}}(λ_{1}^{(j)}), …, a^{m_{ℓ}}(λ_{ℓ}^{(j)})}, j = 1, 2,

λ_{r}^{(1)}(x_{1}) = −\frac{3}{2} − \frac{1}{2π} argλ_{r}(x_{1}) + \frac{i}{2π} log |λ_{r}(x_{1})| + m,

λ_{r}^{(2)}(x_{1}) = −\frac{3}{2} + \frac{1}{2π} argλ_{r}(x_{1}) + \frac{i}{2π} log |λ_{r}(x_{1})| + m,

m = 0, 1, …, n_{s} − 1;

a^{m_{r}}(λ_{r}^{(j)}) = ∥a_{kp}^{m_{r}}(λ_{r}^{(j)}∥_{m_{r}×m_{r}},

where

a_{kp}^{m_{r}}(λ_{r}^{(j)}) = \{ \begin{matrix}−i ∑_{l=k}^{p} \frac{(−1)^{p+k}(2πi)^{l−p}b_{kl}^{m_{r}}(μ_{r}^{(j)})}{(λ_{r}^{(j)} + 1)^{p−l+1}}, & m = 0, k ≤ p, \\ (−1)^{p+k}b_{kp}^{m_{r}}(λ_{r}^{(j)}), & m = 1, 2, …, n_{s} − 1, k ≤ p, \\ 0, & k > p; \end{matrix}

here λ_{r}^{(j)} = −1 + m + μ_{r}^{(j)}, 0 < Reμ_{r}^{(j)} < 1, j = 1, 2, r = 1, …, ℓ, and c_{0}^{(1)}(x_{1}), c_{0}^{(2)}(x_{1}) are defined using the first coefficients of the asymptotic expansion of the functions −L_{+}^{−1}φ_{0}^{(2)} and φ_{0}^{(1)}, respectively (see (4.3)). b) the asymptotics of solutions near the contact boundary ∂S_{0}^{(2)}:

U^{(q)}(x_{1}, x_{2}, x_{3}) = (u^{(q)}, ω^{(q)})(x_{1}, x_{2}, x_{3})

= ∑_{ϑ=±1} ∑_{j=1}^{2} ∑_{s=1}^{ℓ_{0}} Re{ ∑_{m=0}^{n_{s}−1} x_{3}^{m}[d_{sjm}^{(q)}(x_{1}, θ)(z_{z,θ}^{(q)})^{1/2+Δ̃_{j}(x_{1})−m}

×B_{a_{pr}}^{0}( − \frac{1}{2πi} log z_{s,θ}^{(q)})]c_{jm}^{(q)}(x_{1})

+ \begin{matrix}∑_{k,l=0}^{M+2} & ∑_{p+m=0}^{M+2−l} \\ k+l+p+m≠0 \end{matrix} x_{2}^{l}x_{3}^{m}d_{slmpj}^{(q)}(x_{1}, ϑ)(z_{s,ϑ}^{(q)})^{1/2+Δ_{j}(x_{1})+k+p}B_{skmpj}^{(q)}(x_{1}, log z_{s,ϑ}^{(q)})\}

+U_{M+1}^{(q)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s − 1], 0\}, q = 1, 2; (4.15)

here z_{s,+1}^{(q)} = (−1)^{q}[x_{2} + x_{3}τ_{s,+1}^{(q)}], z_{s,−1}^{(q)} = (−1)^{q+1}[x_{2} − x_{3}τ_{s,−1}^{(q)}]. The coefficients d_{sjm}^{(q)}(·, ±1) are calculated similarly as in a) and the coefficients b_{jm}^{(q)} are defined as in a) by using the first coefficients b_{0}^{(q)} (q = 1, 2) of the asymptotic expansion (4.4).

c) the asymptotics of solutions near the cuspidal edge ∂S_{0}: (r_{i}U^{(2)})(x_{1}, x_{2}, x_{3}) = r_{i}(u^{(2)}, ω^{(2)})(x_{1}, x_{2}, x_{3})

= ∑_{ϑ=±1} ∑_{s=1}^{ℓ_{0}} Re\{ ∑_{j=0}^{n_{s}−1} x_{3}^{j}z_{s,θ}^{1/2−j}d_{sj}^{(i)}(x_{1}, θ) + \begin{matrix}∑_{k,l=0}^{M+1} & ∑_{j+p=1}^{M+2−l} \\ l+k+j+p≠1 \end{matrix} x_{2}^{l}x_{3}^{j}z_{s,ϑ}^{−1/2+p+k}d_{slkjp}^{(i)}(x_{1})\}

+U_{M+1}^{(i)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s], 0\}, i = 1, 2,  (4.16)

with the coefficients d_{sj}^{(i)}(·, ±1), d_{slkjp}^{(i)} ∈ C_{0}^{∞}(ℝ) and the remainder U_{M+1}^{(i)} ∈ C_{0}^{M+1}(\overline{ℝ_{±}^{3}}), i = 1, 2; here z_{s,+1} =−x_{2} −x_{3}τ_{s,+1}^{(2)}, z_{s,−1} =x_{2} −x_{3}τ_{s,−1}^{(2)}, −π < Argz_{s,±1} < π, τ_{s,±1} ∈ C_{0}^{∞}(ℝ),

{τ_{s,±1}^{(2)}}_{s=1}^{ℓ_{0}} are all different roots of the polynomial det M^{∘} ^{(2)}((J_{κ}^{⊤}(x_{1}, 0))^{−1} · (0, ±1, τ)) of multiplicity n_{s}, s = 1, …, ℓ_{0}, in the complex lower half-plane. The coefficients d_{sj}^{(i)}(x_{1}, ±1) have the form (see [9, Theorem 2.3]):

d_{sj}^{(1)}(x_{1}, +1) = G_{κ}(x_{1}, 0) 𝕍^{2} _{−1,j}^{(s)}(x_{1}, 0, +1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(2)}}^{−1}(x_{1}, 0, +1)c^{(j)}(x_{1}),

d_{sj}^{(1)}(x_{1}, −1) = −iG_{κ}(x_{1}, 0) 𝕍^{2} _{−1,j}^{(s)}(x_{1}, 0, −1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(2)}}^{−1}(x_{1}, 0, −1)c^{(j)}(x_{1})

d_{sj}^{(2)}(x_{1}, +1) = (−1)^{j+1}G_{κ}(x_{1}, 0) 𝕍^{2} _{−1,j}^{(s)}(x_{1}, 0, +1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(3)}}^{−1}(x_{1}, 0, +1)c^{(j)}(x_{1}),

d_{sj}^{(2)}(x_{1}, −1) = (−1)^{j+1}iG_{κ}(x_{1}, 0) 𝕍^{2} _{−1,j}^{(s)}(x_{1}, 0, −1)σ_{\frac{1}{2} I+𝕍^{∗}_{0}^{(3)}}^{−1}(x_{1}, 0, −1)c^{(j)}(x_{1}),

s = 1, …, l_{0}, j = 0, …, n_{s} − 1,

where G_{κ} is the square root from the Gram determinant of the diffeomorphism κ,

𝕍^{2} _{−1,j}^{(s)}(x_{1}, 0, ±1) = \frac{i^{j+1}}{j!(n_{s} − 1 − j)!} \frac{d^{n_{s}−1−j}}{dτ^{n_{s}−1−j}} (τ − τ_{s,±1}^{(2)})^{n_{s}}

×( M^{∘}^{(2)}((J_{κ}^{⊤}(x_{1}, 0))^{−1} · (0, ±1, τ)))^{−1}|_{τ=τ_{s,±1}^{(2)}},

c^{(j)}(x_{1}) = \frac{i^{j}}{2\sqrt{π}} Γ(j − \frac{1}{2})c_{0}(x_{1})

and c_{0}(x_{1}) is the first coefficient of the asymptotic expansion in (4.6).

5. Investigation of the Properties of Exponents of the First

Terms of an Asymptotic Expansion of Solutions of Problem M_{1} in the Neighbouhoods of Contact Boundaries

We consider the properties of exponents of the first terms of an asymptotic expansion of solutions of the boundary-contact problem M_{1} in the neighbourhood of the contact boundary ∂S_{0}^{(1)}. Analogous properties will be valid in the neighbourhood of the contact boundary ∂S_{0}^{(2)}, too.

For the sake of brevity, by γ_{m} and δ_{m} (m = 1, …, 6) we denote the real and the imaginary part of the first term exponent of the asymptotic expansions (4.14), and (4.15), i.e.,

γ_{m}(x_{1}) = \frac{1}{2} − |\frac{1}{2π} argλ_{m}(x_{1})|, δ_{m}(x_{1}) = −\frac{1}{2π} log |λ_{m}(x_{1})|,

m = 1, …, 6, x_{1} ∈ ∂S_{0}^{(1)},

where λ_{m}(x_{1}) = λ_{R}^{(m)}(x_{1}) m = 1, …, 12, are the eigenvalues of the matrix

b_{R}(x_{1}) = (σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1).

Theorem 5.1. The real parts γ_{m}, m = 1, …, 6, of the exponents of the first terms of the asymptotic expansion of solutions of the boundary-contact problem M_{1} near the contact boundary ∂S_{0}^{(1)} depend on the elastic constants and also on the geometry of the contact boundaries and may take any values from the interval ]0, 1/2[ , i.e.,

(a) if the elastic constants satisfy the limit conditions

\frac{a_{ijlk}^{(2)}}{a_{ijlk}^{(1)}} → 0, \frac{b_{ijlk}^{(2)}}{b_{ijlk}^{(1)}} → 0, \frac{c_{ijlk}^{(2)}}{c_{ijlk}^{(1)}} → 0,

then γ_{m} → 1/2 (m = 1, …, 6); (b) if for

\frac{a_{ijlk}^{(2)}}{a_{ijlk}^{(1)}} → ∞, \frac{b_{ijlk}^{(2)}}{b_{ijlk}^{(1)}} → ∞, \frac{c_{ijlk}^{(2)}}{c_{ijlk}^{(1)}} → ∞

the limiting relations

1. M_{2}^{+} \frac{|ζ_{1}|}{|ζ_{2}|} + M_{1}^{+} \frac{|ζ_{2}|}{|ζ_{1}|} → 0; 2. \frac{|Im⟨ζ_{1}, ζ_{2}⟩|}{|ζ_{1}| |ζ_{2}|} → c (c > 0),

are valid, then γ_{m} → 0 and δ_{m} → 0 (m = 1, …, 6), where ζ = (ζ_{1}, ζ_{2}) is the eigenvector of the matrix b_{R}, while M_{1}^{+} and M_{2}^{+} are maximal eigenvalues of the matrices σ_{N_{1}}^{+} = σ_{N_{1}}(x_{1}, 0, +1) and σ_{N_{2}}^{+} = σ_{N_{2}}(x_{1}, 0, +1).

Here the limits of the coefficients a_{ijlk}^{(q)}, b_{ijlk}^{(q)}, c_{ijlk}^{(q)} are understood in the uniform sense with respect to the indices i, j, l, k.

Proof. Multiply the coefficients a_{ijlk}^{(1)}, b_{ijlk}^{(1)}, c_{ijlk}^{(1)} by α (α > 0), and the coefficients a_{ijlk}^{(2)}, b_{ijlk}^{(2)}, c_{ijlk}^{(2)} by β (β > 0); i.e., we consider the differential equations with the elastic constants αa_{ijlk}^{(1)}, αb_{ijlk}^{(1)}, αc_{ijlk}^{(1)} and βa_{ijlk}^{(2)}, βb_{ijlk}^{(2)}, βc_{ijlk}^{(2)}.

(a) Taking into consideration the estimate obtained in [6] we have

|\frac{1}{2π} argλ_{m}(x_{1})| ≤ \frac{1}{π} arctg(\begin{matrix}√ & \overline{β} \\ √ & \overline{α} \end{matrix} \frac{1}{\sqrt{m_{1}^{+}m_{2}^{+}}} ),

where αm_{1}^{+}, β^{−1}m_{2}^{+} are minimal eigenvalues of the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+}, respectively.

Consequently, as β/α → 0 we get γ_{m} → \frac{1}{2} (m = 1, …, 6).

(b) Since

\frac{1}{2π} arg λ_{m}(x_{1}) = \frac{1}{2π} arctg\frac{2Im⟨ζ_{1}, ζ_{2}⟩}{⟨σ_{N_{2}}^{+}ζ_{1}, ζ_{1}⟩ + ⟨σ_{N_{1}}^{+}ζ_{2}, ζ_{2}⟩}

+ \frac{1}{2π} arctg\frac{2Im⟨ζ_{1}, ζ_{2}⟩}{⟨σ_{N_{2}}^{−}ζ_{1}, ζ_{1}⟩ + ⟨σ_{N_{1}}^{−}ζ_{2}, ζ_{2}⟩},

we obtain the estimate

|\frac{1}{2π} arg λ_{m}(x_{1})| ≥ \frac{1}{π} arctg\frac{\frac{2|Im⟨ζ_{1},ζ_{2}⟩|}{|ζ_{1}| |ζ_{2}|}}{β^{−1}M_{2}^{+} \frac{|ζ_{1}|}{|ζ_{2}|} + αM_{1}^{+} \frac{|ζ_{2}|}{|ζ_{1}|}} .  (5.1)

Further, since

|λ_{m}(x_{1})|^{2} = \frac{(⟨σ_{N_{2}}^{−}ζ_{1}, ζ_{1}⟩ + ⟨σ_{N_{1}}^{−}ζ_{2}, ζ_{2}⟩)^{2} + (2Im⟨ζ_{1}, ζ_{2}⟩)^{2}}{(⟨σ_{N_{2}}^{+}ζ_{1}, ζ_{1}⟩ + ⟨σ_{N_{1}}^{+}ζ_{2}, ζ_{2}⟩)^{2} + (2Im⟨ζ_{1}, ζ_{2}⟩)^{2}} ,

we get

\frac{1}{b} ≤ |λ_{m}(x_{1})|^{2} ≤ b,  (5.2)

where

b = \frac{(β^{−1}M_{2}^{+}|ζ_{1}|^{2} + αM_{1}^{+}|ζ_{2}|^{2})^{2} + (2Im⟨ζ_{1}, ζ_{2}⟩)^{2}}{(β^{−1}m_{2}^{+}|ζ_{1}|^{2} + αm_{1}^{+}|ζ_{2}|^{2})^{2} + (2Im⟨ζ_{1}, ζ_{2}⟩)^{2}} ;

here αM_{1}^{+} and β^{−1}M_{2}^{+} are maximal eigenvalues of the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+} respectively, while the vectors ζ_{1} and ζ_{2} depend in general on α and β.

Thus, using the inequalities (5.1) and (5.2) and taking into consideration the limiting relations 1 and 2 from subsection (b), for β/α → ∞ we get γ_{m} → 0 and δ_{m} → 0 (m = 1, …, 6). □

Remark 5.2. In the centrally symmetric isotropic case the exponents of the first terms of the asymptotic expansion of solutions of the boundary-contact problem M_{1} γ_{j} + iδ_{j} (j = 1, …, 6) are calculated explicitly.

Indeed, the differential operator of the couple-stress elasticity theory for a homogeneous isotropic centrally symmetric medium takes the form (see [26]):

M^{(q)}(∂_{x}) = ( \begin{matrix}M^{1} ^{(q)}(∂_{x}) & M^{2} ^{(q)}(∂_{x}) \\ M^{3} ^{(q)}(∂_{x}) & M^{4} ^{(q)}(∂_{x}) \end{matrix} )_{6×6} ,

M^{k} ^{(q)}(∂_{x}) = ∥ M^{k} _{ij}^{(q)}(∂_{x})∥_{3×3}, k = \overline{1, 4}, q = 1, 2,
M^{1} _{ij}^{(q)}(∂_{x}) = (μ_{q} + α_{q})δ_{ij}Δ + (λ_{q} + μ_{q} − α_{q})∂_{i}∂_{j},
M^{2} _{ij}^{(q)}(∂_{x}) = M^{3} _{ij}^{(q)}(∂_{x}) = −2α_{q} ∑_{k=1}^{3} ε_{ijk}∂_{k},

M^{4} _{ij}^{(q)}(∂_{x}) = δ_{ij}[(ν_{q} + β_{q})Δ − 4α_{q}] + (ε_{q} + ν_{q} − β_{q})∂_{i}∂_{j},

where δ_{ij} and ε_{ijk} are respectively the Kronecker and the Levi-Civita symbol. The coefficients λ_{q}, μ_{q}, α_{q}, ν_{q}, β_{q}, ε_{q}, q = 1, 2, are the elastic constants satisfying the conditions

μ_{q} > 0, 3λ_{q} + 2μ_{q} > 0, α_{q} > 0, ν_{q} > 0, 3ε_{q} + 2ν_{q} > 0, β_{q} > 0.

The stress operator of couple-stress elasticity is written as

N^{(q)}(∂_{z}, n(z)) = ( \begin{matrix}N^{1} ^{(q)}(∂_{z}, n(z)) & N^{2} ^{(q)}(∂_{z}, n(z)) \\ N^{3} ^{(q)}(∂_{z}, n(z)) & N^{4} ^{(q)}(∂_{z}, n(z)) \end{matrix} )_{6×6} ,

N^{k} ^{(q)}(∂_{z}, n(z)) = ∥ N^{k} _{ij}^{(q)}(∂_{z}, n(z))∥_{3×3}, k = \overline{1, 4}, q = 1, 2,
N^{1} _{ij}^{(q)}(∂_{z}, n(z)) = λ_{q}n_{i}(z)∂_{j} + (μ_{q} − α_{q})n_{j}(z)∂_{i} + (μ_{q} + α_{q})δ_{ij} \frac{∂}{∂n(z)} ,

N^{2} _{ij}^{(q)}(∂_{z}, n(z)) = −2α_{q} ∑_{k=1}^{3} ε_{ijk}n_{k}(z), N^{3} _{ij}^{(q)}(∂_{z}, n(z)) = 0,

N^{4} _{ij}^{(q)}(∂_{z}, n(z)) = ε_{q}n_{i}(z)∂_{j} + (ν_{q} − β_{q})n_{i}(z)∂_{i} + (ν_{q} + β_{q})δ_{ij} \frac{∂}{∂n(z)} .

The matrices σ_{N_{q}}^{±} = σ_{N_{q}}(x_{1}, 0, ±1), q = 1, 2, have the following expressions

σ_{N_{1}}^{±} = ( \begin{matrix}σ^{1} _{N_{1}}^{±} & 0 \\ 0 & σ^{2} _{N_{1}}^{±} \end{matrix} )_{6×6} , σ_{N_{2}}^{±} = ( \begin{matrix}σ^{1} _{N_{2}}^{±} & 0 \\ 0 & σ^{2} _{N_{2}}^{±} \end{matrix} )_{6×6} ,

454 O. CHKADUA

where

σ^{1} _{N_{1}}^{±} = ( \begin{matrix}μ_{1} + α_{1} & 0 & 0 \\ 0 & a_{1} & ∓ib_{1} \\ 0 & ±ib_{2} & a_{1} \end{matrix} )_{3×3} , σ^{2} _{N_{1}}^{±} = ( \begin{matrix}ν_{1} + β_{1} & 0 & 0 \\ 0 & c_{1} & ∓id_{1} \\ 0 & ±id_{1} & c_{1} \end{matrix} )_{3×3} ,

σ^{1} _{N_{2}}^{±} = ( \begin{matrix}\frac{1}{μ_{2} + α_{2}} & 0 & 0 \\ 0 & \frac{a_{2}}{a_{2}^{2} − b_{2}^{2}} & ∓i\frac{b_{2}}{a_{2}^{2} − b_{2}^{2}} \\ 0 & ±i\frac{b_{2}}{a_{2}^{2} − b_{2}^{2}} & \frac{a_{2}}{a_{2}^{2} − b_{2}^{2}} \end{matrix} )_{3×3} ,

σ^{2} _{N_{2}}^{±} = ( \begin{matrix}\frac{1}{ν_{2} + β_{2}} & 0 & 0 \\ 0 & \frac{c_{2}}{c_{2}^{2} − d_{2}^{2}} & ∓i\frac{d_{2}}{c_{2}^{2} − d_{2}^{2}} \\ 0 & ±i\frac{d_{2}}{c_{2}^{2} − d_{2}^{2}} & \frac{c_{2}}{c_{2}^{2} − d_{2}^{2}} \end{matrix} )_{3×3} ,

here

a_{q} = \frac{2(λ_{q} + 2μ_{q})(μ_{q} + α_{q})}{λ_{q} + α_{q} + 3μ_{q}} , b_{q} = \frac{2(μ_{q} + α_{q})^{2}}{λ_{q} + α_{q} + 3μ_{q}} , q = 1, 2,

and

c_{q} = \frac{2(ε_{q} + 2ν_{q})(ν_{q} + β_{q})}{ε_{q} + β_{q} + 3ν_{q}} , d_{q} = \frac{2(ν_{q} + β_{q})^{2}}{ε_{q} + β_{q} + 3ν_{q}} , q = 1, 2.

Hence in the considered case the eigenvalues λ_{j} (j = 1, …, 12) of the matrix

b_{R}(x_{1}) = (σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1)

are calculated in the manner as follows:

λ_{1,2} = \frac{(μ_{1} + α_{1}) − (μ_{2} + α_{2}) ± 2i\sqrt{(μ_{1} + α_{1})(μ_{2} + α_{2})}}{μ_{1} + α_{1} + μ_{2} + α_{2}} ,
λ_{3,4} = \{ \begin{matrix}\frac{B±\sqrt{B^{2}−AC}}{A} & for B^{2} − AC ≥ 0, \\ \frac{B±i\sqrt{AC−B^{2}}}{A} & for B^{2} − AC < 0, \end{matrix}
λ_{5,6} = \{ \begin{matrix}\frac{B±\sqrt{B^{2}−AC}}{C} & for B^{2} − AC ≥ 0, \\ \frac{B±i\sqrt{AC−B^{2}}}{C} & for B^{2} − AC < 0, \end{matrix}
λ_{7,8} = \frac{(ν_{1} + β_{1}) − (ν_{2} + β_{2}) ± 2i\sqrt{(ν_{1} + β_{1})(ν_{2} + β_{2})}}{ν_{1} + β_{1} + ν_{2} + β_{2}} ,
λ_{9,10} = \{ \begin{matrix}\frac{B̃±\sqrt{B̃^{2}−ÃC̃}}{Ã} & for B̃^{2} − ÃC̃ ≥ 0, \\ \frac{B̃±i\sqrt{ÃC̃−B̃^{2}}}{Ã} & for B̃^{2} − ÃC̃ < 0, \end{matrix}

λ_{11,12} = \{ \begin{matrix}\frac{B̃±\sqrt{B̃^{2}−ÃC̃}}{C̃} & for B̃^{2} − ÃC̃ ≥ 0, \\ \frac{B̃±i\sqrt{ÃC̃−B̃^{2}}}{C̃} & for B̃^{2} − ÃC̃ < 0, \end{matrix}

where

A = a_{2}a_{1} + b_{1}b_{2} + a_{2}^{2} − b_{2}^{2} + a_{2}b_{1} + b_{2}a_{1}, B = a_{2}a_{1} − b_{1}b_{2} − a_{2}^{2} + b_{2}^{2},
C = a_{2}a_{1} + b_{1}b_{2} + a_{2}^{2} − b_{2}^{2} − a_{2}b_{1} − b_{2}a_{1},
Ã = c_{2}c_{1} + d_{1}d_{2} + c_{2}^{2} − d_{2}^{2} + c_{2}d_{1} + d_{2}c_{1}, B̃ = c_{2}c_{1} − d_{1}d_{2} − c_{2}^{2} + d_{2}^{2},
C̃ = c_{2}c_{1} + d_{1}d_{2} + c_{2}^{2} − d_{2}^{2} − c_{2}d_{1} − d_{2}c_{1}.

Hence we obtain

γ_{1} = \frac{1}{2} − \frac{1}{π} arctg\begin{matrix}√ & \overline{μ_{2} + α_{2}} \\ √ & \overline{μ_{1} + α_{1}} \end{matrix} , δ_{1} = 0,

γ_{2,3} = \{ \begin{matrix}\frac{1}{2} , & B^{2} − AC ≥ 0, \\ \frac{1}{2} − \frac{1}{2π} arctg\frac{\sqrt{AC−B^{2}}}{B} , & B^{2} − AC < 0, B > 0, \\ \frac{1}{2π} arctg\frac{\sqrt{AC−B^{2}}}{|B|} & B^{2} − AC < 0, B < 0, \end{matrix}

δ_{2,3} = \{ \begin{matrix}−\frac{1}{2π} log \frac{B±\sqrt{B^{2}−AC}}{A} , & B^{2} − AC ≥ 0, \\ ∓\frac{1}{2π} log (\frac{A}{C}), & B^{2} − AC < 0, \end{matrix}

γ_{4} = \frac{1}{2} − \frac{1}{π} arctg\begin{matrix}√ & \overline{ν_{2} + β_{2}} \\ √ & \overline{ν_{1} + β_{1}} \end{matrix} , δ_{4} = 0,

γ_{5,6} = \{ \begin{matrix}\frac{1}{2} , & B̃^{2} − ÃC̃ ≥ 0, \\ \frac{1}{2} − \frac{1}{2π} arctg^{\frac{\sqrt{ÃC̃−B̃^{2}}}{B̃}} , & B̃^{2} − ÃC̃ < 0, B̃ > 0, \\ \frac{1}{2π} arctg^{\frac{\sqrt{ÃC̃−B̃^{2}}}{|B̃|}} , & B̃^{2} − ÃC̃ < 0, B̃ < 0, \end{matrix}

δ_{5,6} = \{ \begin{matrix}−\frac{1}{2π} log ^{\frac{B̃±\sqrt{B̃^{2}−ÃC̃}}{Ã}} , & B̃^{2} − ÃC̃ ≥ 0, \\ ∓\frac{1}{2π} log (\frac{Ã}{C̃}), & B̃^{2} − ÃC̃ < 0. \end{matrix}

Remark 5.3. If B^{2} − AC ≠ 0 and B̃^{2} − ÃC̃ ≠ 0, then B_{a_{pr}}^{0}(t) = I, i.e., the first terms of the asymptotic expansion of solutions of the boundary-contact problem M_{1} contain no logarithms near the contact boundary ∂S_{0}^{(i)}, i = 1, 2.

Remark 5.4. Note that in the isotropic case we obtain more exact limit relations (for the elasticity case see [6]):

a) if B > 0, B^{2} − AC ≥ 0 and μ_{2} + α_{2} → 0 or μ_{1} → ∞, then γ_{1} → 1/2, γ_{2} = γ_{3} = 1/2;

b) if B > 0, B^{2} − AC < 0 and μ_{1} → ∞, then γ_{m} → 1/2, m = 1, 2, 3;

c) if B > 0, B^{2} − AC < 0 and μ_{2} + α_{2} → 0, then γ_{1} → 1/2, γ_{2,3} → 1/4;

d) if B < 0 and μ_{1} + α_{1} → 0 or μ_{2} → ∞, then γ_{m} → 0, m = 1, 2, 3;

e) if |μ_{1} − μ_{2}| → 0, |α_{1} − α_{2}| → 0, then γ_{m} → 1/4, m = 1, 2, 3, and

ã) if B̃ > 0, B̃^{2} − ÃC̃ ≥ 0 and ν_{2} + β_{2} → 0 or ν_{1} → ∞, then γ_{4} → 1/2, γ_{5,6} = 1/2;

b̃) if B̃ > 0, B̃^{2} − ÃC̃ < 0 and ν_{1} → ∞, then γ_{m} → 1/2, m = 4, 5, 6; c̃) if B̃ > 0, B̃^{2} − ÃC̃ < 0 and ν_{2} + β_{2} → 0, then γ_{4} → 1/2, γ_{5,6} → 1/4; d̃) if B̃ < 0, ν_{1} + β_{1} → 0 or ν_{2} → ∞, then γ_{m} → 0, m = 4, 5, 6; ẽ) if |ν_{1} − ν_{2}| → 0, |β_{1} − β_{2}| → 0, then γ_{m} → 1/4, m = 4, 5, 6.

It is not difficult to see that if the conditions

B^{2} − AC > 0 and B̃^{2} − ÃC̃ > 0

hold, then the exponent of the first term of the asymptotics of solutions of the boundary-contact problem M_{1} has the form

γ = \frac{1}{2} − \frac{1}{π} arctg max \begin{cases}√ & \overline{μ_{2} + α_{2}} \\ √ & \overline{μ_{1} + α_{1}} \end{cases} , \begin{matrix}√ & \overline{ν_{2} + β_{2}} \\ √ & \overline{ν_{1} + β_{1}} \end{matrix} \}.

Since γ < \frac{1}{2}, γ_{2,3} = γ_{5,6} = \frac{1}{2}, the oscillation of solutions vanishes in some neighbourhood of the contact boundary and therefore solutions describe the real physical process.

Note that this class has been found only in the spatial case since in the plane case it is known that the oscillation does not vanish.

In the general case we have found a class of anisotropic bodies when the oscillation in the asymptotic expansion vanishes and the real parts γ_{j} j = 1, …, 6, of exponents of the first terms of the asymptotic expansion are calculated by simpler formulas.

Let α_{j} > 0 and β_{j} > 0, j = 1, …, 6, be the eigenvalues of the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+}, respectively.

Theorem 5.5. If the conditions

1) rank ( \begin{matrix}σ_{N_{1}}^{+} − α_{j}I \\ σ_{N_{2}}^{+} − β_{j}I \end{matrix} ) < 6, j = 1, …, 6;

2) ⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ ≠ 0, j = 1, …, 6, are satisfied, where ζ_{1}^{(j)} (j = 1, …, 6) is the common eigenvector of the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+}, which correspond to the eigenvalues α_{j} and β_{j}, then the oscillation participanting in the asymptotic expansion of solutions of the boundary-contact problem M_{1} vanishes, i.e., δ_{j} = 0, j = 1, …, 6, and the real parts of the exponents of the first terms of the asymptotic expansion are calculated by a simpler formula

γ_{j} = \frac{1}{2} − \frac{1}{π} arctg\frac{1}{\sqrt{α_{j}β_{j}}} , j = 1, …, 6.

Proof. From the condition 1) we obtain the existence of the common eigenvectors ζ_{1}^{(j)}, j = 1, …, 6, for the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+}, i.e.,

σ_{N_{1}}^{+}ζ_{1}^{(j)} = α_{j}ζ_{1}^{(j)}, σ_{N_{2}}^{+}ζ_{1}^{(j)} = β_{j}ζ_{1}^{(j)}, j = 1, …, 6.  (5.3)

Since \overline{σ}_{N_{1}}^{+} = σ_{N_{1}}^{−} and \overline{σ}_{N_{2}}^{+} = σ_{N_{2}}^{−}, we have

σ_{N_{1}}^{−}\overline{ζ}_{1}^{(j)} = α_{j}\overline{ζ}_{1}^{(j)}, σ_{N_{2}}^{−}\overline{ζ}_{1}^{(j)} = β_{j}\overline{ζ}_{1}^{(j)}, j = 1, …, 6.  (5.4)

Further, we look for an eigenvector ζ^{(j)} of the matrix

(σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1)

in the form ζ^{(j)} = (ζ_{1}^{(j)}, γ̃_{j}ζ_{1}^{(j)}). We obtain

(σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1)ζ^{(j)} = λ_{j}ζ^{(j)}.

Taking into consideration the expression of the matrices σ_{R}(x_{1}, 0, ±1), we get

\{ \begin{matrix}σ_{N_{2}}^{−}ζ_{1}^{(j)} − γ̃_{j}ζ_{1}^{(j)} = λ_{j}σ_{N_{2}}^{+}ζ_{1}^{(j)} + λ_{j}γ̃_{j}ζ_{1}^{(j)}, \\ ζ_{1}^{(j)} + γ̃_{j}σ_{N_{1}}^{−}ζ_{1}^{(j)} = −λ_{j}ζ_{1}^{(j)} + λ_{j}γ̃_{j}σ_{N_{1}}^{+}ζ_{1}^{(j)}. \end{matrix}  (5.5)

Now, by the scalar multiplication of both equations (5.5) by the vector \overline{ζ}^{(j)} we have

\{ \begin{matrix}⟨σ_{N_{2}}^{−}ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ − γ̃_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ = λ_{j}⟨σ_{N_{2}}^{+}ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + λ_{j}γ̃_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩, \\ ⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + γ̃_{j}⟨σ_{N_{1}}^{−}ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ = −λ_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + λ_{j}γ̃_{j}⟨σ_{N_{1}}^{+}ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩. \end{matrix} (5.6)

Substituting (5.3) and (5.4) into (5.6), we get

\{ \begin{matrix}β_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ − γ̃_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ = λ_{j}β_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + λ_{j}γ̃_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩, \\ ⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + γ̃_{j}α_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ = −λ_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ + λ_{j}γ̃_{j}α_{j}⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩. \end{matrix}

Taking into account the condition 2), i.e., ⟨ζ_{1}^{(j)}, \overline{ζ}_{1}^{(j)}⟩ ≠ 0, we obtain the following system of equations with respect to λ_{j} and γ̃_{j} (j = 1, …, 6):

\{ \begin{matrix}β_{j} − γ̃_{j} = λ_{j}β_{j} + λ_{j}γ̃_{j}, \\ 1 + γ̃_{j}α_{j} = −λ_{j} + λ_{j}γ̃_{j}α_{j}. \end{matrix}

Further,

λ_{j} = \frac{\sqrt{α_{j}β_{j}} ∓ i}{\sqrt{α_{j}β_{j}} ± i} , γ̃_{j} = ±i\sqrt{\frac{β_{j}}{α_{j}}} , j = 1, …, 6.

Clearly,

|λ_{j}| = 1, j = 1, …, 6.

Hence

δ_{j} = 0, j = 1, …, 6.

Calculating γ_{j} (j = 1, …, 6), we get

γ_{j} = \frac{1}{2} − \frac{1}{π} arctg\frac{1}{\sqrt{α_{j}β_{j}}} , j = 1, …, 6. □

If under the conditions of Theorem 5.5

α_{i}β_{i} ≠ α_{j}β_{j} for all i ≠ j, i, j = 1, …, 6,

then B_{a_{pr}}^{0}(t) = I, i.e., the first terms of the asymptotic expansions (4.14), (4.15) contain no logarithms.

Remark 5.6. Let us assume that in the neighborhood of the boundary the contact surfaces S_{0}^{(1)} and S_{0}^{(2)} are parallel to the isotropic plane. We consider the case, where the coefficients b_{ijlk}^{(q)} = 0 and, instead of a_{ijlk}^{(q)} and c_{ijlk}^{(q)}, we have the elastic constants of transversally-isotropic elastic bodies, i.e., instead of a_{ijlk}^{(q)} we have

a_{11}^{(q)}, a_{33}^{(q)}, a_{13}^{(q)}, a_{55}^{(q)}, a_{66}^{(q)}, q = 1, 2,

and, instead of c_{ijlk}^{(q)}, we have

c_{11}^{(q)}, c_{33}^{(q)}, c_{13}^{(q)}, c_{55}^{(q)}, c_{66}^{(q)}, q = 1, 2,

which satisfy the following conditions:

\{ \begin{matrix}a_{11}^{(q)} − a_{66}^{(q)} > 0, a_{55}^{(q)} > 0, a_{66}^{(q)} > 0, \\ a_{33}^{(q)} > \frac{(a_{13}^{(q)})^{2}}{a_{11}^{(q)} − a_{66}^{(q)}} , q = 1, 2, \end{matrix} \{ \begin{matrix}c_{11}^{(q)} − c_{66}^{(q)} > 0, c_{55}^{(q)} > 0, c_{66}^{(q)} > 0, \\ c_{33}^{(q)} > \frac{(c_{13}^{(q)})^{2}}{c_{11}^{(q)} − c_{66}^{(q)}} , q = 1, 2. \end{matrix}

It is not difficult to see that in this case the eigenvalues α_{j} and β_{j}, j = 1, …, 6, of the matrices σ_{N_{1}}^{+} and σ_{N_{2}}^{+} are calculated explicitly.

Let a_{11}^{(q)} ≠ a_{33}^{(q)} and c_{11}^{(q)} ≠ c_{33}^{(q)} (q = 1, 2); then the conditions 1)–2) of Theorem 5.5 have the form

\frac{C_{1} − D_{1}}{D_{2} − C_{2}} = \frac{B_{1}}{B_{2}} , \frac{C̃_{1} − D̃_{1}}{D̃_{2} − C̃_{2}} = \frac{B̃_{1}}{B̃_{2}} ,

where

B_{q} = \frac{a_{55}^{(q)} (\sqrt{a_{11}^{(q)}}\sqrt{a_{33}^{(q)}} − a_{13}^{(q)})}{a_{55}^{(q)} + \sqrt{a_{11}^{(q)}}\sqrt{a_{33}^{(q)}}}, C_{q} = \frac{1}{2} \frac{a_{11}^{(q)}a_{55}^{(q)} (\sqrt{a_{2}^{(q)}} + \sqrt{a_{3}^{(q)}})}{a_{55}^{(q)} + \sqrt{a_{11}^{(q)}}\sqrt{a_{33}^{(q)}}},

D_{q} = \frac{1}{2} \frac{\sqrt{a_{33}^{(q)}}\sqrt{a_{11}^{(q)}}a_{55}^{(q)} (\sqrt{a_{2}^{(q)}} + \sqrt{a_{3}^{(q)}})}{a_{55}^{(q)} + \sqrt{a_{11}^{(q)}}\sqrt{a_{33}^{(q)}}}, q = 1, 2,

a_{2}^{(q)} and a_{3}^{(q)} are the roots of the equation (see [26])

a_{11}^{(q)}a_{55}^{(q)}a^{2} + [(a_{13}^{(q)} + a_{55}^{(q)})^{2} − a_{11}^{(q)}a_{33}^{(q)} − (a_{55}^{(q)})^{2}]a + a_{33}^{(q)}a_{55}^{(q)} = 0, q = 1, 2,

and

B̃_{q} = \frac{c_{55}^{(q)} (\sqrt{c_{11}^{(q)}}\sqrt{c_{33}^{(q)}} − c_{13}^{(q)})}{c_{55}^{(q)} + \sqrt{c_{11}^{(q)}}\sqrt{c_{33}^{(q)}}}, C̃_{q} = \frac{1}{2} \frac{c_{11}^{(q)}c_{55}^{(q)} (\sqrt{c_{2}^{(q)}} + \sqrt{c_{3}^{(q)}})}{c_{55}^{(q)} + \sqrt{c_{11}^{(q)}}\sqrt{c_{33}^{(q)}}},

D̃_{q} = \frac{1}{2} \frac{\sqrt{c_{33}^{(q)}}\sqrt{c_{11}^{(q)}}c_{55}^{(q)} (\sqrt{c_{2}^{(q)}} + \sqrt{c_{3}^{(q)}})}{c_{55}^{(q)} + \sqrt{c_{11}^{(q)}}\sqrt{c_{33}^{(q)}}}, q = 1, 2,

where c_{2}^{(q)} and c_{3}^{(q)} are the roots of the equation

c_{11}^{(q)}c_{55}^{(q)}c^{2} + [(c_{13}^{(q)} + c_{55}^{(q)})^{2} − c_{11}^{(q)}c_{33}^{(q)} − (c_{55}^{(q)})^{2}]c + c_{33}^{(q)}c_{55}^{(q)} = 0, q = 1, 2.

If we assume that a_{13}^{(q)} = −a_{55}^{(q)} and c_{13}^{(q)} = −c_{55}^{(q)}, q = 1, 2, then the conditions 1)–2) of Theorem 5.5 can be rewritten in a simpler form:

\frac{\sqrt{a_{11}^{(1)}} − \sqrt{a_{33}^{(1)}}}{\sqrt{a_{33}^{(2)}} − \sqrt{a_{11}^{(2)}}} = \begin{matrix}√ & \overline{a_{55}^{(1)}} \\ √ & \overline{a_{55}^{(2)}} \end{matrix} and \frac{\sqrt{c_{11}^{(1)}} − \sqrt{c_{33}^{(1)}}}{\sqrt{c_{33}^{(2)}} − \sqrt{c_{11}^{(2)}}} = \begin{matrix}√ & \overline{c_{55}^{(1)}} \\ √ & \overline{c_{55}^{(2)}} \end{matrix} .

Now let us consider the case, where the domains D_{q}, q = 1, 2, are filled with the same material. The following theorem is fulfilled (for analogous results in the case of elasticity theory see [7], [8]; see also [14]).

Theorem 5.7. If the domains D_{q}, q = 1, 2, are filled with the same material, then the asymptotic expansion of solutions of the boundary-contact problem M_{1} near the contact boundary ∂S_{0}^{(1)} takes the form

U^{(q)}(x_{1}, x_{2}, x_{3}) = (u^{(q)}, ω^{(q)})(x_{1}, x_{2}, x_{3})

= ∑_{ϑ=±1} ∑_{j=1}^{2} ∑_{s=1}^{l_{0}} Re\{ ∑_{m=0}^{n_{s}−1} x_{3}^{m}[d_{sjm}^{(q)}(x_{1}, θ)(z_{s,theta}^{(q)})^{\frac{1}{4} +Δ_{j}(x_{1})−m}]c_{jm}^{(q)}(x_{1})

+ \begin{matrix}∑_{k,l=0}^{M+2} & ∑_{p+m=0}^{M+2−l} \\ k+l+p+m≠0 \end{matrix} x_{2}^{l}x_{3}^{m}d_{slmpj}^{(q)}(x_{1}, ϑ)(z_{s,ϑ}^{(q)})^{\frac{1}{4} +Δ_{j}(x_{1})+k+p}B_{skmpj}^{(q)}(x_{1}, log z_{s,ϑ}^{(q)})\}

+U_{M+1}^{(q)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s − 1], 1\}, q = 1, 2,  (5.7)

with the coefficients d_{sjm}^{(q)}(· , ±1), c_{jm}^{(q)}, d_{slmpj}^{(q)}(· , ±1) ∈ C_{0}^{∞}(ℝ) and the remainder U_{M+1}^{(q)} ∈ C_{0}^{M+1}(\overline{ℝ}_{±}^{3}), q = 1, 2,

z_{s,+1}^{(q)} = (−1)^{q}(x_{2} +x_{3}τ_{s,+1}^{(q)}), z_{s,−1}^{(q)} = (−1)^{q+1}(x_{2} −x_{3}τ_{s,−1}^{(q)}), −π < Argτ_{s,±1}^{(q)} < π.

In this case the parameters Δ_{j} are calculated by

Δ_{j}(x_{1}) = (δ_{1}^{(j)}(x_{1}), …, δ_{6}^{(j)}(x_{1})), j = 1, 2,

where

δ_{k}^{(1)}(x_{1}) = −\frac{i}{2π} log |λ_{k}(x_{1})|, δ_{k}^{(2)}(x_{1}) = \frac{1}{2} − \frac{i}{2π} log |λ_{k}(x_{1})|, k = 1, …, 6.

Proof. When the domains D_{q}, q = 1, 2, are filled with the same material, we can show, like in [7], that the eigenvalues λ_{k}(x_{1}), k = 1, …, 12, of the matrix

(σ_{R}(x_{1}, 0, +1))^{−1}σ_{R}(x_{1}, 0, −1), x_{2} ∈ ∂S_{0}^{(1)},

are calculated by means of the eigenvalues β_{k}(x_{1}), k = 1, 2, 3, of the matrix σ_{𝕍^{∗}}_{0}, i.e.,

λ_{k}(x_{1}) = \{ \begin{matrix}i\sqrt{\frac{1 − 2β_{k}(x_{1})}{1 + 2β_{k}(x_{1})}} , & if k = 1, …, 6, \\ −i\sqrt{\frac{1 − 2β_{k−6}(x_{1})}{1 + 2β_{k−6}(x_{1})}} , & if k = 7, …, 12, \end{matrix}

where −1/2 < β_{k} < 1/2, k = 1, …, 6.

Hence the exponent of the first term of the asymptotic expansion of solutions has the form

\frac{1}{2} ∓ \frac{1}{4} − \frac{i}{2π} log |λ_{k}(x_{1})|, k = 1, …, 6. □

Note that in this case the asymptotic expansion has step equal to one half (see [8]).

The same asymptotic expansion can also be obtained near the contact boundary ∂S_{0}^{(2)}.

6. Solvability and Asymptotics of Solutions of the Boundary-Contact Wedge-Type Problem M_{2}

We will formulate theorems of the uniqueness and existence of solutions of problem M_{2}.

Assume that the following compatibility conditions are fulfilled on the curves ∂S_{0}^{(1)}, ∂S_{0}^{(2)} ∂S_{0}:

∃Φ_{0}^{(i)}, i = 1, 2, 3, Φ_{0}^{(i)} ∈ 𝔹_{p,r}^{s−1}(∂D_{1}), Φ_{0}^{(2)} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(1)}), Φ_{0}^{(3)} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(2)}),

\begin{matrix}h_{1} − π_{S_{0}^{(1)}}Φ_{0}^{(1)} + π_{S_{0}^{(1)}}Φ_{0}^{(2)} ∈ 𝔹̃_{p,r}^{s−1}(S_{0}^{(1)}), \\ h_{2} − π_{S_{0}^{(2)}}Φ_{0}^{(1)} + π_{S_{0}^{(2)}}Φ_{0}^{(3)} ∈ 𝔹̃_{p,r}^{s−1}(S_{0}^{(2)}) \\ π_{S_{0}}Φ_{0}^{(2)} + π_{S_{0}}Φ_{0}^{(3)} ∈ 𝔹̃_{p,r}^{s−1}(S_{0}) \end{matrix}  (6.1)

hold for φ_{1} ∈ 𝔹_{p,r}^{s−1}(∂D_{1}), φ_{2} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(1)}), φ_{3} ∈ 𝔹_{p,r}^{s−1}(∂D_{2}^{(2)}), 1 < p < ∞, 1 ≤ r ≤ ∞, 1/p − 1/2 < s < 1/p + 1/2.

Here Φ_{0}^{(i)}, i = 1, 2, 3, are some fixed extensions of the functions φ_{i}, i = 1, 2, 3, on ∂D_{1}, ∂D_{2}^{(1)}, ∂D_{2}^{(2)}, respectively.

Theorem 6.1. Let 4/3 < p < 4 and the compatibility conditions (6.1) be fulfilled on ∂S_{0}^{(1)}, ∂S_{0}^{(2)} ∂S_{0} for s = 1 − 1/p. Then the boundary-contact wedgetype problem M_{2} has solutions of the classes W_{p}^{1}(D_{q}), q = 1, 2, if and only if the condition

∫_{S_{1}} φ_{1}([a × z] + b, a) ds − ∫_{S_{2}^{(1)}} φ_{2}([a × z] + b, a) ds − ∫_{S_{2}^{(2)}} φ_{3}([a × z] + b, a) ds

+ ∫_{S_{0}^{(1)}} h_{1}([a × z] + b, a) ds + ∫_{S_{0}^{(2)}} h_{2}([a × z] + b, a) ds = 0

is fulfilled, where a and b are arbitrary three-dimensional constant vectors.

Solutions of the boundary-contact problem M_{2} are given by the potential-type functions

U^{(1)} = 𝕍^{(1)}(𝕍_{−1}^{(1)})^{−1}(B_{2M+1}^{(1)})^{−1}φ_{0}^{(1)} + 𝕍^{(1)}(𝕍_{−1}^{(1)})^{−1}(B_{2M+1}^{(1)})^{−1}ψ_{0}^{(1)} + R_{1}, (6.2)
r_{1}U^{(2)} = 𝕍^{(2)}(𝕍_{−1}^{(2)})^{−1}(B_{2M+1}^{(2)})^{−1}φ_{0}^{(2)} + 𝕍_{−1}^{(2)}(𝕍_{−1}^{(2)})^{−1}(B_{2M+1}^{(2)})^{−1}ψ_{0}^{(2)} + R_{2}, (6.3)
r_{2}U^{(2)} = 𝕍^{(3)}(𝕍_{−1}^{(3)})^{−1}(B_{2M+1}^{(3)})^{−1}φ_{0}^{(3)} + 𝕍_{−1}^{(3)}(𝕍_{−1}^{(3)})^{−1}(B_{2M+1}^{(3)})^{−1}ψ_{0}^{(3)} + R_{3}, (6.4)

with R_{1} ∈ C^{M+1}(\overline{D}_{1}), R_{2} ∈ C^{M+1}(\overline{D}_{2}^{(1)}), R_{3} ∈ C^{M+1}(\overline{D}_{2}^{(2)}), while the functions (φ_{0}^{(1)}, φ_{0}^{(2)}), (ψ_{0}^{(1)}, φ_{0}^{(3)}) and (ψ_{0}^{(2)}, ψ_{0}^{(3)}) are solutions of the systems

\{ \begin{matrix}π_{S_{0}^{(1)}}N_{1}φ_{0}^{(1)} = Ψ_{1}, \\ φ_{0}^{(2)} = φ_{0}^{(1)} + G_{1}, \end{matrix} \{ \begin{matrix}π_{S_{0}^{(2)}}N_{2}ψ_{0}^{(1)} = Ψ_{2}, \\ φ_{0}^{(3)} = ψ_{0}^{(1)} + G_{2}, \end{matrix}  \{ \begin{matrix}π_{S_{0}}N_{3}ψ_{0}^{(2)} = Ψ_{3}, \\ ψ_{0}^{(3)} = −ψ_{0}^{(2)} + G_{3}, \end{matrix}

respectively. Here

N_{1} = (B_{2M+1}^{(1)})^{−1} − (B_{2M+1}^{(2)})^{−1},
N_{2} = (B_{2M+1}^{(1)})^{−1} − (B_{2M+1}^{(3)})^{−1},
N_{3} = −(B_{2M+1}^{(2)})^{−1} − (B_{2M+1}^{(3)})^{−1},

where

B_{2M+1}^{(1)} = −(𝕍_{−1}^{(1)})^{2M+1} + ( − \frac{1}{2}I+ 𝕍^{∗} _{0}^{(1)})(𝕍_{−1}^{(1)})^{−1},

B_{2M+1}^{(i)} = (𝕍_{−1}^{(i)})^{2M+1} + (\frac{1}{2}I+ 𝕍^{∗} _{0}^{(i)})(𝕍_{−1}^{(i)})^{−1}, i = 2, 3.

The operators N_{j}, j = 1, 2, 3, are positive-definite ΨDOs.

Let

Δ_{N_{j}}(x_{1}) = ( \underbrace{δ_{j}^{(1)}(x_{1}), …, δ_{j}^{(1)}(x_{1})}_{m_{1}-times}, …, \underbrace{δ_{j}^{(ℓ)}(x_{1}), …, δ_{j}^{(ℓ)}(x_{1})}_{m_{ℓ}-times} ),

δ_{j}^{(k)}(x_{1}) = −\frac{i}{2π} log |λ_{N_{j}}^{(k)}(x_{1})|, j = 1, 2, 3, k = 1, …, ℓ

(generally speaking, m_{k} and ℓ depend on j),

where λ_{N_{j}}^{(k)}, j = 1, 2, k = 1, …, ℓ, are the eigenvalues of the matrix

b_{N_{j}}(x_{1}) = (σ_{N_{j}}(x_{1}, 0, +1)^{−1}σ_{N_{j}}(x_{1}, 0, −1)

of multiplicity m_{k}, k = 1, …, ℓ, ∑_{k=1}^{6} m_{k} = 6; here m_{k}, k = 1, …, ℓ, and ℓ

depend on j, and the eigenvalues of the matrix b_{N_{3}} = I are trivial, λ_{N_{3}}^{(k)} = 1, k = 1, …, 6.

Note that the boundary data of the problem M_{2} are assumed to be sufficiently smooth, i.e., φ_{1} ∈ ℍ_{r}^{(∞,S+2M),∞}(S_{1}), φ_{2} ∈ ℍ_{r}^{(∞,S+2M),∞}(S_{2}^{(1)}), φ_{3} ∈ ℍ_{r}^{(∞,S+2M),∞}(S_{2}^{(2)}), f_{i} ∈ ℍ_{r}^{(∞,S+2M+1),∞}(S_{0}^{(i)}), h_{i} ∈ ℍ_{r}^{(∞,S+2M),∞}(S_{0}^{(i)}), i = 1, 2, \frac{1}{r} − \frac{1}{2} < s < \frac{1}{r} + \frac{1}{2}.

Recalling that solutions of the problem M_{2} are represented by potential-type functions (see (6.2), (6.3), (6.4)) and using the asymptotic expansion of such functions (see [9, Theorems 2.2 and 2.3]), we obtain the following asymptotic expansions of problem M_{2} in terms of some local coordinate systems of curves ∂S_{0}^{(1)}, ∂S_{0}^{(2)}, ∂S_{0}:

a) Asymptotic expansion near the contact boundaries ∂S_{0}^{(j)}, j = 1, 2:

U^{(q)}(x_{1}, x_{2}, x_{3}) = (u^{(q)}, ω^{(q)})(x_{1}, x_{2}, x_{3})

= ∑_{ϑ=±1} ∑_{s=1}^{ℓ_{0}} Re\{ ∑_{j=0}^{n_{s}−1} x_{3}^{j}[d_{sj}^{(q)}(x_{1}, θ)(z_{s,θ}^{(q)})^{\frac{1}{2} −j+iΔ(x_{1})}]c^{(j)}(x_{1})

+ \begin{matrix}∑_{l,k=0}^{M+1} & ∑_{p+j=1}^{M+2−l} \\ k+l+p+j≠1 \end{matrix} x_{2}^{l}x_{3}^{j}d_{sljp}^{(q)}(x_{1}, ϑ)z_{s,ϑ}^{− \frac{1}{2} +iΔ(x_{1})+p+k}B_{skjp}^{(q)}(x_{1}, z_{s,ϑ}^{(q)})\}

U_{M+1}^{(q)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s], 0\}, q = 1, 2,  (6.5)

with U_{M+1}^{(q)} ∈ C_{0}^{M+1}(\overline{ℝ}_{±}^{3}), q = 1, 2. Here Δ = Δ_{N_{j}}, j = 1, 2, and

z_{s,+1}^{(q)} = (−1)^{q}[x_{2} + x_{3}τ_{s,+1}^{(q)}], z_{s,−1}^{(q)} = (−1)^{q}[x_{2} − x_{3}τ_{s,−1}^{(q)}], −π < Argz_{s,±1} < π.

The first coefficients d_{sj}^{(q)}(·, ±1), c^{(j)} are calculated as in(4.14) (see [9]).

b) Asymptotic expansions near the cuspidal edge ∂S_{0} (see (4.16)):

r_{i}U^{(2)}(x_{1}, x_{2}, x_{3}) = r_{i}(u^{(2)}, ω^{(2)})(x_{1}, x_{2}, x_{3})

= ∑_{ϑ=±1} ∑_{s=1}^{ℓ_{0}} Re\{ ∑_{j=0}^{n_{s}−1} x_{3}^{j}z_{s,θ}^{\frac{1}{2} −j}d_{sj}^{(i)}(x_{1}, θ) + \begin{matrix}∑_{l,k=0}^{M+1} & ∑_{j+p=1}^{M+2−l} \\ l+k+j+p≠1 \end{matrix} x_{2}^{l}x_{3}^{j}z_{s,ϑ}^{− \frac{1}{2} +p+k}d_{slkjp}^{(i)}(x_{1})\}

+U_{M+1}^{(i)}(x_{1}, x_{2}, x_{3}) for M > \frac{2}{r} − min\{[s], 0\}, i = 1, 2,  (6.6)

with U_{M+1}^{(i)} ∈ C_{0}^{M+1}(\overline{ℝ}_{±}^{3}), i = 1, 2. Here

z_{s,+1} = −x_{2} − x_{3}τ_{s,+1}^{(2)}, z_{s,−1} = −x_{2} − x_{3}τ_{s,−1}^{(2)}, −π < Argz_{s,±1} < π.

The coefficients d_{sj}^{(i)}(x_{1}, ±1) are calculated as in (4.16).

Remark 6.2. Note that if we consider the boundary-contact problem M_{2} for the nonhomogeneous equations

M^{(q)}(∂_{x})U^{(q)} + F^{(q)} = 0 in D_{q}, q = 1, 2,

and use the boundary and boundary-contact data

φ_{1} = 0, φ_{2} = 0, φ_{3} = 0, f_{i} = h_{i} = 0, i = 1, 2,

then the compatibility conditions (6.1) are fulfilled automatically on ∂S_{0}^{(1)}, ∂S_{0}^{(2)}, ∂S_{0}.

Remark 6.3. It is easy to see that if the matrices b_{N_{j}}, j = 1, 2, are unitary (b_{N_{j}}^{∗} = b_{N_{j}}^{−1}), then the oscillation in the asymptotic expansion (42) vanishes, i.e., Δ_{j} = 0, j = 1, 2.

In the case of transversally isotropic bodies we obtain the following necessary and sufficient conditions under which the oscillation vanishes:

\frac{B_{1}}{B_{2}} = \frac{4C_{1}D_{1} − B_{1}^{2}}{4C_{2}D_{2} − B_{2}^{2}} , \frac{B̃_{1}}{B̃_{2}} = \frac{4C̃_{1}D̃_{1} − B̃_{1}^{2}}{4C̃_{2}D̃_{2} − B̃_{2}^{2}} .  (6.7)

For the centrally symmetric isotropic case condition (6.7) tales the form

\begin{matrix}λ_{1} + μ_{1} − α_{1} = λ_{2} + μ_{2} − α_{2}, \\ ε_{1} + ν_{1} − β_{1} = ε_{2} + ν_{2} − β_{2}. \end{matrix}  (6.8)

Condition (6.8) written in terms of the Poisson constants was found in [16], when investigating the asymptotic properties of solutions of boundary positivedefinite pseudodifferential equations of crack-type problems of elasticity (see also [38]).

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(Received 27.05.2003)

Author’s address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 0193 Georgia E-mail: chkadua@rmi.acnet.ge