GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 6, 1995, 631-652

MIXED INTERFACE PROBLEMS FOR ANISOTROPIC ELASTIC BODIES

D. NATROSHVILI

Abstract. Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where on one part of the interface, rigid contact conditions are given (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.

1. Introduction

The investigation deals with the problems of anisotropic elasticity for composite bodies which have piecewise homogeneous structure. From the mathematical point of view these problems can be considered as problems involving a system of partial differential equations with discontinuous coefficients.

The most general structure of the elastic body under consideration mathematically can be described as follows. In three-dimensional Euclidean space ℝ^{3} we have some closed smooth two-dimensional connected surface S_{1} which involves other closed smooth surfaces S_{2}, … S_{m} (S_{j} ∩ S_{k} = ∅, j ≠ k). By these surfaces the space ℝ^{3} is divided into several connected domains Ω_{1}, …, Ω_{μ}. Each domain Ω_{l} is assumed to be filled up by an anisotropic material with the corresponding (in general, different) elastic coefficients

c^{l}_{kjpq} = c^{l}_{pqkj} = c^{l}_{jkpq}, l = 1, …, μ, k, j, p, q = 1, 2, 3.  (1.1)

The common boundaries of two different materials are called contact boundaries (surfaces) of the piecewise homogeneous body. If some domains

631 represent empty inclusions, then the surfaces corresponding to them together with S_{1} are called boundary surfaces of the piecewise homogeneous body in question.

1991 Mathematics Subject Classification. 35C15, 73C35.

Key words and phrases. Mixed problems, interface problems, anisotropic problems.

1072-947X/95/1100-0631$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Such piecewise homogeneous bodies are encountered in many physical, mechanical, and engineering applications.

Classical and nonclassical mathematical problems for isotropic piecewise homogeneous bodies are studied in [1, 2, 3] by means of potential methods, while similar problems for anisotropic piecewise homogeneous bodies are investigated in [4, 5].

By using the functional methods analogous problems have been considered in [6].

In the present paper we treat the mixed boundary-contact problems with discontinuous boundary conditions on contact surfaces. Such types of problems have not been investigated even for isotropic piecewise homogeneous bodies. Our study is based on the potential methods and on the theory of pseudodifferential equations on manifolds with boundary.

A displacement vector corresponding to the domain Ω_{l} will be denoted by

u^{l} = (u^{l}_{1}, u^{l}_{2}, u^{l}_{3})^{T}; T^{l} (D_{x}, n)u^{l}(x) denotes the corresponding stress vector calculated on the surface element with the unit normal vector n = (n_{1}, n_{2}, n_{3}):

[T^{l} (D, n)u^{l}(x)]_{k} = c^{l}_{kjpq}D_{q}u^{l}_{p}(x), D_{q} = ∂/∂x_{q}.

Here and in what follows, summation over repeated indices is from 1 to 3. The symbol [·]^{T} denotes transposition.

Components of the stress τ^{l}_{kj} and of the strain e^{l}_{kj} tensors are related by Hooke’s law

τ^{l}_{kj} = c^{l}_{kjpq}e^{l}_{pq}, e^{l}_{pq} = 2^{−1}(D_{p}u^{l}_{q} + D_{q}u^{l}_{p}).

The potential energy

E^{l}(u^{l}, u^{l}) = e^{l}_{kj}τ^{l}_{kj} = c^{l}_{kjpq}e^{l}_{kj}e^{l}_{pq}  (1.2)

is assumed to be a positive-definite quadratic form in symmetric variables e^{l}_{kj} = e^{l}_{jk}. Therefore there exists positive δ_{0} > 0 such that

E^{l}(u^{l}, u^{l}) ≥ δ_{0}e^{l}_{kj}e^{l}_{kj}.  (1.3)

The basic homogeneous equation of statics reads as (provided bulk forces are equal to zero)

A^{l}(D_{x})u^{l}(x) = 0, x ∈ Ω_{l},  (1.4)

MIXED INTERFACE PROBLEMS FOR ANISOTROPIC ELASTIC BODIES 633

where

A^{l} = ∥A^{l}_{kp}∥, A^{l}_{kp}(D) = c^{l}_{kjpq}D_{j}D_{q}.

It follows from (1.1)–(1.3) that A^{l} is a formally self-adjoint strongly elliptic matrix differential operator (cf. [6]), and consequently for any ξ ∈ ℝ^{3} and for arbitrary complex vector η ∈ ℂ^{3} the inequality

Re (A^{l}(ξ)η, η) = A^{l}(ξ)η · η ≥ δ_{1}|ξ|^{2}|η|^{2}

holds with δ_{1} = const > 0. As usual (a, b) = a · b = a_{k}\overline{b}_{k} denotes a scalar product of two vectors.

2. Formulation of the Problems

For simplicity we consider the following model problems. The piecewise homogeneous anisotropic body is assumed to consist of two connected domains Ω_{1} = Ω^{+} and Ω_{2} = Ω^{−}, provided Ω^{+} is bounded (diam Ω^{+} < ∞) and Ω^{−} = ℝ^{3}\\overline{Ω}^{+}; \overline{Ω}^{+} = Ω^{+} ∪ S, S = ∂Ω^{±}. Thus we have only one contact surface S and the whole space ℝ^{3} = \overline{Ω}_{1}∪\overline{Ω}_{2} can be considered as a piecewise homogeneous anisotropic body with the interface S. For domains of general structure described in Section 1, all problems can be investigated similarly with slight modifications.

Let a smooth, connected, non-self-intersecting curve γ ⊂ S divide the surface S into two parts S_{1} and S_{2} : S = S_{1} ∪ S_{2} ∪ γ; \overline{S}_{l} = S_{l} ∪ γ.

The basic mixed contact (interface) problems can be formulated in the following way.

C–D Problem. Find the vectors u^{1} and u^{2} satisfying equations (1.4) in Ω_{1} and Ω_{2}, respectively, and the contact conditions

\begin{matrix}[u^{1}(x)]^{+} − [u^{2}(x)]^{−} = f(x), \\ [T^{1} (D_{x}, n(x))u^{1}(x)]^{+} − [T^{2} (D_{x}, n(x))u^{2}(x)]^{−} = F (x), \end{matrix} \} x ∈ S_{1}, \begin{matrix}(2.1) \\ (2.2) \end{matrix}

\begin{matrix}[u^{1}(x)]^{+} = f_{+}(x), \\ [u^{2}(x)]^{−} = f_{−}(x), \end{matrix} \} x ∈ S_{2}; \begin{matrix}(2.3) \\ (2.4) \end{matrix}

C–N Problem. Find the vectors u^{1} and u^{2} satisfying equations (1.4) in Ω_{1} and Ω_{2}, respectively, contact conditions (2.1), (2.2) on S_{1}, and

\begin{matrix}[T^{1} (D_{x}, n(x))u^{1}(x)]^{+} = F_{+}(x), \\ [T^{2} (D_{x}, n(x))u^{2}(x)]^{−} = F_{−}(x), \end{matrix} \} x ∈ S_{2}; \begin{matrix}(2.5) \\ (2.6) \end{matrix}

here the symbols [·]^{±} denote limiting values on S from Ω^{±}, and f, f_{±}, F, and F_{±} are the given vector functions. In addition, in both problems we suppose that

u^{2}(x) = o(1)  (2.7)

as |x| → +∞. Condition (2.7) implies (see [7,8])

D^{α}u^{2}(x) = O(|x|^{−1−|α|}) as |x| → +∞  (2.8)

for an arbitrary multi-index α = (α_{1}, α_{2}, α_{3}), |α| = α_{1} + α_{2} + α_{3}.

From particular problems of mathematical physics and mechanics (see [9]) it is well known that, in general, solutions to mixed boundary value problems or the derivatives of the solutions possess singularities in the vicinity of curves of discontinuity of boundary conditions (curve γ) and they do not belong to the class of regular functions C^{1}(\overline{Ω}^{±}). Moreover, for C^{∞}-regular data of problems they do not possess even C^{α}-smoothness with α > 1/2 in the neighbourhood of γ while being infinitely differentiable elsewhere (i.e., in \overline{Ω}^{±}\γ).

Because of this fact we consider both mixed problems formulated above in Sobolev spaces

u^{1} ∈ W_{p}^{1}(Ω^{+}), u^{2} ∈ W_{p,loc}^{1}(Ω^{−})  (2.9)

with a view to involve a wider class of boundary data.

For conditions (2.1)–(2.6) to make sense we need some functional spaces. By B_{p,q}^{ν}(Ω^{+}), B_{p,q,loc}^{ν}(Ω^{−}), B_{p,q}^{ν}(S) and H_{p}^{ν}(Ω^{+}), H_{p,loc}^{ν}(Ω^{−}), H_{p}^{ν}(S) are

meant the Besov and the Bessel potential spaces, respectively, with ν ∈ ℝ, 1 < p < ∞, 1 ≤ q ≤ ∞ (see [10,11,12]).

The definition of regular function spaces C^{k+α}(\overline{Ω}^{±}) and C^{k+α}(S) with integer k ≥ 0 and 0 < α < 1 can be found, e.g., in [1].

Let us introduce the following functional spaces on S_{j} ⊂ S (j = 1, 2) :

B_{p,q}^{ν}(S_{j}) = {f|_{S_{j}} : f ∈ B_{p,q}^{ν}(S)},
B̃_{p,q}^{ν}(S_{j}) = \{f ∈ B_{p,q}^{ν}(S) : supp f ⊂ \overline{S}_{j}\} ⊂ B_{p,q}^{ν}(S),
H_{p}^{ν}(S_{j}) = {f|_{S_{j}} : f ∈ H_{p}^{ν}(S)},
H̃_{p}^{ν}(S_{j}) = \{f ∈ H_{p}^{ν}(S) : supp f ⊂ \overline{S}_{j}\} ⊂ H_{p}^{ν}(S)

(cf. [13,14]).

For vector function (2.9) the boundary and contact conditions for the displacement vectors can be considered in terms of traces. The first-order derivatives of the functions from W_{p}^{1}(Ω^{+}) and W_{p,loc}^{1}(Ω^{−}) belong to L_{p}(Ω^{+}) and L_{p,loc}(Ω^{−}), and they have no traces on S. For the boundary and contact conditions (2.2), (2.5), and (2.6) to make sense, we proceed as follows.

For regular solutions u^{l} ∈ C^{1}(\overline{Ω}_{l}) ∩ C^{2}(Ω_{l}) of equation (1.4) and for

arbitrary regular vectors v^{l} ∈ C^{1}(\overline{Ω}_{l}) (with diam supp v^{2} < +∞) the following Green’s formulas are valid (see, e.g., [15])

∫_{Ω^{+}} A^{1}(D)u^{1} · v^{1} dx = − ∫_{Ω^{+}} E^{1}(u^{1}, v^{1}) dx + ∫_{S}[T^{1} (D, n)u^{1}]^{+} · [v^{1}]^{+}dS, (2.10)
∫_{Ω^{−}} A^{2}(D)u^{2} · v^{2} dx = − ∫_{Ω^{−}} E^{2}(u^{2}, v^{2}) dx − ∫_{S}[T^{2} (D, n)u^{2}]^{+} · [v^{2}]^{+}dS, (2.11)

where n is an exterior unit normal vector on S,

E^{l}(u^{l}, v^{l}) = c^{l}_{kjpq}D_{k}u^{l}_{j}D_{p}v^{l}_{q}.  (2.12)

We can rewrite (2.10) and (2.11) as follows (provided that u^{1} and u^{2} satisfy (2.9) and equation (1.4) (in the distributional sense))

⟨[T^{1} (D, n)u^{1}]^{+}, [v^{1}]^{+}⟩_{S} = ∫_{Ω^{+}} E^{1}(u^{1}, v^{1})dx,  (2.13)
⟨[T^{2} (D, n)u^{2}]^{−}, [v^{2}]^{−}⟩_{S} = − ∫_{Ω^{−}} E^{2}(u^{2}, v^{2})dx  (2.14)

with

v^{1} ∈ W_{p′}^{1}(Ω^{+}), v^{2} ∈ W_{p′,comp}^{1}(Ω^{−}), p′ = \frac{p}{p − 1}.

It is evident that

[v^{l}]_{S}^{±} ∈ B_{p′,p′}^{1−1/p}(S) = B_{p′,p′}^{1/p}(S), l = 1, 2,

and the symbol ⟨·, ·⟩_{S} defines the duality between B_{p,p}^{−1/p}(S) and B_{p′,p′}^{1/p}(S), which for the smooth functions f and g has the form

⟨f, g⟩_{S} = (f, g)_{L^{2}(S)} = ∫_{S} f · g dS.

Due to (2.12), dualities (2.13) and (2.14) define [T^{1} u^{1}]^{+} and [T^{2} u^{2}]^{−} on S correctly, and

[T^{1} u^{1}]^{+}, [T^{2} u^{2}]^{−} ∈ B_{p,p}^{−1/p}(S).

The conditions on stresses in (2.2), (2.5), and (2.6) are to be understood in the sense just described.

Now we can write precisely conditions for boundary data in the above formulated problems: u^{1} and u^{2} satisfy (2.7), (2.9) and

\begin{matrix}f ∈ B_{p,p}^{1−1/p}(S_{1}), f_{±} ∈ B_{p,p}^{1−1/p}(S_{2}), \\ F ∈ B_{p,p}^{−1/p}(S_{1}), F_{±} ∈ B_{p,p}^{−1/p}(S_{2}). \end{matrix}  (2.15)

It is evident that the compatibility conditions are

f^{0} = \begin{cases}f & on S_{1} \\ f_{+} − f_{−} & on S_{2} \end{cases} ∈ B_{p,p}^{1−1/p}(S)  (2.16)

in the C–D Problem, and

F^{0} = \begin{cases}F & on S_{1} \\ F_{+} − F_{−} & on S_{2} \end{cases} ∈ B_{p,p}^{−1/p}(S)  (2.17)

in the C–N Problem.

In what follows these conditions are assumed to be fulfilled. Moreover, for simplicity we assume that S and γ possess the C^{∞}-smooth-

ness (in fact it suffices to have some finite regularity).

3. Potentials and Their Properties

Let Γ^{l} be the fundamental matrix of the operator A^{l}(D)

A^{l}(D)Γ^{l}(x) = Iδ(x),

where δ(·) is the Dirac distribution and I is the unit matrix I = ∥δ_{kj}∥_{3×3}. It can be proved that (see [15])

Γ^{l}(x) = (2π)^{−3} ∫_{ℝ^{3}} e^{−ixξ}A^{−1}(−iξ) dξ = −\frac{1}{8π|x|} ∫_{0}^{2π} A^{l}^{−1}(aη̃) dφ, (3.1)

where A^{l}^{−1}(ξ) is the matrix reciprocal to A^{l}(ξ) and a = ∥a_{kj}∥_{3×3} is an orthogonal matrix with the property a^{T} x = (0, 0, |x|)^{T}, η̃ = (cos φ, sin φ, 0).

Obviously, (3.1) implies

Γ^{l}(tx) = t^{−1}Γ^{l}(x), Γ^{l}(x) = Γ^{l}(−x) = [Γ^{l}(x)]^{T} , t > 0.

Let us introduce the single- and double-layer potentials

(V^{l} g)(x) = ∫_{S} Γ^{l}(x − y)g(y) dS_{y},
(U^{l})(x) = ∫_{S}[T^{l} (D_{y}, n(y))Γ^{l}(y − x)]^{T} g(y) dS_{y}.

The superscript l indicates that the potential corresponds to the funda-

mental matrix Γ^{l}.

The properties of these potentials are studied in [4,15] in regular C^{k+α} spaces, and in [14] in the Bessel potential H_{p}^{ν} and the Besov B_{p,q}^{ν} spaces.

We need some results obtained in the above cited works. We shall formulate them in the form of the following theorems.

Theorem 3.1 ([4,15]). Let k ≥ 0 be an integer and 0 < α < 1. Then

V^{l} : C^{k+α}(S) → C^{k+1+α}(\overline{Ω}^{±}),  (3.2)
U^{l} : C^{k+α}(S) → C^{k+α}(\overline{Ω}^{±}).  (3.3)

For any g ∈ C^{k+α}(S) and x ∈ S

[(V^{l} g)(x)]^{+} = [(V^{l} g)(x)]^{−} = (V^{l} g)(x) ≡ H^{l}g(x),  (3.4)
[T^{l} (D_{x}, n(x))(V g)(x)]^{±} = ( ∓ \frac{1}{2} I + K^{l})g(x),  (3.5)
[(U^{l}g)(x)]^{±} = ( ± \frac{1}{2} I + K^{l}^{∗})g(x),  (3.6)
[T^{l} (D_{x}, n(x))(Ug)(x)]^{+} =[T^{l} (D_{x}, n(x))(Ug)(x)]^{−} ≡L^{l}g(x),  (3.7)
k ≥ 1,

where n(x) is an exterior to Ω^{+} unit normal vector at the point x ∈ S; I stands for the unit operator,

H^{l}g(x) = ∫_{S} Γ^{l}(x − y)g(y) dS_{y},

K^{l}g(x) = ∫_{S} T^{l} (D_{x}, n(x))Γ^{l}(x − y)g(y) dS_{y},

K^{l}^{∗}g(x) = ∫_{S}[T^{l} (D_{y}, n(y))Γ^{l}(y − x)]^{T} g(y) dS_{y},

L^{l}g(x) = lim_{Ω^{±}∋z→x∈S} T^{l} (D_{z}, n(x)) ∫_{S}[T^{l} (D_{y}, n(y))Γ^{l}(y − z)]^{T} g(y) dS_{y}.

Theorem 3.2 ([4,15]). Let k ≥ 0 be an integer and 0 < α < 1. Then

\begin{aligned}(i) & H^{l} : C^{k+α}(S) → C^{k+1+α}(S), \\ & K^{l}, K^{l}^{∗} : C^{k+α}(S) → C^{k+α}(S), \\ & L^{l} : C^{k+1+α}(S) → C^{k+α}(S); \end{aligned}  \begin{matrix}(3.8) \\ (3.9) \\ (3.10) \end{matrix}

(ii) operators (3.5) and (3.6) are mutually adjoint singular integral operators (SIO) of normal type (i.e., their symbol matrices are not degenerated) and their indices are equal to zero; operators (3.8) and

\frac{1}{2}I + K^{l}, \frac{1}{2}I + K^{l}^{∗} : C^{k+α}(S) → C^{k+α}(S)

are invertible; moreover

H^{l}^{−1} : C^{k+1+α}(S) → C^{k+α}(S)

is a singular integro-differential operator; (iii) for h ∈ C^{α}(S) and g ∈ C^{1+α}(S)

( − H^{l}h, h)_{L^{2}(S)} ≥ 0, (L^{l}g, g) ≥ 0

with equality only for h = 0 and g(x) = [a × x] + b, where a and b are arbitrary three-dimensional constant vectors and [· × ·] stands for a vector product;

(iv) the general solution for the homogeneous equation

( − \frac{1}{2}I + K^{∗})g = 0

is g(x) = [a×x]+b, x ∈ S, with arbitrary three-dimensional constant vectors a and b.

Theorem 3.3 ([15]). Operator L^{l} is a singular integro-differential operator, and the equations

K^{l}^{∗}H^{l} = H^{l}K^{l}, K^{l}L^{l} = L^{l}K^{l}^{∗},

H^{l}L^{l} = −\frac{1}{4}I + (K^{l}^{∗})^{2}, L^{l}H^{l} = −\frac{1}{4}I + (K^{l})^{2}

hold.

Theorem 3.4 ([14]). Operators (3.8)– (3.10) can be extended by continuity to the following bounded operators:

\begin{matrix}H^{l} : H_{p}^{ν}(S) → H_{p}^{ν+1}(S) & [B_{p,q}^{ν}(S) → B_{p,q}^{ν+1}(S)], \\ K^{l}, K^{l}^{∗} : H_{p}^{ν}(S) → H_{p}^{ν}(S) & [B_{p,q}^{ν}(S) → B_{p,q}^{ν}(S)], \\ L^{l} : H_{p}^{ν+1}(S) → H_{p}^{ν}(S) & [B_{p,q}^{ν+1}(S) → B_{p,q}^{ν}(S)], \\ 1 < p < ∞, 1 ≤ q ≤ ∞, & ν ∈ ℝ. \end{matrix}  (3.11)

Operators H^{l}, K^{l}, K^{l}^{∗}, and L^{l} are pseudodifferential operators of order −1, 0, 0, and 1, respectively.

Operators (3.11) and

\frac{1}{2}I + K^{l}, \frac{1}{2}I + K^{l}^{∗} : H_{p}^{ν}(S) → H_{p}^{ν}(S) [B_{p,q}^{ν}(S) → B_{p,q}^{ν}(S)]

are invertible.

Theorem 3.5 ([14]). Operators (3.2) and (3.3) can be extended by continuity to the following bounded operators:

V^{l} : B_{p,p}^{ν}(S) → H_{p}^{ν+1+1/p}(Ω^{+}) [B_{p,q}^{ν}(S) → B_{p,q}^{ν+1+1/p}(Ω^{+})], (3.12)
U^{l} : B_{p,p}^{ν}(S) → H_{p}^{ν+1/p}(Ω^{+}) [B_{p,q}^{ν}(S) → B_{p,q}^{ν+1/p}(Ω^{+})],  (3.13)
V^{l} : B_{p,p}^{ν}(S) → H_{p,loc}^{ν+1+1/p}(Ω^{−}) [B_{p,q}^{ν}(S) → B_{p,q,loc}^{ν+1+1/p}(Ω^{−})], (3.14)
U^{l} : B_{p,p}^{ν}(S) → H_{p,loc}^{ν+1/p}(Ω^{−}) [B_{p,q}^{ν}(S) → B_{p,q,loc}^{ν+1/p}(Ω^{−})],  (3.15)
1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ.

Jump relations (3.4)– (3.7) on S remain valid for operators (3.12)– (3.15), respectively, in the corresponding spaces.

Theorem 3.6 ([14]). Operators −H^{l} and L^{l} are formally self-adjoint pseudodifferential operators with positive-definite principal symbol matrices whose entries are homogeneous functions of order −1 and 1, respectively. Inequalities (iii) of Theorem 3.2 hold for any h ∈ H_{2}^{−1/2}(S) and any g ∈ H_{2}^{1/2}(S) with the same conclusion (they are to be understood as dualities).

Theorem 3.7 ([14]). The operators

H^{l} : B̃_{p,q}^{ν}(S_{j}) → B_{p,q}^{ν+1}(S_{j}),  (3.16)
H^{l} : H̃_{p}^{ν}(S_{j}) → H_{p}^{ν+1}(S_{j}),  (3.17)
L^{l} : B̃_{p,q}^{ν+1}(S_{j}) → B_{p,q}^{ν}(S_{j}),  (3.18)
L^{l} : H̃_{p}^{ν+1}(S_{j}) → H_{p}^{ν}(S_{j}), j = 1, 2,  (3.19)

are bounded for any 1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ.

Operators (3.16) and (3.18) are Fredholm operators if

1/p − 3/2 < ν < 1/p − 1/2  (3.20)

holds.

Operators (3.17) and (3.19) are Fredholm operators if and only if condition (3.20) holds.

Operators (3.16)– (3.19) are invertible for all ν satisfying (3.20).

4. One Auxiliary Contact Problem (C–Problem)

Let us consider the following contact (interface) problem for a piecewise homogeneous space ℝ^{3} = \overline{Ω}_{1} ∪ \overline{Ω}_{2} with an interface S = ∂Ω_{1} = ∂Ω_{2} (see Section 2).

C–Problem. Find regular vectors u^{1} and u^{2} satisfying equations (1.4) in Ω_{1} and Ω_{2}, respectively, the contact conditions on S

[u^{1}(x)]^{+} − [u^{2}(x)]^{−} = φ(x),  (4.1)
[T^{1} (D_{x}, n(x))u^{1}(x)]^{+} − [T^{2} (D_{x}, n(x))u^{2}(x)]^{−} = Φ(x), x ∈ S,  (4.2)

and conditions (2.7) at infinity; here φ and Φ are the given vector-functions on S

φ ∈ C^{1+α}(S), Φ ∈ C^{α}(S).

Formulas (2.10) and (2.11) together with (1.3) imply

Lemma 4.1. The homogeneous C-Problem has only the trivial solution.

Let us look for the vectors u^{1} and u^{2} in the form of single layer potentials with special densities

u^{l}(x) = V [H^{l}^{−1}g^{l}](x), x ∈ Ω_{l},  (4.3)

where H^{l}^{−1} is the operator inverse to H^{l} (see Theorem 3.2.ii)).

Then by Theorem 3.1 due to (4.1) and (4.2) for the unknown vector densi-

ties g^{l} we obtain the following equations on S:  \{ \begin{matrix}g^{1} − g^{2} = φ, \\ ( − \frac{1}{2}I + K^{1})H^{1}^{−1}g^{1} − (\frac{1}{2}I + K^{2})H^{2}^{−1}g^{2} = Φ. \end{matrix}

\begin{matrix}(4.4) \\ (4.5) \end{matrix}

Denote

N^{1} = ( − \frac{1}{2}I + K^{1})H^{1}^{−1}, N^{2} = −(\frac{1}{2}I + K^{2})H^{2}^{−1}, N = N^{1} + N^{2}. (4.6)

By (4.4), (4.5), and (4.6) we have

\{ \begin{matrix}g^{1} = φ + g^{2}, \\ Ng^{2} = Φ − N^{1}φ. \end{matrix}  \begin{matrix}(4.7) \\ (4.8) \end{matrix}

Lemma 4.2. Operators N^{1}, N^{2}, and N are singular integro-differential operators with the following properties:

(i) N^{1}, N^{2}, N : C^{k+1+α}(S) → C^{k+α}(S); (4.9)

for f, g, h ∈ C^{1+α}(S)

(N^{1}f, f)_{L^{2}(S)} ≥ 0, (N^{2}g, g)_{L^{2}(S)} ≥ 0, (Nh, h)_{L^{2}(S)} ≥ 0 (4.10)

with equality only for f(x) = [a × x] + b, g = 0, and h = 0; a and b are arbitrary three-dimensional constant vectors.

(ii) Operators N^{1}, N^{2}, and N are formally self-adjoint pseudodifferential operators of order 1 and their principal (homogeneous of order 1) symbol matrices are positive definite;

(iii) Operators (4.9) can be extended by continuity to the following bounded operators:

N^{1}, N^{2}, N : H_{p}^{ν+1}(S) → H_{p}^{ν}(S) [B_{p,q}^{ν+1}(S) → B_{p,q}^{ν}(S)].  (4.11)

Inequalities (4.10) remain valid for any f, g, h ∈ H_{2}^{1/2}(S) with the same conclusion.

The operator N defined by (4.9) and (4.11) is invertible.

Proof. The fact that N^{1}, N^{2} and N are singular integro-differential oper-

ators follows directly from representation (4.6), since H^{l}^{−1} is a singular

integro-differential operator and (±\frac{1}{2}I + K^{l}) are singular integral operators. Therefore (4.9) holds.

To prove inequalities (4.10) we proceed as follows. If f and g have the

C^{1+α} smoothness, then the single-layer potentials V^{1} [H^{1}^{−1}f] and V^{2} [H^{2}^{−1}g] are regular solutions of equations (1.4) in Ω_{1} and Ω_{2}, respectively, satisfying condition (2.8). Due to formulas (2.10) and (2.11) we have

∫_{Ω^{+}} E^{1}(V^{1} [H^{1}^{−1}f], V^{1} [H^{1}^{−1}f]) dx =

= ∫_{S} [( − \frac{1}{2}I + K^{1})H^{1}^{−1}f] · f dS ≥ 0,  (4.12)

∫_{Ω^{−}} E^{2}(V^{2} [H^{2}^{−1}f], V^{2} [H^{2}^{−1}f]) dx =

= − ∫_{S} [(\frac{1}{2}I + K^{2})H^{2}^{−1}g] · g dS ≥ 0,  (4.13)

where E^{l} is defined by (2.12) and satisfies condition (1.3).

If in (4.12) and (4.13) we have equalities, then (see [15])

V^{1} [H^{1}^{−1}f](x) = [a^{1} × x] + b^{1}, x ∈ Ω_{1} = Ω^{+},

V^{2} [H^{2}^{−1}g](x) = [a^{2} × x] + b^{2}, x ∈ Ω_{2} = Ω^{−};

and applying (2.8), we conclude

f(x) = [a^{1} × x] + b^{1}, g = 0, x ∈ S.

Thus (4.10) holds and assertion (i) is proved.

Formally self-adjointness of the operators N^{1}, N^{2}, and N readily follows from Theorem 3.3. Indeed,

N^{∗} = (H^{1}^{−1})^{∗}( − \frac{1}{2}I + K^{1}^{∗}) − (H^{2}^{−1})^{∗}(\frac{1}{2}I + K^{2}^{∗}) =

− \frac{1}{2} H^{1}^{−1} + H^{1}^{−1}K^{1}^{∗} − \frac{1}{2} H^{2}^{−1} − H^{2}^{−1}K^{2}^{∗} =

= −\frac{1}{2} H^{1}^{−1} + K^{1}H^{1}^{−1} − \frac{1}{2} H^{2}^{−1} − K^{2}H^{2}^{−1} = N.

The remaining part of assertion (ii) follows from Theorems 3.4, 3.2.(ii), 3.6, and inequality (4.10).

The proof of assertion (iii) is quite similar to that of Theorems 3.4 and 3.6. Due to the general theory of elliptic pseudodifferential equations on mani-

folds without boundary, the invertibility of the operator N follows both from the properties of its symbol matrix and from the positiveness of N.

Remark 4.3. Note that the operator N^{−1} inverse to operators (4.9) and (4.11) is a pseudodifferential operator of order −1 with the following mapping properties:

N^{−1} : C^{k+α}(S) → C^{k+1+α}(S), k ≥ 0, 0 < α < 1,
N^{−1} : H_{p}^{ν}(S) → H_{p}^{ν+1}(S) [B_{p,q}^{ν}(S) → B_{p,q}^{ν+1}(S)],
1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ.

The principal symbol matrix of N^{−1} is a positive-definite symmetric matrix with homogeneous (of order −1) entries.

Now from (4.3), (4.7), (4.8) and Lemmas 4.1 and 4.2 we get the representation of the unique solution of the C–problem:

u^{1}(x) = V^{1} [H^{1}^{−1}N^{−1}(Φ + N^{2}φ)](x), x ∈ Ω_{1},  (4.14)
u^{2}(x) = V^{2} [H^{2}^{−1}N^{−1}(Φ − N^{1}φ)](x), x ∈ Ω_{2},  (4.15)

where φ and Φ are vector functions contained in (4.1) and (4.2).

These formulas imply that if

φ ∈ C^{k+1+α}(S), Φ ∈ C^{k+α}(S),

then

u^{l} ∈ C^{k+1+α}(\overline{Ω}_{l}), l = 1, 2

(see Lemma 4.2 and Theorem 3.1).

Remark 4.4. Note that if

φ ∈ B_{p,p}^{ν+1}(S) [B_{p,q}^{ν+1}(S)], Φ ∈ B_{p,p}^{ν}(S) [B_{p,q}^{ν}(S)],

then by (4.14) and (4.15)

u^{1} ∈ H_{p}^{ν+1+1/p}(Ω_{1}) [B_{p,q}^{ν+1+1/p}(Ω_{1})],

u^{2} ∈ H_{p,loc}^{ν+1+1/p}(Ω_{2}) [B_{p,q,loc}^{ν+1+1/p}(Ω_{2})],

1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ

(see Theorems 3.4 and 3.5 and Lemma 4.2).

5. Investigation of the C–D Problem

Let us consider the C–D problem (2.1)–(2.4), (2.9), (2.15), (2.16). First we substitute conditions (2.3) and (2.4) by the equivalent equations

\begin{matrix}[u^{1}(x)]^{+} − [u^{2}(x)]^{−} = f_{+}(x) − f_{−}(x), \\ [u^{1}(x)]^{+} + [u^{2}(x)]^{−} = f_{+}(x) + f_{−}(x), \end{matrix} \} x ∈ S_{2}. \begin{matrix}(5.1) \\ (5.2) \end{matrix}

Due to (2.16) it is evident that the difference

[u^{1}(x)]^{+} − [u^{2}(x)]^{−} = f^{0}(x) ∈ B_{p,p}^{1−1/p}(S)

is a known vector on S.

Let F̃ be some fixed extension of the vector F from S_{1} onto S_{2} which preserves the space, i.e.,

F̃ ∈ B_{p,p}^{−1/p}(S), F̃|_{S_{1}} = F.

Any other extension Φ of the vector F from S_{1} onto S_{2} preserving the space can be represented in the form

Φ = F̃ + Ψ ∈ B_{p,p}^{−1/p}(S)

with arbitrary

Ψ ∈ B̃_{p,p}^{−1/p}(S_{2}).

Let us now look for the solution of the C–D problem in the form (cf. (4.14), (4.15))

u^{1}(x) = V^{1} [H^{1}^{−1}N^{−1}(F̃ + Ψ + N^{2}f^{0})](x), x ∈ Ω_{1},  (5.3)
u^{2}(x) = V^{2} [H^{2}^{−1}N^{−1}(F̃ + Ψ − N^{1}f^{0})](x), x ∈ Ω_{2},  (5.4)

with the known vector-functions f^{0}, F̃ and the unknown vector-function Ψ.

It is evident that equation (1.4) and conditions (2.1), (2.2), and (5.1) are satisfied. Condition (5.2) leads to the following pseudodifferential equation for Ψ:

N^{−1}Ψ = \frac{1}{2}(f_{+} + f_{−}) − N^{−1}F̃ − \frac{1}{2}N^{−1}(N^{2} − N^{1})f^{0} on S_{2}. (5.5)

Note that by virtue of Lemma 4.2 and Remark 4.3

Q ≡ −N^{−1}F̃ − \frac{1}{2}N^{−1}(N^{2} − N^{1})f^{0} ∈ B_{p,p}^{1−1/p}(S)  (5.6)

and

q ≡ \frac{1}{2}[f_{+} + f_{−}] + Q|_{S_{2}} ∈ B_{p,p}^{1−1/p}(S_{2}).  (5.7)

Let r_{2} be the restriction operator to S_{2}. Then (5.5) can be rewritten in the form

r_{2}N^{−1}Ψ = q on S_{2}  (5.8)

with q defined by (5.7), (5.6).

We have to investigate the solvability of (5.8) in the Besov space B̃_{p,p}^{−1/p}(S_{2}).

To this end denote by σ(x; ξ), x ∈ S, ξ ∈ ℝ^{2} the principal symbol matrix of the operator N^{−1} whose entries are homogeneous functions of order −1 with respect to ξ.

To establish the Fredholm property of equation (5.8) due to the general theory of pseudodifferential equations on manifold with boundary, we must investigate eigenvalues of the matrix

M = [σ(x; 0, −1)]^{−1}[σ(x; 0, 1)]

(see [16]–[20]).

Lemma 5.1. Eigenvalues of the matrix M for any x ∈ S are positive numbers.

Proof of the Lemma follows from positive definiteness of the matrix σ(x; ξ) for any x ∈ S and |ξ| = 1. Indeed, if λ is an eigenvalue of the matrix M, then there exists η ∈ ℂ^{3}\{0} such that Mη = λη, whence

σ(x; 0, 1)η = λσ(x; 0, −1)η,

i.e.,

λ = (σ(x; 0, 1)η · η)[(σ(x; 0, −1)η · η)]^{−1} > 0.

Lemma 5.1 implies (see [16]–[18], 20])

Theorem 5.2. Let 1 < p < ∞, 1 ≤ q ≤ ∞. Then the operators

r_{2}N^{−1} : B̃_{p,q}^{ν}(S_{2}) → B_{p,q}^{ν+1}(S_{2}),  (5.9)
r_{2}N^{−1} : H̃_{p}^{ν}(S_{2}) → H_{p}^{ν+1}(S_{2})  (5.10)

are bounded for all ν ∈ ℝ.

Operator (5.9) is a Fredholm operator if (3.20) holds.

Operator (5.10) is a Fredholm operator if and only if conditions (3.20) hold.

Both operators (5.9) and (5.10) are invertible if ν and p satisfy inequality (3.20).

It is evident that Theorem 5.2 yields unique solvability of equation (5.8) for 4/3 < p < 4. Consequently, we obtain the following

Theorem 5.3. Let 4/3 < p < 4 and conditions (2.15), (2.16) be fulfilled. Then the C–D problem has a unique solution satisfying (2.9). The solution is representable in the form (5.3), (5.4) with Ψ defined by the uniquely solvable pseudodifferential equation (5.8).

The method described above makes it possible to improve the regularity property of the solution by increasing the smoothness of boundary data.

Theorem 5.4. Let 4/3 < p < 4, 1 < t < ∞, 1 ≤ q ≤ ∞, 1/t − 3/2 < ν < 1/t − 1/2, and u^{1}, u^{2} be the solution of the C–D problem satisfying conditions (2.9). In addition, if

f ∈ B_{t,t}^{ν+1}(S_{1}), f_{±} ∈ B_{t,t}^{ν+1}(S_{2}), F ∈ B_{t,t}^{ν}(S_{1}),

f^{0} = \begin{cases}f & on S_{1} \\ f_{+} − f_{−} & on S_{2} \end{cases} ∈ B_{t,t}^{ν+1}(S),

then

u^{1} ∈ H_{t}^{ν+1+1/t}(Ω_{1}), u^{2} ∈ H_{t,loc}^{ν+1+1/t}(Ω_{2}).

If

f ∈ B_{t,q}^{ν+1}(S_{1}), f_{±} ∈ B_{t,q}^{ν+1}(S_{2}), F ∈ B_{t,q}^{ν}(S_{1}), f^{0} ∈ B_{t,q}^{ν+1}(S), then

u^{1} ∈ B_{t,q}^{ν+1+1/t}(Ω_{1}), u^{2} ∈ B_{t,q,loc}^{ν+1+1/t}(Ω_{2}).

In particular, if

f ∈ C^{α}(S_{1}), f_{±} ∈ C^{α}(S_{2}), F ∈ B_{∞,∞}^{α−1}(S_{1}), f^{0} ∈ C^{α}(S),

then

u^{1} ∈ C^{α′}(\overline{Ω}_{1}), u^{2} ∈ C^{α′}(\overline{Ω}_{2})

for any α′ ∈ (0, α_{0}), α_{0} = min{α, 1/2}.

Proof of the theorem follows from Theorems 5.2, 3.5 and the well-known embedding theorems for the Besov and the Bessel potential spaces (cf. similar theorems in [14, 20]).

6. Investigation of the C–N Problem

Now let us consider the C–N problem (2.1), (2.2), (2.5), (2.6), (2.9), (2.15), (2.17). We shall study this problem by applying the approach described in Section 5. First we substitute conditions (2.5) and (2.6) by the equivalent ones

[T^{1} (D_{x}, n(x))u^{1}(x)]^{+} − [T^{2} (D_{x}, n(x))u^{2}(x)]^{−} = F_{+}(x) − F_{−}(x), (6.1)
[T^{1} (D_{x}, n(x))u^{1}(x)]^{+} + [T^{2} (D_{x}, n(x))u^{2}(x)]^{−} = F_{+}(x) + F_{−}(x), (6.2)
x ∈ S_{2}.

Condition (6.1) combined with (2.2) gives (on S)

[T^{1} (D_{x}, n(x))u^{1}]^{+} − [T^{2} (D_{x}, n(x))u^{2}]^{−} = F^{0}(x) ∈ B_{p,p}^{−1/p}(S)

due to (2.17).

Denote by f̃ some fixed extension of the vector f from S_{1} onto S_{2} preserving the functional space

f̃ ∈ B_{p,p}^{1−1/p}(S), f̃|_{S_{1}} = f.

Any other extension φ of the vector f from S_{1} onto S_{2} preserving the functional space can be represented in the form

φ = f̃+ ψ ∈ B_{p,p}^{1−1/p}(S)

with

ψ ∈ B̃_{p,p}^{1−1/p}(S_{2}).

Now if we look for the solution of the C–N problem in the form

u^{1}(x) = V^{1} {H^{1}^{−1}N^{−1}[F^{0} + N^{2}(f̃+ ψ)]}(x), x ∈ Ω_{1},  (6.3)
u^{2}(x) = V^{2} {H^{2}^{−1}N^{−1}[F^{0} − N^{1}(f̃+ ψ)]}(x), x ∈ Ω_{2},  (6.4)

then equation (1.4) and conditions (2.1), (2.2), (6.1) are satisfied automatically. Condition (6.2) implies the following equation on S_{2} for the unknown vector ψ ∈ B̃_{p,p}^{1−1/p}(S_{2})

N^{2}N^{−1}N^{1}ψ = \frac{1}{2}(F_{+} + F_{−}) + \frac{1}{2}(N^{2} − N^{1})N^{−1}F^{0} − N^{2}N^{−1}N^{1} f̃. (6.5)

This equation is obtained with the help of the equality

N^{1}N^{−1}N^{2} = N^{2}N^{−1}N^{1}  (6.6)

which can be easily verified.

It is evident that

Q_{1} ≡ \frac{1}{2}(N^{2} − N^{1})N^{−1}F^{0} − N^{2}N^{−1}N^{1} f̃ ∈ B̃_{p,p}^{−1/p}(S),  (6.7)
q_{1} ≡ \frac{1}{2}(F_{+} + F_{−}) + Q_{1}|_{S_{2}} ∈ B_{p,p}^{−1/p}(S_{2}).  (6.8)

Using the notation

P = N^{2}N^{−1}N^{1},

from (6.5), (6.7), and (6.8) we get

r_{2}P ψ = q_{1} on S_{2},  (6.9)

where r_{2} is again the restriction operator (to S_{2}).

Solvability of equation (6.9) will be studied by the method applied in the previous section.

To this end we have to examine the properties of the operator P.

Lemma 6.1. Operator P is a formally self-adjoint and non-negative operator with positive-definite principal symbol matrix. For arbitrary h ∈ C^{1+α}(S)

(P h, h)_{L^{2}(S)} ≥ 0

with equality only for h = [a × x] + b, where a and b are arbitrary constant vectors.

Operators

P : C^{k+1+α}(S) → C^{k+α}(S), k ≥ 0, 0 < α < 1,  (6.10)
P : B_{p,q}^{ν+1}(S) → B_{p,q}^{ν}(S),  (6.11)
P : H_{p}^{ν+1}(S) → H_{p}^{ν}(S),  (6.12)
1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ,

are bounded.

Proof. Mapping properties (6.10)–(6.12) follow from Lemma 4.2. Self-ad-jointness is the consequence of equality (6.6) and Lemma 4.2.ii).

Let h ∈ C^{1+α}(S) and consider single layer potentials

v^{1}(x) = V^{1} [H^{1}^{−1}N^{−1}N^{2}h](x), x ∈ Ω_{1},

v^{2}(x) = −V^{2} [H^{2}^{−1}N^{−1}N^{1}h](x), x ∈ Ω_{2}.

It is easy to show that these potentials are regular vectors in Ω_{1} and Ω_{2}, respectively, and relation (2.8) holds for them. Therefore, making use of Green’s formulas (2.10) and (2.11), we get

∫_{Ω^{+}} E^{1}(v^{1}, v^{1})dx = ∫_{S} N^{1}N^{−1}N^{2}h · N^{−1}N^{2}h dS,

∫_{Ω^{−}} E^{2}(v^{2}, v^{2})dx = ∫_{S} N^{2}N^{−1}N^{1}h · N^{−1}N^{1}h dS.

Upon taking the sum and using (6.6) and (4.6), we obtain

∫_{Ω^{+}} E^{1}(v^{1}, v^{1})dx + ∫_{Ω^{−}} E^{2}(v^{2}, v^{2})dx = ∫_{S} N^{1}N^{−1}N^{2}h · h dS ≥ 0.

In the case of the equality to zero we conclude that v^{1} and v^{2} are rigid displacements

v^{1}(x) = [a′ × x] + b′, x ∈ Ω_{1}; v^{2}(x) = [a′′ × x] + b′′, x ∈ Ω_{2}.

Condition (2.8) for v^{2} implies: v^{2}(x) = 0, x ∈ Ω_{2}. Therefore H^{2}^{−1}N^{−1}N^{1}h =

0. Invertibility of the operators H^{2}^{−1} and N^{−1} (see Theorem 3.2.ii) and

Lemma 4.2) yields N^{1}h = 0, whence by Theorem 3.2.iv) h(x) = [a × x] + b. Now it is easy to verify that [v^{1}(x)]_{S}^{+} = h(x), i.e., a′ = a, b′ = b.

Positive definiteness of the principal symbol matrix of the operator P can be shown in the following way.

Denoting by σ, σ_{1}, σ_{2}, and σ_{0} the principal symbol matrices of the op-

erators N^{−1}, N^{1}, N^{2}, and P, respectively, we have (see (6.6))

σ_{0}(x; ξ) = σ_{2}(x; ξ)σ(x; ξ)σ_{1}(x; ξ) =
= σ_{1}(x; ξ)σ(x; ξ)σ_{2}(x; ξ), x ∈ S, ξ ∈ ℝ^{2}\{0}.  (6.13)

By Lemma 4.2.(ii) all matrices on the right-hand side of (6.13) (and their reciprocal matrices) are positive definite. Moreover, the matrix σ(x; ξ) has the form (see (4.6))

σ(x; ξ) = [σ_{1}(x; ξ) + σ_{2}(x; ξ)]^{−1}.

Hence

σ_{0}^{−1}(x; ξ) = σ_{1}^{−1}(x; ξ)[σ_{1}(x; ξ) + σ_{2}(x; ξ)]σ_{2}^{−1}(x; ξ) : ℂ^{3} → ℂ^{3}

is the isomorphism on ℂ^{3} for any x ∈ S and |ξ| = 1.

Let

η = σ_{0}^{−1}(x; ξ)ζ, ζ ∈ ℂ^{3}.

Due to the positive definiteness of the matrices σ_{l}(x; ξ), l = 1, 2, we have

(σ_{0}(x; ξ)η, η) = (ζ, σ_{0}^{−1}(x; ξ)ζ) =
= \overline{(σ_{1}^{−1}(x; ξ)ζ, ζ))} + \overline{(σ_{2}^{−1}(x; ξ)ζ, ζ)} ≥ δ′|ζ|^{2} > 0

for any x ∈ S, |ξ| = 1 and η ∈ ℂ^{3}\{0}.

Therefore σ_{0}(x; ξ) is a positive-definite matrix (since it is positive and nondegenerated).

Remark 6.2. From the proof of Lemma 6.1 it follows that the general solution of equation P h(x) = 0, x ∈ S, is representable by the formula h(x) = [a × x] + b, x ∈ S, with arbitrary tree-dimensional vectors a and b. This implies that if h solves the above homogeneous equation and supp h ≠ S, then h(x) = 0, x ∈ S.

Lemma 6.3. Eigenvalues of the matrix

[σ_{0}(x; 0, −1)]^{−1}[σ_{0}(x; 0, 1)],

where σ_{0}(x; ξ) is the principal symbol of P, are positive numbers.

Proof. It is quite similar to that of Lemma 5.1.

Thus the operator P belongs to the class of pseudodifferential operators for which the equations of type (6.9) on manifolds with boundary were studied in [16]–[18]. As in Section 5 the results obtained in the above-cited papers allow us to formulate the following theorems.

Theorem 6.4. Let 1 < p < ∞, 1 ≤ q ≤ ∞. Then the operators

r_{2}P : B̃_{p,q}^{ν+1}(S_{2}) → B_{p,q}^{ν}(S_{2}),  (6.14)
r_{2}P : H̃_{p}^{ν+1}(S_{2}) → H_{p}^{ν}(S_{2})  (6.15)

are bounded for all ν ∈ ℝ.

Operator (6.14) is a Fredholm operator if (3.20) holds.

Operator (6.15) is a Fredholm operator if and only if (3.20) holds.

Both operators (6.14) and (6.15) are invertible if ν satisfies inequality (3.20).

Theorem 6.5. Let 4/3 < p < 4 and conditions (2.15), (2.17) be fulfilled. Then the C–N problem has a unique solution of the class (2.9). The solution is representable in the form of (6.3) and (6.4) with ψ defined by the uniquely solvable pseudodifferential equation (6.9).

Theorem 6.6. Let 4/3 < p < 4, 1 < t < ∞, 1 ≤ q ≤ ∞, 1/t − 3/2 < ν < 1/t − 1/2 and let u^{1}, u^{2} be the solution of the C–N problem satisfying (2.9). In addition, if

f ∈ B_{t,t}^{ν+1}(S_{1}), F ∈ B_{t,t}^{ν}(S_{1}), F_{±} ∈ B_{t,t}^{ν}(S_{2})

and F^{0} defined by (2.17) belongs to B_{t,t}^{ν}(S), then

u^{1} ∈ H_{t}^{ν+1+1/t}(Ω_{1}), u^{2} ∈ H_{t,loc}^{ν+1+1/t}(Ω_{2}).

If

f ∈ B_{t,q}^{ν+1}(S_{1}), F ∈ B_{t,q}^{ν}(S_{1}), F_{±} ∈ B_{t,q}^{ν}(S_{2}), F^{0} ∈ B_{t,q}^{ν}(S),

then

u^{1} ∈ B_{t,q}^{ν+1+1/t}(Ω_{1}), u^{2} ∈ B_{t,q,loc}^{ν+1+1/t}(Ω_{2}).

In particular, if

f ∈ C^{α}(S_{1}), F ∈ B_{∞,∞}^{α−1}(S_{1}), F_{±} ∈ B_{∞,∞}^{α−1}(S_{2}), F^{0} ∈ B_{∞,∞}^{α−1}(S),

then

u^{1} ∈ C^{α′}(\overline{Ω}_{1}), u^{2} ∈ C^{α′}(\overline{Ω}_{2})

for any α′ ∈ (0, α_{0}), α_{0} = min{α, 1/2}.

Remark 6.7. The C–D and C–N problems involve as particular cases the screen and the crack type problems. In fact, if both elastic materials occupying Ω_{1} and Ω_{2} are the same and conditions (2.1) and (2.2) are homogeneous (f = 0, F = 0), then the surface S_{1} becomes a formal interface; the displacement vector satisfies equation (1.4) for the points x ∈ S_{1}, and we obtain either the screen type or the crack type problems for ℝ^{3} with a cut along the surface S_{2}.

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(Received 24.11.1993)

Author’s address:

Department of Mathematics (99)

Georgian Technical University

77, M. Kostava St., Tbilisi 380075

Republic of Georgia