GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 111-122

THE BOUNDARY-CONTACT PROBLEM OF ELASTICITY FOR HOMOGENEOUS ANISOTROPIC MEDIA WITH A

CONTACT ON SOME PART OF THE BOUNDARIES

O. CHKADUA

Abstract. The existence and uniqueness of solutions of the boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of their boundaries are investigated in the Besov and Bessel potential classes using the methods of the potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.

1. Introduction. The paper is dedicated to the investigation of the boundary-contact problem of the static theory of elasticity for homogeneous anisotropic media when the contact of two bounded domains occurs from the outside on some part of the boundaries. In what follows such problems will be called nonclassical.

Boundary-contact problems of this kind, i.e., when the contact of two bounded domains occurs from the outside on some part of the boundaries, were considered for one differential equation in the Sobolev spaces H_{2}^{s} by Schechter in [1].

Classical boundary-contact problems for isotropic homogeneous elastic media were completely investigated by the method of the potential theory and multidimensional singular integral equations on manifolds without boundary in the monograph by Kupradze, Gegelia, Basheleishvili, and Burchuladze [2].

Classical boundary-contact problems (i.e., problems with a contact all over the boundary) for anisotropic homogeneous elastic media were investigated by the method of the potential theory and pseudodifferential equations on compact manifolds without boundary in the monograph by Burchuladze and Gegelia [3] and in the papers by Chkadua [4] and Natroshvili [5].

111

1991 Mathematics Subject Classification. 73C02, 35J55, 47G30.

Key words and phrases. Boundary-contact problem of elasticity, homogeneous anisotropy, pseododifferential equations, potential methods.

1072-947X/95/0300-0111$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

In this paper the methods of the potential theory and the general theory of pseudodifferential equations on manifolds with boundary are used to investigate the existence and uniqueness of solutions of the nonclassical boundary-contact problem in Besov and Bessel potential classes. The smoothness of solutions is studied. The solution possesses the C^{α}-smooth-ness for any α < \frac{1}{2}.

2. Statement of the Problem. Let D_{1} and D_{2} be the bounded domains in the three-dimensional Euclidean space ℝ^{3} with the boundaries ∂D_{1}, ∂D_{2} ∈ C^{∞}, D_{1} ∩ D_{2} = ∅, ∂D_{1} ∩ ∂D_{2} = \overline{S}_{0}; \overline{S}_{0} is the closure of the nonempty open (in the topology of ∂D_{1} and ∂D_{2}) set S_{0}. Then ∂D_{1} = S_{1} ∪ \overline{S}_{0}, ∂D_{2} = S_{2} ∪ \overline{S}_{0}, ∂S_{0} ∈ C^{∞}.

The basic static equations of elasticity for anisotropic homogeneous elastic media in terms of displacement components have the form (see [6], [7], [8])

A^{(q)}(∂x)u^{(q)} + F^{(q)} = 0 in D_{q}, q = 1, 2,

where u^{(q)}=(u_{1}^{(q)}, u_{2}^{(q)}, u_{3}^{(q)}) is the displacement vector, F^{(q)}=(F_{1}^{(q)}, F_{2}^{(q)}, F_{3}^{(q)}) is the mass force applied to D_{q}, and A^{(q)}(∂x) is the matrix differential operator

\begin{matrix}A^{(q)}(∂x) = ∥A_{jk}^{(q)}(∂x)∥_{3×3}, A_{jk}^{(q)}(∂x) = a_{ijlk}^{(q)}∂_{i}∂_{l}, \\ ∂_{i} = \frac{∂}{∂x_{i}} , q = 1, 2. \end{matrix}  (1)

a_{ijlk}^{(q)} are the elastic constants satisfying the conditions

a_{ijlk}^{(q)} = a_{lkij}^{(q)} = a_{ijkl}^{(q)}.

In (1) and below the repeated indices imply summation from 1 to 3. Assume that the quadratic forms

a_{ijlk}^{(q)}ξ_{ij}ξ_{lk}, ξ_{ij} = ξ_{ji},  (2)

with respect to the variables ξ_{ij} are positively definite. Let us introduce the differential stress operator

T^{(q)}(∂z, n(z)) = ∥T_{jk}^{(q)}(∂z, n(z))∥_{3×3},
T_{jk}^{(q)}(∂z, n(z)) = a_{ijlk}^{(q)}n_{i}(z)∂_{l}, q = 1, 2,

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

From the symmetry of coefficients a_{ijlk}^{(q)} and the positive definiteness of the quadratic forms (2) it follows (see [7]) that the operators A^{(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^{3} the relations

Re (A^{(q)}(ξ)η, η) = (A^{(q)}(ξ)η, η) ≥ P_{0}^{(q)}|ξ|^{2}|η|^{2}, q = 1, 2,

are valid, where P_{0}^{(q)} = const > 0 depends only on the elastic constants. Thus, the matrices A^{(q)}(ξ) which are the symbols of A^{(q)}(∂x) are positively definite for ξ ∈ ℝ^{3}\{0}.

In what follows the functional spaces will be denoted as in [8], [9]. For a sufficiently smooth surface M with boundary embedded into a sufficiently smooth compact surface M_{0} without boundary we introduce the following Besov spaces:

B_{p,t}^{s}(M) = {f|_{M} : f ∈ B_{p,t}^{s}(M_{0})},
B̃_{p,t}^{s}(M) = {g : g ∈ B_{p,t}^{s}(M_{0}), supp g ⊂ M̄}.

The notations H_{p}^{s}(M), H̃_{p}^{s}(M) have a similar meaning for the space of Bessel potentials.

We shall consider the following basic nonclassical boundary-contact problem:

In the domains D_{q}, q = 1, 2, find the vector-functions u^{(q)} : D_{q} → ℝ^{3} belonging to the class W_{p}^{1}(D_{q}) = H_{p}^{1}(D_{q}), q = 1, 2, and satisfying the conditions

\{ \begin{matrix}A^{(q)}(∂x)u^{(q)} = 0 in D_{q}, q = 1, 2, \\ \{u^{(1)}\}^{+} = φ_{1} on S_{1}, \\ \{u^{(2)}\}^{−} = φ_{2} on S_{2}, \\ \{u^{(1)}\}^{+} − \{u^{(2)}\}^{−} = g on S_{0}, \\ \{T^{(1)}(∂z, n(z))u^{(1)}\}^{+} − \{T^{(2)}(∂z, n(z))u^{(2)}\}^{−} = f on S_{0}, \end{matrix}  (3) (4) (5) (6) (7)

where φ_{1} ∈ B_{p,p}^{1/p′}(S_{1}), φ_{2} ∈ B_{p,p}^{1/p′}(S_{2}), g ∈ B_{p,p}^{1/p′}(S_{0}), f ∈ B_{p,p}^{−1/p}(S_{0}), p′ = \frac{p}{p−1}, 1 < p < ∞.

From the trace theorem it follows (see [8]) that the trace of any function u^{(1)} ∈ W_{p}^{1}(D_{1}) (u^{(2)} ∈ W_{p}^{1}(D_{2})) is defined on ∂D_{1} (∂D_{2}) : {u^{(1)}}^{+} ∈

B_{p,p}^{1/p′}(∂D_{1}) ({u^{(2)}}^{−} ∈ B_{p,p}^{1/p′}(∂D_{2})). Let u^{(1)} ∈ W_{p}^{1}(D_{1}) (u^{(2)} ∈ W_{p}^{1}(D_{2})) be such that A^{(1)}(∂x)u^{(1)} ∈ 𝕃_{p}(D_{1}) (A^{(2)}(∂x)u^{(2)} ∈ 𝕃_{p}(D_{2})). Then {T^{(1)}(∂z, n(z))u^{(1)}}^{+} ({T^{(2)}(∂z, n(z))u^{(2)}}^{−}) is well defined by the equality (see [10], [11])

∫_{D_{q}} [v^{(q)}A^{(q)}(∂x)u^{(q)} + E^{(q)}(u^{(q)}, v^{(q)})]dx =

= ±⟨{T^{(q)}(∂z, n(z))u^{(q)}}^{±}, {v^{(q)}}^{±}⟩_{∂D_{q}},  (8)

for any v^{(q)} ∈ W_{p′}^{1}(D_{q}), E^{(q)}(u^{(q)}, v^{(q)}) = a_{ijlk}^{(q)}∂_{i}u_{j}^{(q)}∂_{l}\overline{v}_{k}^{(q)}, q = 1, 2; the sym-

bol ⟨·, ·⟩ denotes the duality between the spaces B_{p,p}^{−1/p}(∂D_{q}) and B_{p′,p′}^{1/p}(∂D_{q}). In (8) the sign + is used when q = 1 and the sign − when q = 2.

Consider the fundamental matrix-function (see [4])

H^{(q)}(x) = F_{ξ′→x′}^{−1}( ± \frac{1}{2π} ∫_{±} (A^{(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′, x_{3}), ξ′ = (ξ_{1}, ξ_{2}); ∫_{±} denotes integration over the contour L^{±} where L^{+} (L^{−}) has the positive orientation and covers all roots of the polynomial det A^{(q)}(iξ′, iτ) with respect to τ in the upper (resp., lower) τ-halfplane; F^{−1} is the inverse Fourier transform.

Then the simple- and double-layer potentials will be written as

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

U^{(q)}(g_{2})(x) = ∫_{∂D_{q}} [T^{(q)}(∂y, n(y))H^{(q)}(x − y)]′g_{2}(y)d_{y}S, x ∉ ∂D_{q},

∂D_{q} = S_{q} ∪ S̄_{0}, q = 1, 2.

The symbol [ ]′ denotes the matrix transposition.

For these potentials the theorems below are valid.

Theorem 1 (see [10], [11]). Let s ∈ ℝ, 1 < p < ∞, 1 ≤ t ≤ ∞. Then the operators V^{(q)}, U^{(q)}, q = 1, 2, admit extensions to operators which are continuous in the following spaces:

V^{(q)} : B_{p,t}^{s}(∂D_{q}) → B_{p,t}^{s+1+1/p}(D_{q}) (B_{p,p}^{s}(∂D_{q}) → H_{p}^{s+1+1/p}(D_{q})),

U^{(q)} : B_{p,t}^{s}(∂D_{q}) → B_{p,t}^{s+1/p}(D_{q}) (B_{p,p}^{s}(∂D_{q}) → H_{p}^{s+1/p}(D_{q})),
q = 1, 2.

Theorem 2 (see [10], [11]). Let 1 < p < ∞, 1 ≤ t ≤ ∞, ε > 0, g_{1} ∈ B_{p,t}^{−1+ε}(∂D_{q}), g_{2} ∈ B_{p,t}^{ε}(∂D_{q}), q = 1, 2. Then

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

THE BOUNDARY-CONTACT PROBLEM OF ELASTICITY 115

\{U^{(q)}(g_{2})(z)\}^{±} =(−1)^{q+1} \frac{1}{2}g_{2}(z) +

+ ∫_{∂D_{q}}[T^{(q)}(∂y, n(y))H^{(q)}(z−y)]′g_{2}(y)d_{y}S, z ∈∂D_{q}, q =1, 2.

Theorem 3 (see [10], [11]). Let 1 < p < ∞, g_{1} ∈ B_{p,p}^{−1/p}(∂D_{q}), g_{2} ∈ B_{p,p}^{1/p′}(∂D_{q}), p′ = \frac{p}{p−1}, q = 1, 2. Then

\{T^{(q)}(∂z, n(z))V^{(q)}(g_{1})(z)\}^{±} = (−1)^{q} \frac{1}{2}g_{1}(z) +

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

{T^{(q)}(∂z, n(z))U^{(q)}(g_{2})(z)}^{+} = {T^{(q)}(∂z, n(z))U^{(q)}(g_{2})(z)}^{−}, (9)
z ∈ ∂D_{q}, q = 1, 2.

We introduce the notations

V_{−1}^{(q)}(g_{1})(z) = ∫_{∂D_{q}} H^{(q)}(z − y)g_{1}(y)d_{y}S,

V_{0}^{(q)}(g_{2})(z) = ∫_{∂D_{q}} [T^{(q)}(∂y, n(y))H^{(q)}(z − y)]′g_{2}(y)d_{y}S,

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

V_{1}^{(1)}(g_{2})(z) = {T^{(1)}(∂z, n(z))U^{(1)}(g_{2})(z)}^{+}, V_{1}^{(2)}(g_{2})(z) = {T^{(2)}(∂z, n(z))U^{(2)}(g_{2})(z)}^{−}.

Theorem 4 (see [10], [11]). Let s ∈ ℝ, 1 < p < ∞, 1 ≤ t ≤ ∞. Then the operators V_{−1}^{(q)}, V_{0}^{(q)} ,V^{∗} _{0}^{(q)}, V_{1}^{(q)} admit extensions onto operators which are continuous in the following spaces:

V_{−1}^{(q)} : H_{p}^{s}(∂D_{q}) → H_{p}^{s+1}(∂D_{q}) (B_{p,t}^{s}(∂D_{q}) → B_{p,t}^{s+1}(∂D_{q})),

V_{0}^{(q)}, V^{∗} _{0}^{(q)} : H_{p}^{s}(∂D_{q}) → H_{p}^{s}(∂D_{q}) (B_{p,t}^{s}(∂D_{q}) → B_{p,t}^{s}(∂D_{q})),

V_{1}^{(q)} : H_{p}^{s}(∂D_{q}) → H_{p}^{s−1}(∂D_{q}) (B_{p,t}^{s}(∂D_{q}) → B_{p,t}^{s−1}(∂D_{q})),

q = 1, 2.

The operators V_{−1}^{(q)}, ±\frac{1}{2}I + V_{0}^{(q)}, ±\frac{1}{2}I+ V^{∗} _{0}^{(q)}, V_{1}^{(q)}, q = 1, 2 (I is the

identity operator) are the pseudodifferential operators of orders −1, 0, 0, 1, respectively. On the manifold ∂D_{q} their symbols have the form (see [4], [5], [12])

σ(V_{−1}^{(q)})(z, ξ)=−\frac{1}{2π} ∫_{+} (A^{(q)}(ξ + nτ))^{−1} dτ, q = 1, 2,

σ( − \frac{1}{2}I+V^{∗} _{0}^{(1)})(z, ξ)= \frac{1}{2πi} ∫_{+} T^{(1)}(ξ + nτ, n)(A^{(1)}(ξ + nτ))^{−1} dτ,

σ(\frac{1}{2}I +V_{0}^{(1)})(z, ξ)=−\frac{1}{2πi} ∫_{+} (T^{(1)}(ξ + nτ, n)(A^{(1)}(ξ + nτ))^{−1})′ dτ,

σ(\frac{1}{2}I+V^{∗} _{0}^{(2)})(z, ξ)=−\frac{1}{2πi} ∫_{−} T^{(2)}(ξ + nτ, n)(A^{(1)}(ξ + nτ))^{−1} dτ,

σ( − \frac{1}{2}I +V_{0}^{(2)})(z, ξ)= \frac{1}{2πi} ∫_{−} (T^{(2)}(ξ+nτ, n)(A^{(2)}(ξ+nτ))^{−1})′ dτ,

σ(V_{1}^{(q)})(z, ξ)=−\frac{1}{2π} ∫_{+} T^{(q)}(ξ + nτ, n)(T^{(q)}(ξ + nτ, n) ×

× (A^{(q)}(ξ + nτ))^{−1})′ dτ, q = 1, 2,

where n = n(z), ξ is the cotangential vector of the manifold ∂D_{q} at the point z.

These pseudodifferential operators are investigated in the Hölder spaces C^{1,β}(S) (S is the compact manifold without boundary) in [4], [5], [12].

Equality (9) is a generalization of theorems of the Liapunov–Tauber type (see [4], [5]).

The principal homogeneous symbols of the operators V_{−1}^{(q)} and V_{1}^{(q)} are even matrix-functions. One can easily verify that this is so for the operator V_{−1}^{(q)}, while for the operator V_{1}^{(q)} this follows from (9).

The pseudodifferential operators V_{−1}^{(q)} and V_{1}^{(q)} are also formally selfadjoint operators (see [4]).

Let us consider the question whether the Dirichlet boundary value problems are solvable in the classes W_{p}^{1}(D_{q}), q = 1, 2.

The Dirichlet problem in the domain D_{q} is

\begin{cases}A^{(q)}(∂x)u^{(q)} = 0 & in D_{q} , \\ \{u^{(q)}\}^{±} = Φ^{(q)} & on ∂D_{q}, \end{cases} q = 1, 2,  (10_{q})

where Φ^{(q)} ∈ B_{p,p}^{1/p′}(∂D_{q}), 1 < p < ∞; the sign + is used when q = 1 and the sign − when q = 2.

For problem (10)_{q} we have

Lemma (see [11]). The boundary value problem (10)_{q} has a unique solution in the class W_{p}^{1}(D_{q}), this solution being given by the formula u^{(q)} = V^{(q)}(V_{−1}^{(q)})^{−1}Φ^{(q)}, q = 1, 2.

The operators A_{1} = (−\frac{1}{2}I+ V^{∗} _{0}^{(1)})(V_{−1}^{(1)})^{−1} and A_{2} = (\frac{1}{2}I+ V^{∗} _{0}^{(2)})×

×(V_{−1}^{(2)})^{−1} are the pseudodifferential operators of order 1.

Theorem 5. For the pseudodifferential operators A_{1} = (−\frac{1}{2}I+ V^{∗} _{0}^{(1)})×

(V_{−1}^{(1)})^{−1} and A_{2} = (\frac{1}{2}I+ V^{∗} _{0}^{(2)})(V_{−1}^{(2)})^{−1} the following relations are valid:

a) ⟨A_{1}φ, φ⟩_{∂D_{1}} ≥ 0, for ∀φ ∈ H_{2}^{1/2}(∂D_{1}), the equality being fulfilled only if φ = [a_{1}^{(1)} × x] + b_{1}^{(1)} + i([a_{2}^{(1)} × x] + b_{2}^{(1)}), where a_{1}^{(1)}, b_{1}^{(1)}, a_{2}^{(1)}, b_{2}^{(1)} ∈ ℝ^{3} are arbitrary vectors;

b) ⟨A_{2}ψ, ψ⟩_{∂D_{2}} ≤ 0, for ∀ψ ∈ H_{2}^{1/2}(∂D_{2}), the equality being fulfilled only if ψ = [a_{1}^{(2)} × x] + b_{1}^{(2)} + i([a_{2}^{(2)} × x] + b_{2}^{(2)}) for ∀a_{1}^{(2)}, b_{1}^{(2)}, a_{2}^{(2)}, b_{2}^{(2)} ∈ ℝ^{3}.

In Theorem 5 the symbol < ·, · >_{∂D_{q}} denotes the duality between the spaces H_{2}^{±1/2}(∂D_{q}), q = 1, 2, defined by the formula

⟨f, g⟩_{∂D_{q}} = ∫_{∂D_{q}} f · \overline{g} dS for f, g ∈ C^{1}(∂D_{q})

and the symbol [· × ·] denotes the vector product.

The solution of the considered boundary-contact problem (3)–(7) will be sought for in the form of simple-layer potentials in the domains D_{q}, q = 1, 2:

u^{(q)} = V^{(q)}g_{q} in D_{q}, q = 1, 2.

From the boundary and boundary-contact conditions of the problem we obtain

\{ \begin{matrix}π_{1}V_{−1}^{(1)}g_{1} = φ_{1} on S_{1}, \\ π_{2}V_{−1}^{(2)}g_{2} = φ_{2} on S_{2}, \\ π_{0}V_{−1}^{(1)}(g_{1}) − π_{0}V_{−1}^{(2)}g_{2} = g on S_{0}, \\ π_{0}( − \frac{1}{2}I+ V^{∗} _{0}^{(1)})g_{1} − π_{0}(\frac{1}{2}I+ V^{∗} _{0}^{(2)})g_{2} = f on S_{0}, \end{matrix}  (11) (12) (13) (14)

where π_{i} denotes the operator of restriction on S_{i}, i = 0, 1, 2.

Let Φ_{0}^{(q)} ∈ B_{p,p}^{1/p′}(∂D_{q}) be an extention of the function φ_{q} ∈ B_{p,p}^{1/p′}(S_{q}) on the entire boundary ∂D_{q} = S_{q} ∪ S̄_{0}, q = 1, 2. It is easy to verify that any extension Φ^{(q)} ∈ B_{p,p}^{1/p′}(∂D_{q}) of the function φ_{q} has the form Φ^{(q)} = Φ_{0}^{(q)} + φ_{0}^{(q)}, where φ_{0}^{(q)} ∈ B̃_{p,p}^{1/p′}(S_{0}), q = 1, 2.

Now (11) and (12) imply

V_{−1}^{(q)}g_{q} = Φ_{0}^{(q)} + φ_{0}^{(q)}, q = 1, 2.

Since V_{−1}^{(q)}, q = 1, 2, are invertible operators (see [11], [12]), we have

g_{q} = (V_{−1}^{(q)})^{−1}(Φ_{0}^{(q)} + φ_{0}^{(q)}), q = 1, 2.  (15)

The substitution of (15) in (13) and (14) gives a system of pseudodifferential equations with respect to φ_{0}^{(1)} and φ_{0}^{(2)} defined on the manifold with the boundary S_{0}:

\begin{cases}φ_{0}^{(1)} − φ_{0}^{(2)} = g^{∗}, \\ π_{0}( − \frac{1}{2}I+ V^{∗} _{0}^{(1)})(V_{−1}^{(1)})^{−1}φ_{0}^{(1)}− \\ −π_{0}(\frac{1}{2}I+ V^{∗} _{0}^{(2)})(V_{−1}^{(2)})^{−1}φ_{0}^{(2)} = f^{∗}, \end{cases}

where

g^{∗} = g − π_{0}Φ_{0}^{(1)} + π_{0}Φ_{0}^{(2)},
f^{∗} = f − π_{0}( − \frac{1}{2}I+ V^{∗} _{0}^{(1)})(V_{−1}^{(1)})^{−1}Φ_{0}^{(1)} +

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

In what follows, demanding φ_{k} ∈ B_{p,t}^{s}(S_{k}), k = 1, 2, g ∈ B_{p,t}^{s}(S_{0}) (in particular, for s = \frac{1}{p′}, t = p) we will assume that the following condition holds:

∃Φ_{0}^{(k)} ∈ B_{p,t}^{s}(∂D_{k}), k = 1, 2 : g − (π_{0}Φ_{0}^{(1)} − π_{0}Φ_{0}^{(2)}) ∈ B̃_{p,t}^{s}(S_{0}).

Note that in the case \frac{1}{p} − 1 < s < \frac{1}{p} this condition holds automatically (see [8], Theorem 2.10.3(c)). In the case \frac{1}{p} < s < \frac{1}{p} + 1 it becomes g|_{S_{0}} = φ_{1}|_{S_{0}} − φ_{2}|_{S_{0}} (see [8], Theorems 2.3.3(b), 2.10.3(b)). In the case s = \frac{1}{p} it looks more complicated (see [8], Remark 4.3.2-2, and also 2.9.1, 4.2.2, and 4.2.3).

This system of pseudodifferential equations can be rewritten as

\begin{cases}φ_{0}^{(1)} − φ_{0}^{(2)} = g^{∗}, \\ π_{0}A_{1}φ_{0}^{(1)} − π_{0}A_{2}φ_{0}^{(2)} = f^{∗}, \end{cases}  (16)

where π_{0}A_{1} = π_{0}(−\frac{1}{2}I+ V^{∗} _{0}^{(1)})(V_{−1}^{(1)})^{−1}, and π_{0}A_{2} = π_{0}(\frac{1}{2}I+ V^{∗} _{0}^{(2)})(V_{−1}^{(2)})^{−1}

are the pseudodifferential operators on the manifold with boundary S_{0}:

π_{0}A_{q} : H̃_{p}^{s}(S_{0}) → H_{p}^{s−1}(S_{0}) (B̃_{p,t}^{s}(S_{0}) → B_{p,t}^{s−1}(S_{0})), q = 1, 2.

For the operators π_{0}A_{q}, q = 1, 2, we have

Theorem 6 (see [11]). Let 1 < p < ∞, 1 ≤ t ≤ ∞, \frac{1}{p} − \frac{1}{2} < s < \frac{1}{p} + \frac{1}{2}. Then the operators

π_{0}A_{q} : H̃_{p}^{s}(S_{0}) → H_{p}^{s−1}(S_{0}) (B_{p,t}^{s}(S_{0}) → B_{p,t}^{s−1}(S_{0})), q = 1, 2,

are invertible.

We define φ_{0}^{(2)} from the first equation of system (16)

φ_{0}^{(2)} = φ_{0}^{(1)} − g^{∗}.  (17)

The substitution of (17) into the second equation of system (16) gives

π_{0}A_{1}φ_{0}^{(1)} − π_{0}A_{2}φ_{0}^{(1)} = f^{∗} − π_{0}A_{2}g^{∗}.

Thus we have to investigate the equation

(π_{0}A_{1} − π_{0}A_{2})φ_{0}^{(1)} = Ψ.  (18)

Recalling that a function of the form [a_{1} × x] + b_{1} + i([a_{2} × x] + b_{2}) equal to zero on S_{q}, q = 1, 2, is identically zero, from Theorem 5 it follows that the operator

π_{0}A_{1} − π_{0}A_{2} : H̃_{2}^{1/2}(S_{0}) → H_{2}^{−1/2}(S_{0})

is invertible.

This pseudodifferential operator is an elliptic operator on the manifold with the boundary S_{0}.

We introduce the notation

P = π_{0}A_{1} − π_{0}A_{2}.

Assume that z is an arbitrary point of ∂S_{0} and choose some local coordinate system in its neighborhood. Denote by σ_{P}(z, ξ′), ξ′ = (ξ_{1}, ξ_{2}), the value, at the point z, of the principal homogeneous symbol of the operator P written in terms of the chosen local coordinate system.

The eigenvalues of the matrix

(σ_{P}(z, 0, −1))^{−1} · σ_{P}(z, 0, +1)

play an essential role in the investigation of the Noetherity of the operator P in the corresponding Besov and Bessel potential spaces.

Since the matrix σ_{P}(z, ξ′) is positively definite, all the eigenvalues of the matrix (σ_{P}(z, 0, −1))^{−1} · σ_{P}(z, 0, +1) are positive.

Due to the foregoing arguments, using the general theorems on elliptic pseudodifferential operators on the manifold with boundary obtained in [13], [14], [15], we can prove the next statement (see also [16], [17]).

Theorem 7. Let 1 < p < ∞, 1 ≤ t ≤ ∞. Then for the operator P : H̃_{p}^{s}(S_{0}) → H_{p}^{s−1}(S_{0}) to be Noetherian it is necessary that the inequality

\frac{1}{p} − \frac{1}{2} < s < \frac{1}{p} + \frac{1}{2}  (19)

be fulfilled.

If (19) is fulfilled, then the operator

P : H̃_{p}^{s}(S_{0}) → H_{p}^{s−1}(S_{0}) (B̃_{p,t}^{s}(S_{0}) → B_{p,t}^{s−1}(S_{0}))

is invertible.

Theorem 7 implies that there exist unique extensions Φ^{(q)} = Φ_{0}^{(q)} + φ_{0}^{(q)} (Φ^{(q)} ∈ B_{p,p}^{1/p′}(∂D_{q})) of the function φ_{q} (φ_{q} ∈ B_{p,p}^{1/p′}(S_{1})) onto the entire boundary ∂D_{q}, q = 1, 2, such that solutions of the corresponding Dirichlet problems are solutions of the boundary-contact problem (3)-(7). Here φ_{0}^{(1)} is the solution of equation (18), while φ_{0}^{(2)} is defined by equality (17).

From the above discussion it follows that if u^{(q)}, q = 1, 2, is the solution of the boundary-contact prolem (3)-(7), then u^{(1)} and u^{(2)} are the solutions of the corresponding Dirichlet boundary value problems (10)_{q}, q = 1, 2. In the latter problems the boundary conditions are the unique extensions Φ^{(q)} = Φ_{0}^{(q)}+φ_{0}^{(q)} of the function φ_{q} onto ∂D_{q}, q = 1, 2, where φ_{0}^{(q)}, q = 1, 2, is the solution of system (16).

Theorem 8. Let 4/3 ≤ p < 4. Then the boundary-contact problem (3)- (7) has the unique solution in the classes W_{p}^{1}(D_{q}), q = 1, 2, this solution being given by the formula

u^{(q)} = V^{(q)}(V_{−1}^{(q)})^{−1}(Φ_{0}^{(q)} + φ_{0}^{(q)})), q = 1, 2,  (20)

where Φ_{0}^{(q)} ∈ B_{p,p}^{1/p′}(∂D_{q}) is some extension of φ_{q} onto ∂D_{q} and φ_{0}^{(q)} ∈ B̃_{p,p}^{1/p′}(S_{0}) is the solution of system (16).

From Theorems 1, 7 and the embedding theorem (see [8]) we obtain

Theorem 9. Let 4/3 ≤ p < 4, 1 < t < ∞, 1 ≤ r ≤ ∞, \frac{1}{t}−\frac{1}{2} < s < \frac{1}{t}+\frac{1}{2}, u^{(q)} ∈ W_{p}^{1}(D_{q}), q = 1, 2, be the solution of the boundary-contact problem (3)–(7). Then:

If φ_{1} ∈ B_{t,t}^{s}(S_{1}), φ_{2} ∈ B_{t,t}^{s}(S_{2}), g ∈ B_{t,t}^{s}(S_{0}), f ∈ B_{t,t}^{s−1}(S_{0}), then u^{(q)} ∈ H_{t}^{s+1/t}(D_{q}), q = 1, 2;

If φ_{1} ∈ B_{t,r}^{s}(S_{1}), φ_{2} ∈ B_{t,r}^{s}(S_{2}), g ∈ B_{t,r}^{s}(S_{0}), f ∈ B_{t,r}^{s−1}(S_{0}), then u^{(q)} ∈ B_{t,r}^{s+1/t}(D_{q}), q = 1, 2;

If φ_{1} ∈ C^{α}(S̄_{1}), φ_{2} ∈ C^{α}(\overline{S}_{2}), g ∈ C^{α}(S̄_{0}), f ∈ B_{∞,∞}^{a−1}(S_{0}), α ∈]0, 1/2], then u^{(q)} ∈ ∩_{α′<α}C^{α′}(D̄_{q}), q = 1, 2.

Analogous theorems are proved in [11] and [18] for the mixed problems of elasticity and in [10] for the boundary value problems in the theory of cracks.

References

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3. T. V. Burchuladze and T. G. Gegelia, Development of the potential method in the theory of elasticity. (Russian) Metsniereba, Tbilisi, 1985.

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16. M. S. Agranovich, Elliptic singular integrodifferential operators. (Russian) Uspekhi Mat. Nauk 20(1965), No. 5(131), 3-120.

17. R. V. Kapanadze, On some properties of singular operators in normed spaces. (Russian) Proc. Tbilisi Univ. Math. Mech. Astron. 129(1968), 17-26.

18. E. M. Shargorodsky, On the continuity of generalized solutions of mixed problems for the Lamé system. (Russian) Bull. Acad. Sci. Georgian SSR, 135(1989), No.3 493-496.

(Received 04.10.1993)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhadze St., Tbilisi 380093 Republic of Georgia