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
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
428
O. CHKADUA
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 D1 be a finite domain, D2 be a domain that can be both finite or infinite
in the Euclidean space R3 with compact boundaries ∂D1 , ∂D2 (∂D1 ∈ C ∞ ),
and let there exist a surface S0 of the class C ∞ of dimension two, which divides
(1)
(2)
(1)
the domain D2 into two subdomains D2 and D2 with C ∞ boundaries ∂D2
(1)
(2)
(2)
(1)
(2)
and ∂D2 (D2 ∩ D2 = ∅, D2 ∩ D2 = S 0 ). Then ∂S0 is the boundary of
the surface S0 (∂S0 ⊂ ∂D2 ), representing one-dimensional closed cuspidal edge,
where ∂S0 is the crack edge.
Let the domains D1 and D2 have the contact on the two-dimensional mani(1)
(2)
(1)
(2)
folds S 0 and S 0 of the class C ∞ , i.e., ∂D1 ∩ ∂D2 = S 0 ∪ S 0 , D1 ∩ D2 = ∅,
(1)
(2)
(1)
(2)
(1)
(1)
(1)
S 0 ∩ S 0 = ∅, and S1 = ∂D1 \ (S 0 ∪ S 0 ). Then ∂D2 = S2 ∪ S 0 ∪ S 0 ,
(2)
(2)
(2)
∂D2 = S2 ∪ S 0 ∪ S 0 .
Suppose that the domains Dq , q = 1, 2, are filled with anisotropic homogeneous elastic materials.
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
429
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 Dq , q = 1, 2,
(1.1)
(q)
(q)
(q)
where U (q) = (u(q) , ω (q) ), u(q) = (u1 , u2 , u3 ) is the displacement vector, ω (q) =
(q)
(q)
(q)
(q)
(q)
(ω1 , ω2 , ω3 ) is the rotation vector, F (q) = (F1 , . . . , F6 ) is the mass force
applied to Dq , and M(q) (∂x ) is the matrix differential operator


1
2
(q)
(q)
M (∂x ) M (∂x ) 
,
M(q) (∂x ) =  3
4
M (q) (∂x ) M (q) (∂x )
6×6
l
M
(q)
(∂x ) = k
1 (q)
l (q)
M jk (∂x )k3×3 ,
2 (q)
(q)
l = 1, 4, q = 1, 2,
(q)
(q)
M jk (∂x ) = bijlk ∂i ∂l − εlrk aijlr ∂i ,
M jk (∂x ) = aijlk ∂i ∂l ,
3 (q)
(q)
(q)
(1.2)
M jk (∂x ) = blkij ∂i ∂l + εirj airlk ∂l ,
4 (q)
(q)
(q)
(q)
(q)
M jk (∂x ) = cijlk ∂i ∂l − blrij εlrk ∂i + εirj birlk ∂l − εipj εlrk aiplr ;
(q)
(q)
(q)
εikj is the Levi-Civita symbol, aijlk , bijlk , cijlk are the elastic constants satisfying
the conditions
(q)
(q)
(q)
(q)
aijlk = alkij , cijlk = clkij , 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
(q)
(q)
(q)
aijlk ξij ξlk + 2bijlk ξij ηlk + cijlk ηij ηlk , q = 1, 2,
with respect to variables ξij , ηij are positive-definite, i.e., ∃M > 0
(q)
(q)
(q)
aijlk ξij ξlk+2bijlk ξij ηlk+cijlk ηij ηlk ≥ M (ξij ξij+ηlk ηlk ) for all ξij , ηlk , q = 1, 2. (1.3)
We introduce the differential stress operator


1
2
(q)
(q)
N (∂z , n(z)) N (∂z , n(z)) 
N (q) (∂z , n(z)) =  3
4
N (q) (∂z , n(z)) N (q) (∂z , n(z))
l
N
(q)
(∂z , n(z)) = k
l (q)
N jk (∂z , n(z))k3×3 ,
,
6×6
l = 1, 4, q = 1, 2,
2 (q)
(q)
(q)
4 (q)
(q)
(q)
1 (q)
(q)
N jk (∂z , n(z)) = bijlk ni (z)∂l − aijlk εlrk ni (z),
3 (q)
(q)
N jk (∂z , n(z)) = cijlk ni (z)∂l − blrij εlrk ni (z),
N jk (∂z , n(z)) = aijlk ni (z)∂l ,
N jk (∂z , n(z)) = blkij ni (z)∂l ,
where n(z) = (n1 (z), n2 (z), n3 (z)) is the unit normal of the manifold ∂D1 at a
point z ∈ ∂D1 (external with respect to D1 ) and a point z ∈ ∂D2 (internal with
respect to D2 ).
430
O. CHKADUA
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]):
s
s
s
s
es = B
es = H
ep,r
ep,r
e ps × H
e ps .
Bsp,r = Bp,r
× Bp,r
, B
×B
, Hsp = Hps × Hps , H
p,r
p
(q)
(q)
From the symmetry of the coefficients aijlk , cijlk 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 ∈ R3 and any complex vector η ∈ C6 the relations
◦
◦
(q)
Re(M (q) (ξ)η, η) = (M (q) (ξ)η, η) ≥ P0 |ξ|2 |η|2
(q)
are valid, where P0
◦
= const > 0 depends only on the elastic constants. Thus
the matrices M (q) (ξ) are positive-definite for ξ ∈ R3 \ {0}.
Taking into account the property of the Levi-Civita symbol


δil δir δik
εipj εlrk = det  δpl δpr δpk  (δpl is the Kronecker symbol),
δjl δjr δjk
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 ∈ R3 \ {0}.
Since
detM(q) (ξ) 6= 0 for ξ 6= 0.
(1)
1
Let U (1) ∈ Wp1 (D1 ), U (2) ∈ Wp,loc
(D2 ). Then r1 U (2) = rD(1) U (2) ∈ Wp1 (D2 )
(2)
2
1
and r2 U (2) = rD(2) U (2) ∈ Wp,loc
(D2 ), where ri is the restriction operator on
(i)
2
D2 , i = 1, 2. From the theorem on traces (see [39]) it follows that the trace of
(i)
the functions U (i) , ri U (2) , i = 1, 2, exists on ∂D1 , ∂D2 , i = 1, 2, and {U (1) }± ∈
0
0
1/p
1/p
(i)
Bp,p (∂D1 ), {ri U (2) }± ∈ Bp,p (∂D2 ), i = 1, 2, p0 = p/(p − 1). Let U (1) ∈
1
(D2 ) be such that M(1) (∂x )U (1) ∈ Lp (D1 ), M(2) (∂x )U (2) ∈
Wp1 (D1 ), U (2) ∈ Wp,loc
Lp,comp (D2 ). Then {N (1) U (1) }± and {N (2) (ri U (2) )}± , i = 1, 2, are correctly
defined by the equalities
Z
¤
£ (1) (1)
V M (∂x )U (1) + E (1) (U (1) , V (1) ) dx
D1
(1)
= ±h{N
U (1) }± , {V (1) }± i∂D1 for all V (1) ∈ Wp10 (D1 )
(1.4)
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
and
Z
£
(i)
D2
431
(i)
(i) ¤
V 2 M(2) (∂x )(ri U (2) ) + E (2) (ri U (2) , V2 ) dx
(i)
= ∓h{N (2) (ri U (2) )}± , {V2 }± i∂D(i) ,
2
(1)
(1)
(2)
i = 1, 2,
(1.5)
(2)
1
for all V2 ∈ Wp10 (D2 ) (V2 ∈ Wp,comp
(D2 )),
where
(1)
E (1) (U (1) , V (1) ) = aijlk ξij (U (1) )ξlk (V
(1)
+ bijlk ξij (V
(1)
(1)
(1)
) + bijlk ξij (U (1) )ηlk (V
(1)
(1)
)ηlk (U (1) ) + cijlk ηij (U (1) )ηlk (V
)
(1)
);
here U (1) = (u(1) , ω (1) ),
(1)
ξij (U (1) ) = ∂j ui − εijk ωk is the deformation component,
ηij (U (1) ) = ∂j ωi is the bending torsion component.
(i)
The quadratic form E (2) (ri U (2) , V2 ), i = 1, 2, is defined analogously.
In the case of the infinite domain D2 , 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 M1 and M2 .
We will study the solvability and asymptotics of solutions U (q) ∈ Wp1 (Dq ), q =
1
1, 2, (U (2) ∈ Wp,loc
(D2 ) with condition (1.6) at infinity) the following boundarycontact problems of couple-stress elasticity:
Problem M1 :

M(q) (∂x )U (q) = 0






πS1 {U (1) }+ = ϕ1





 πS (1) {N (2) (r1 U (2) )}+ = ϕ2
in Dq , q = 1, 2,
on S1 ,
(1)
on S2 ,
2
(2)

πS (2) {N (2) (r2 U (2) )}+ = ϕ3
on S2 ,


2


(i)


πS (i) {U (1) }+ − πS (i) {ri U (2) }+ = fi
on S0 ,


0
0


 π (i) {N (1) U (1) }+ − π (i) {N (2) (r U (2) )}+ = h on S (i) , i = 1, 2,
i
i
0
S
S
0
0
where
1/p0
−1/p
(1)
−1/p
(2)
ϕ1 ∈ Bp,p (S1 ), ϕ2 ∈ Bp,p (S2 ), ϕ3 ∈ Bp,p (S2 ),
1/p0
(i)
fi ∈ Bp,p (S0 ),
−1/p
(i)
hi ∈ Bp,p (S0 ), i = 1, 2, 1 < p < ∞, p0 = p/(p − 1).
If D2 is a finite domain, then we have the following wedge-type problem:
432
O. CHKADUA
Wedge-type Problem M2 :

M(q) (∂x )U (q) = 0






πS1 {N (1) U (1) }+ = ϕ1





 π (1) {N (2) (r1 U (2) )}+ = ϕ2
in Dq , q = 1, 2,
on S1 ,
(1)
on S2 ,
S2
(2)

on S2 ,
πS (2) {N (2) (r2 U (2) )}+ = ϕ3


2


(i)

(1) +

π
} − πS (i) {ri U (2) }+ = fi
on S0 ,
(i) {U

S0

0


 π (i) {N (1) U (1) }+ − π (i) {N (2) (r U (2) )}+ = h on S (i) , i = 1, 2,
i
i
0
S
S
0
0
where
1/p0
1/p0
(1)
1/p0
(2)
ϕ1 ∈ Bp,p (S1 ), ϕ2 ∈ Bp,p (S2 ), ϕ3 ∈ Bp,p (S2 ),
−1/p
1/p0
(i)
(i)
fi ∈ Bp,p (S0 ), hi ∈ Bp,p (S0 ), i = 1, 2, 1 < p < ∞.
2. Fundamental Solutions and Potentials
Consider the fundamental matrix-functions
µ Z
¶
¡ (q) 0
¢−1 −iτ x3
−1
(q)
H (x) = Fξ0 →x0 ±
M (iξ , iτ ) e
dτ ,
q = 1, 2,
L±
where the sign “−” refers to the case x3 > 0 and
R the sign “+” to the case
0
0
x3 < 0; x = (x1 , x2 , x3 ), x = (x1 , x2 ), ξ = (ξ1 , ξ2 ); L± denotes integration over
the contour L± , where L+ (L− ) has the positive orientation and covers all roots
of the polynomial detM(q) (iξ 0 , 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
Z
(1)
V (g1 )(x) =
H(1) (x − y)g1 (y)dy S, x ∈
/ ∂D1 ,
∂D1
Z
(1)
(2)
V (g2 )(x) =
H(2) (x − y)g2 (y)dy S, x ∈
/ ∂D2 ,
(1)
∂D
Z 2
(2)
V(3) (g3 )(x) =
H(2) (x − y)g3 (y)dy S, x ∈
/ ∂D2 .
(2)
∂D2
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:
¢
¡
s+1+1/p
(D1 ) ,
(D1 ) Bsp,p (∂D1 ) → Hs+1+1/p
V(1) : Bsp,r (∂D1 ) → Bp,r
p
(1) ¢
(1)
(1) ¡
(1)
(D2 ) ,
(D2 ) Bsp,p (∂D2 ) → Hs+1+1/p
V(2) : Bsp,r (∂D2 ) → Bs+1+1/p
p
p,r
(2)
s+1+1/p
(2) ¢
(2)
s+1+1/p
(2) ¡
V(3) : Bsp,r (∂D2 ) → Bp,r,loc (D2 ) Bsp,p (∂D2 ) → Hp,loc (D2 ) .
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
433
Theorem 2.2. Let 1 < p < ∞, 1 ≤ r ≤ ∞, ε > 0, g1 ∈ B−1+ε
p,r (∂D1 ),
(1)
(2)
−1+ε
−1+ε
g2 ∈ Bp,r (∂D2 ), g3 ∈ Bp,r (∂D2 ). Then
Z
(1)
±
{V (g1 )(z)} =
H(1) (z − y)g1 (y)dy S, z ∈ ∂D1 ,
Z∂D1
(1)
{V(2) (g2 )(z)}± =
H(2) (z − y)g2 (y)dy S, z ∈ ∂D2 ,
(1)
∂D
Z 2
(2)
H(2) (z − y)g3 (y)dy S, z ∈ ∂D2 .
{V(3) (g3 )(z)}± =
(2)
∂D2
−1/p
−1/p
(1)
Theorem 2.3. Let 1 < p < ∞, g1 ∈ Bp,p (∂D1 ), g2 ∈ Bp,p (∂D2 ), g3 ∈
−1/p
(2)
Bp,p (∂D2 ). Then
© (1) (1)
ª±
1
N V (g1 )(z) = ∓ g1 (z)
2
Z
+
N (1) (∂z , n(z))H(1) (z − y)g1 (y)dy S, z ∈ ∂D1 ,
©
∂D1
ª±
1
N (2) V(2) (g2 )(z) = ± g2 (z)
2
Z
(1)
+
N (2) (∂z , n(z))H(2) (z − y)g2 (y)dy S, z ∈ ∂D2 ,
(1)
©
∂D2
ª±
1
N (2) V(3) (g3 )(z) = ± g3 (z)
2
Z
(2)
(2)
+
N (∂z , n(z))H(2) (z − y)g3 (y)dy S, z ∈ ∂D2 .
(2)
∂D2
Let us introduce the following notation:
Z
(1)
V−1 (g1 )(z) =
H(1) (z − y)g1 (y)dy S, z ∈ ∂D1 ,
Z∂D1
(2)
(1)
V−1 (g2 )(z) =
H(2) (z − y)g2 (y)dy S, z ∈ ∂D2 ,
(1)
∂D
Z 2
(3)
(2)
V−1 (g3 )(z) =
H(2) (z − y)g3 (y)dy S, z ∈ ∂D2 ,
(2)
∂D
Z 2
∗ (1)
N (1) (∂z , n(z))H(1) (z − y)g1 (y)dy S, z ∈ ∂D1 ,
V 0 (g1 )(z) =
Z∂D1
∗ (2)
(1)
N (2) (∂z , n(z))H(2) (z − y)g2 (y)dy S, z ∈ ∂D2 ,
V 0 (g2 )(z) =
(1)
∂D
Z 2
∗ (3)
(2)
N (2) (∂z , n(z))H(2) (z − y)g3 (y)dy S, z ∈ ∂D2 .
V 0 (g3 )(z) =
(2)
∂D2
(i)
Theorem 2.4. Let 1 < p < ∞, 1 ≤ r ≤ ∞. Then the operators V−1 ,
∗ (i)
V 0 , i = 1, 2, 3, admit extensions to the operators which are continuous in the
434
O. CHKADUA
following spaces:
(i)
V−1 : Hsp (∂Ωi ) → Hs+1
p (∂Ωi )
¡ s
¢
Bp,r (∂Ωi ) → Bs+1
(∂Ω
)
, i = 1, 2, 3,
i
p,r
∗ (i)
V0
: Hsp (∂Ωi ) → Hsp (∂Ωi )
¡ s
¢
Bp,r (∂Ωi ) → Bsp,r (∂Ωi ) , i = 1, 2, 3,
(1)
(2)
here Ω1 = D1 , Ω2 = D2 , Ω3 = D2 .
3. Uniqueness, Existence and Smoothness Theorems for
Problem M1
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 D2
is an analytic function in D2 (see [17]). Then we see that the equalities
(
{r1 U (2) }+ − {r2 U (2) }+ = 0
on S0 ,
{N (2) (r1 U (2) )}+ + {N (2) (r2 U (2) )}+ = 0 on S0
(3.1)
are valid on S0 .
Let us study the uniqueness of a solution of the boundary-contact problem
1
M1 in the classes W21 (Dq ), q = 1, 2 (W2,loc
(D2 ) with condition (1.6) at infinity).
Lemma 3.1. A solution of the boundary-contact problem M1 is unique in the
1
classes W21 (Dq ), q = 1, 2 (W2,loc
(D2 ) and satisfying condition (1.6) at infinity).
Proof. Let U (q) , q = 1, 2, be a solution of the homogeneous problem M1 .
(2)
(1)
We write the Green formulae (see (1.4), (1.5)) in the domains D1 , D2 , D2
for the vector-functions U (1) , r1 U (2) , r2 U (2) as
Z
E (1) (U (1) , U (1) )dx = h{N (1) U (1) }+ , {U (1) }+ i∂D1 ,
D1
Z
(3.2)
E (2) (ri U (2) , ri U (2) )dx = −h{N (2) (ri U (2) )}+ , {ri U (2) }+ i∂D(i) , i = 1, 2.
(i)
2
D2
Taking into account the boundary and boundary-contact conditions, the
Green formulas (3.2) can be rewritten as
Z
E (1) (U (1) , U (1) )dx = h{N (1) U (1) }+ , {U (1) }+ iS (1) ∪S (2) ,
0
0
D1
Z
(3.3)
(2)
(2)
(2)
(2)
(2) +
(2) +
E (ri U , ri U )dx = −h{N (ri U )} , {ri U } iS (i) ∪S0 , i = 1, 2.
(i)
D2
0
Since equalities (3.1) are fulfilled for the function U (2) , we have
h{N (2) (r1 U (2) )}+ , {r1 U (2) }+ iS0 = −h{N (2) (r2 U (2) )}+ , {r2 U (2) }+ iS0 .
(3.4)
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
435
Summing now the Green formulas (3.3) and taking into account (3.4), we
obtain
Z
Z
(1)
(1)
(1)
E (U , U )dx +
E (2) (r1 U (2) , r1 U (2) )dx
(1)
D1
D2
Z
+
E (2) (r2 U (2) , r2 U (2) )dx = 0.
(3.5)
(2)
D2
From equality (3.5) and the positive-definiteness of forms (1.3) we have
(
(q)
(q)
∂j ui − εijk ωk = 0,
(q)
∂j ω i
= 0, q = 1, 2.
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 S1 , we have
U (q) (x) = 0, x ∈ Dq ,
¤
q = 1, 2.
1/p0
Any extension Φ(1) ∈ Bp,p (∂D1 ) of the function ϕ1 onto the whole boundary
∂D1 has the form
(1)
(1)
(1)
Φ(1) = Φ0 + ϕ0 + ψ0 ,
0
0
(2)
(1)
(1)
(1)
(1)
e 1/p
e 1/p
).
), ψ ∈ B
where Φ is some fixed extension of ϕ1 and ϕ ∈ B
p,p (S
p,p (S
0
−1/p
0
0
(1)
0
0
Any extension Φ(2) ∈ Bp,p (∂D2 ) of the function ϕ2 onto the whole bound(1)
ary ∂D2 has the form
(2)
(2)
(2)
Φ(2) = Φ0 + ϕ0 + ψ0 ,
(2)
(2)
(1)
(2)
e −1/p
e −1/p
where Φ0 is some fixed extension of ϕ2 and ϕ0 ∈ B
p,p (S0 ), ψ0 ∈ Bp,p (S0 ).
−1/p
(2)
Any extension Φ(3) ∈ Bp,p (∂D2 ) of the function ϕ3 onto the whole bound(2)
ary ∂D2 has the form
(3)
(3)
(3)
Φ(3) = Φ0 + ϕ0 + ψ0 ,
e p,p (S ), ψ ∈ B
e p,p (S0 ).
where Φ0 is some fixed extension of ϕ3 and ϕ0 ∈ B
0
0
A solution of the boundary-contact problem M1 will be sought for in the
form of the simple-layer potentials
(
(1)
V(2) g2 in D2 ,
U (1) = V(1) g1 in D1 ,
U (2) =
(2)
V(3) g3 in D2 .
(3)
(3)
−1/p
(2)
(3)
−1/p
436
O. CHKADUA
Taking into account the boundary and boundary-contact conditions of problem M1 and equality (3.1) we obtain a system of equations with respect to
(1)
(1)
(2)
(2)
(3)
(3)
(g1 , g2 , g3 , ϕ0 , ψ0 , ϕ0 , ψ0 , ϕ0 , ψ0 ):
 (1)
(1)
(1)
(1)

V−1 g1 − ϕ0 − ψ0 = Φ0
on ∂D1 ,




∗

(2)
(2)
(2)
(2)
(1)


on ∂D2 ,
( 21 I+ V 0 )g2 − ϕ0 − ψ0 = Φ0




∗ (3)

(3)
(3)
(3)
(2)
1

on ∂D2 ,
I+
(

V
0 )g3 − ϕ0 − ψ0 = Φ0

2



(2)
(1)
(1)
(1)

−πS (1) V−1 g2 + ϕ0 = f1 − πS (1) Φ0
on S0 ,


0
0

∗ (1)
(2)
(2)
(1)
(3.6)
πS (1) (− 12 I+ V 0 )g1 − ϕ0 = h1 + πS (1) Φ0 on S0 ,

0
0



(3)
(1)
(1)
(2)

−πS (2) V−1 g3 + ψ0 = f2 − πS (2) Φ0
on S0 ,


0
0


∗ (1)

(3)
(3)
(2)

1

πS (2) (− 2 I+ V 0 )g1 − ϕ0 = h2 + πS (2) Φ0 on S0 ,


0
0



(2)
(3)

π
V
g
−
π
V
g
=
0
on S0 ,

S0 −1 2
S0 −1 3



 (2)
(3)
(2)
(3)
ψ0 + ψ0 = −πS0 Φ0 − πS0 Φ0
on S0 .
It is almost obvious that system (3.6) has a solution if and only if the compatibility conditions on ∂S0
(2)
(3)
(2)
(2)
(3)
e s−1
Φ0 ∈ Bs−1
p,r (∂D2 ) : πS0 Φ0 +πS0 Φ0 ∈ Bp,r (S0 ) (3.7)
(1)
∃Φ0 ∈ Bs−1
p,r (∂D2 ),
(1)
(2)
(2)
(3)
s−1
s−1
hold for ϕ2 ∈ Bs−1
p,r (S2 ), ϕ3 ∈ Bp,r (S2 ), πS0 Φ0 + πS0 Φ0 ∈ Bp,r (S0 ), 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
(1)
(2)
(2)
¡ (1) s
¢
s
A : H sp →H sp
B p,r → B p,r ,
where
(1)
(1)
(2)
s
s−1
s−1
s−1
e s (S (1) ) ⊕ H
e s (S (2) )
H p = Hp (∂D1 ) ⊕ Hp (∂D2 ) ⊕ Hp (∂D2 ) ⊕ H
0
0
p
p
(1)
(2)
s−1
s−1
s−1
s−1
e
e
e
e
⊕H (S ) ⊕ H (S0 ) ⊕ H (S ) ⊕ H (S0 ),
p
0
(2)
p
p
0
p
(1)
(2)
(1)
(1)
s−1
s
s−1
s−1
s
s
H p = Hp (∂D1 ) ⊕ Hp (∂D2 ) ⊕ Hp (∂D2 ) ⊕ Hp (S0 ) ⊕ Hp (S0 )
(2)
(2)
s−1
s
⊕Hsp (S0 ) ⊕ Hs−1
p (S0 ) ⊕ Hp (S0 ) ⊕ Hp (S0 ),
(1)
(2)
(1)
s−1
s−1
s−1
s
e s (S (2) )
e s (S (1) ) ⊕ B
B p,r = Bp,r (∂D1 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (∂D2 ) ⊕ B
0
0
p,r
p,r
(2)
(1)
e s−1 (S0 ),
e s−1 (S ) ⊕ B
e s−1 (S0 ) ⊕ B
e s−1 (S ) ⊕ B
⊕B
p,r
0
(2)
p,r
p,r
(1)
0
(2)
p,r
(1)
(1)
s
s
s−1
s−1
s
s−1
B p,r = Bp,r (∂D1 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (S0 ) ⊕ Bp,r (S0 )
(2)
(2)
s
s−1
⊕Bsp,r (S0 ) ⊕ Bs−1
p,r (S0 ) ⊕ Bp,r (S0 ) ⊕ Bp,r (S0 );
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
437
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
(2)
(3)
D = diag{I, −V−1 , −V−1 , I, . . . , I}54×54 .
Consider now the operator
AM = TM + D ◦ A,
where
M = 2, 3, . . . ,
(2)
(3)
TM = diag{0, (−V−1 )M , (−V−1 )M , 0, . . . , 0}54×54 .
Since the operator AM differs from the operator D ◦ A in a compact operator,
it is sufficient to investigate the operator AM acting in the spaces
(1)
(3)
(3)
¡ (1) s
¢
s
AM : H sp →H sp
B p,r → B p,r ,
where
(3)
(1)
(2)
(1)
(1)
s
s
s
s
s
s−1
H p = Hp (∂D1 ) ⊕ Hp (∂D2 ) ⊕ Hp (∂D2 ) ⊕ Hp (S0 ) ⊕ Hp (S0 )
(2)
(2)
s
s−1
⊕Hsp (S0 ) ⊕ Hs−1
p (S0 ) ⊕ Hp (S0 ) ⊕ Hp (S0 ),
(3)
(1)
(2)
(1)
(1)
s
s
s
s
s
s−1
B p,r = Bp,r (∂D1 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (S0 ) ⊕ Bp,r (S0 )
(2)
(2)
s
s−1
⊕Bsp,r (S0 ) ⊕ Bs−1
p,r (S0 ) ⊕ Bp,r (S0 ) ⊕ Bp,r (S0 ).
Now consider the system of equations that corresponds to
given by
 (1)
(1)
(1)
(1)

V−1 ge1 − ϕ
e0 − ψe0 = Ψ0




∗ (2)

(2) (2)
(2) (2)
(2)
(2) M
(2) 1


I+
e0 +V−1 ψe0 = Ψ0
)
(
g2 +V−1 ϕ
[(−V
−V
V
0 )]e
−1
−1

2



∗ (3)

(3) M
(3) 1
(3) (3)
(3) (3)
(3)

I+
g3 +V−1 ϕ
e0 +V−1 ψe0 = Ψ0
(−V
)
−V
(

V
−1
−1 2
0 )e




(2)
(1)

−πS (1) V−1 ge2 + ϕ
e0 = G1


0

∗ (1)
(2)
πS (1) (− 12 I+ V 0 )e
g1 − ϕ
e0 = G2

0



(3)
(1)

−πS (2) V−1 ge3 + ψe0 = F1


0


∗ (1)

(3)

1

π
g1 − ϕ
e0 = F2
(2) (− I+ V 0 )e

2
S

0



(2)
(3)

π
e2 − πS0 V−1 ge3 = E1

S0 V−1 g



 e(2) e(3)
ψ0 + ψ0 = E2
where
(1)
Ψ0 ∈ Hsp (∂D1 )
¡
¢
Bsp,r (∂D1 ) ,
(3)
Ψ0
(2)
(1)
Ψ0 ∈ Hsp (∂D2 )
(2) ¢
(2) ¡
∈ Hsp (∂D2 ) Bsp,r (∂D2 ) ,
¡
the operator AM
on ∂D1 ,
(1)
on ∂D2 ,
(2)
on ∂D2 ,
(1)
on S0 ,
(1)
on S0 ,
(2)
on S0 ,
(2)
on S0 ,
on S0 ,
on S0 ,
(1) ¢
Bsp,r (∂D2 ) ,
(3.8)
438
O. CHKADUA
(1) ¢
(1) ¡ s−1
(1) ¢
Bsp,r (S0 ) , G2 ∈ Hs−1
Bp,r (S0 ) ,
p (S0 )
(2) ¡
(2) ¢
(2) ¡ s−1
(2) ¢
F1 ∈ Hsp (S0 ) Bsp,r (S0 ) , F2 ∈ Hs−1
Bp,r (S0 ) ,
p (S0 )
¡
¢
¡ s−1
¢
E1 ∈ Hsp (S0 ) Bsp,r (S0 ) , E2 ∈ Hs−1
Bp,r (S0 ) .
p (S0 )
(1)
G1 ∈ Hsp (S0 )
¡
(1)
∗ (i)
(i)
The ΨDO −V−1 is positive and the operators −V−1 ( 12 I+ V 0 ), i = 2, 3, are
nonnegative, i.e.,
(1)
−1/2
h−V−1 ϕ, ϕi∂D1 > 0 for all ϕ ∈ H2 (∂D1 ), ϕ 6= 0,
D
³
E
∗ (2) ´
(2) 1
−1/2
(1)
− V−1 I+ V 0 ψ, ψ
≥ 0 for all ψ ∈ H2 (∂D2 )
(1)
2
∂D2
and
D
(3)
− V−1
³1
∗ (3) ´
I+ V 0
E
−1/2
ψ, ψ
(2)
> 0 for all ψ ∈ H2 (∂D2 ),
(2)
2
∂D2
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
³
∗ (i) ´
(i)
(i)
(i) 1
BM = (−V−1 )M − V−1 I+ V 0 , i = 2, 3,
2
are positive operators, i.e.,
(2)
−1/2
hBM ϕ, ϕi∂D(1) > 0 for all ψ ∈ H2
2
(3)
−1/2
hBM ψ, ψi∂D(2) > 0 for all ψ ∈ H2
2
(1)
(1)
(∂D2 ), ϕ 6= 0,
(2)
(∂D2 ), ψ 6= 0.
(i)
Hence the ΨDOs V−1 and BM , i = 2, 3, are invertible (which is proved as in
[32], [7]). The first, second and third equations of system (3.8) imply
ge1 = (V−1 )−1 ϕ
e0 + (V−1 )−1 ψe0 + (V−1 )−1 Ψ0 ,
(i)
(i) (i)
(i)
(i) (i)
(i)
(i)
gei = −(BM )−1 V−1 ϕ
e0 − (BM )−1 V−1 ψe0 + (BM )−1 Ψ0 , i = 2, 3.
(1)
(1)
(1)
(1)
(1)
(1)
After substituting ge1 , ge2 , ge3 into the remaining equations of system (3.8), we
obtain a system of equations whose corresponding operator has the form


πS (1) A(x, D)
0
0
0

0
πS (2) B(x, D)
0
+ T−∞ ,
P=
0
0
0
πS0 C(x, D) 36×36
where

A(x, D) = 

B(x, D) = 
(2)
(2)
(2)
πS (1) V−1 (BM )−1 V−1
0
πS (1) (− 12 I+
0
−I
(3)
(3)

I
(3)
∗ (1)
(1)
V 0 )(V−1 )−1

πS (2) V−1 (BM )−1 V−1
I
−I
πS (2) (− 12 I+ V 0 )(V−1 )−1
0
∗ (1)
0
,
(1)
,
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
Ã
C(x, D) =
(2)
(2)
(2)
(3)
(3)
439
!
(3)
−πS0 V−1 (BM )−1 V−1 πS0 V−1 (BM )−1 V−1
I
I
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(x0 , D0 ) and B(x0 , D0 ) be ΨDOs with the symbols σA (x0 , ξ 0 ) and
σB (x0 , ξ 0 ) (ξ 0 = (ξ1 , ξ2 )) “frozen” at the points and written in terms of some
(1)
(2)
local coordinate system of the manifolds S0 and S0 , respectively.
Denote
µ
¶
µ
¶
L− 0
L+ 0
0
0
0
0
R(x , D ) =
◦ A(x , D ) ◦
0 I
0 I
and
µ
0
0
Q(x , D ) =
L− 0
0 I
¶
µ
0
0
◦ B(x , D ) ◦
L+ 0
0 I
¶
,
where L+ = diagΛ+ , L− = diagπ+ Λ− ` are 6 × 6 matrix operators, Λ± is a ΨDO
operator with the symbol Λ± (ξ 0 ) = ξ2 ± i ± i|ξ1 |, π+ denotes the operator of
restriction onto R2+ , and ` is an extension operator.
The operators
µ
¶
L± 0
0 I
are invertible in the respective spaces (see [39]).
The principal homogeneous symbols of the ΨDOs R(x0 , D0 ) and Q(x0 , D0 ) are
written as
Ã
!
00
0 0
00
00
(ξ
−i|ξ
|)σ
(x
,
ξ
)(ξ
+i|ξ
|)
(ξ
−i|ξ
|)I
n−1
N
n−1
n−1
2
(1)
σR (x0 , ξ 0 ) =
, x0 ∈ S 0 ,
00
0 0
−(ξn−1 + i|ξ |)I
σN1 (x , ξ )
Ã
!
00
0 0
00
00
(ξ
−i|ξ
|)σ
(x
,
ξ
)(ξ
+i|ξ
|)
(ξ
−i|ξ
|)I
n−1
N
n−1
n−1
3
(2)
σQ (x0 , ξ 0 ) =
, x0 ∈ S 0 ,
00
0 0
−(ξn−1 + i|ξ |)I
σN1 (x , ξ )
where σN1 (x0 , ξ 0 ), σN2 (x0 , ξ 0 ) and σN3 (x0 , ξ 0 ) are the principal homogeneous symbols of the ΨDOs
³ 1
∗ (1) ´
(1)
(2)
(2)
(2)
(3)
(3)
(3)
N1 = − I+ V 0 (V−1 )−1 , N2 = V−1 (BM )−1 V−1 , N3 = V−1 (BM )−1 V−1 ,
2
respectively, written in terms of a given local coordinate system, and I is the
identity matrix.
(k)
Let λR , k = 1, . . . , 12, be the eigenvalues of the matrix
¡
¢−1
(1)
σR (x1 , 0, +1) σR (x1 , 0, −1), x1 ∈ ∂S0 ,
where
µ
σR (x1 , 0, −1) =
σN2 (x1 , 0, −1)
−I
I
σN1 (x1 , 0, −1)
¶
,
440
O. CHKADUA
µ
σR (x1 , 0, +1) =
σN2 (x1 , 0, +1)
I
−I
σN1 (x1 , 0, +1)
¶
,
(k)
and let λQ , k = 1, . . . , 12, be the eigenvalues of the matrix
¢−1
¡
(2)
σQ (x1 , 0, +1) σQ (x1 , 0, −1), x1 ∈ ∂S0 ,
where
µ
σQ (x1 , 0, −1) =
µ
σQ (x1 , 0, +1) =
σN3 (x1 , 0, −1)
−I
I
σN1 (x1 , 0, −1)
σN3 (x1 , 0, +1)
I
−I
σN1 (x1 , 0, +1)
Introduce the notation
¯
¯1
¯
¯
(j)
δR = sup ¯ arg λR (x1 )¯,
1≤j≤12 2π
δQ =
(1)
¶
,
¶
¯1
¯
¯
¯
(j)
sup ¯ arg λQ (x1 )¯,
1≤j≤12 2π
(2)
x1 ∈∂S0
x1 ∈∂S0
δ = max(δR , δQ ).
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
e s (R2 ) ⊕ H
e s (R2 ) → Hs−1 (R2 ) ⊕ Hs−1 (R2 )
R(x0 , D0 ) : H
p
+
p
+
p
+
p
+
¢
¡ s
e (R2 ) ⊕ B
e s (R2 ) → Bs−1 (R2 ) ⊕ Bs−1 (R2 ) ,
B
p,r
+
p,r
+
p,r
+
p,r
+
e s (R2 ) ⊕ H
e s (R2 ) → Hs−1 (R2 ) ⊕ Hs−1 (R2 )
Q(x0 , D0 ) : H
p
+
p
+
p
+
p
+
¡ s
¢
2
s
2
s−1
2
s−1
e (R ) ⊕ B
e (R ) → B (R ) ⊕ B (R2 )
B
p,r
+
p,r
+
p,r
+
p,r
+
are Fredholm.
Note that the ΨDOs R(x0 , D0 ) and Q(x0 , D0 ) are Fredholm in the anisotropic
Bessel potential spaces with weight
e (µ,s),k (R2 ) ⊕ H
e (µ,s),k (R2 ) → H(µ,s−1),k (R2 ) ⊕ H(µ,s−1),k (R2 )
H
p
+
p
+
p
+
p
+
for all µ ∈ R and k = 0, 1, . . . (see [10]).
Since the operators πS (1) N1 , πS (1) N2 , πS (2) N1 and πS (2) N2 are positive-definite,
0
0
0
0
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
e s−1 (S (1) ) ⊕ H
e s (S (1) ) → Hs (S (1) ) ⊕ Hs−1 (S (1) )
πS (1) A(x, D) : H
0
0
0
0
p
p
p
p
0
¡ s−1 (1)
¢
(1)
(1)
e (S ) ⊕ B
e s (S ) → Bs (S ) ⊕ Bs−1 (S (1) )
B
0
0
0
0
p,r
p,r
p,r
p,r
e s−1 (S (2) ) ⊕ H
e s (S (2) ) → Hs (S (2) ) ⊕ Hs−1 (S (2) )
πS (2) A(x, D) : H
0
0
0
0
p
p
p
p
0
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
¡
441
¢
e s−1 (S (2) ) ⊕ B
e s (S (2) ) → Bs (S (2) ) ⊕ Bs−1 (S (2) )
B
0
0
0
0
p,r
p,r
p,r
p,r
are invertible.
Let us consider the operator C(x, D). The corresponding system
Ã
! µ
¶
(2)
e1
ϕ0
E
πS0 C(x, D)
=
(3)
E2
ϕ0
is reduced to pseudodifferential equation on the open manifold S0
(3)
e1 ,
πS0 Nψe0 = E
(2)
(3)
ψe0 = −ψe0 + E2 ,
where N = N2 + N3 .
The ΨDO πS0 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
e s−1 (S0 ) → Hs (S0 )
πS0 N : H
p
p
¡ s−1
¢
e (S0 ) → Bs (S0 )
B
p,r
p,r
and
e s−1 (S0 ) ⊕ H
e s−1 (S0 ) → Hs (S0 ) ⊕ Hs (S0 )
πS0 C(x, D) : H
p
p
p
p
¢
¡ s−1
s−1
s
e (S0 ) ⊕ B
e (S0 ) → B (S0 ) ⊕ Bs (S0 )
B
p,r
p,r
p,r
p,r
are invertible.
Note that the ΨDO πS0 N is invertible in anisotropic Bessel potential spaces
(µ,s),k
e (µ,s−1),k
(S0 ) (see [10]).
→ Hp
with weight H
p
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
(1)
(1)
e s−1 (S (1) ) ⊕ H
e s (S (1) )
H
Hsp (S0 ) ⊕ Hs−1
0
0
p
p
p (S0 )
⊕
⊕
e s−1 (S (2) ) ⊕ H
e s (S (2) ) → Hs (S (2) ) ⊕ Hs−1 (S (2) )
P : H
0
0
p
p
0
0
p
p
⊕
⊕
e s−1 (S0 )
e s−1 (S0 ) ⊕ H
Hsp (S0 ) ⊕ Hsp (S0 )
H
p
p

(1)
e s (S (1) )
e s−1 (S (1) ) ⊕ B
e s−1 (S (1) )
B
Bsp,r (S0 ) ⊕ B
0
p,r
 p,r 0 ⊕ p,r 0
⊕

 e s−1 (2)
e s (S (2) ) → Bs (S (2) ) ⊕ B
e s−1 (S (2) )
 Bp,r (S0 ) ⊕ B
0
p,r
0
0
p,r
p,r

⊕

⊕
e s−1 (S0 )
e s−1 (S0 ) ⊕ B
Bsp,r (S0 ) ⊕ Bsp,r (S0 )
B
p,r
p,r
is invertible.







442
O. CHKADUA
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
(1)
(3)
¡ (1) s
¢
s
B p,r → B p,r
(3)
AM : H sp →H sp
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
(1)
(2)
A : H sp →H sp
¡
(1)
B
s
p,r
(2)
→B
s
p,r
¢
is invertible.
If we take s = 1/p0 in the condition for the operator A to be invertible (see
Theorem 3.7), we conclude that p must satisfy the equality
4
4
<p<
.
3 − 2δ
1 + 2δ
Theorem 3.7 and the above reasoning imply the validity of the solution existence and uniqueness for problem M1 .
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 M1 has
1
a unique solution in the classes Wp1 (Dq ), q = 1, 2, (Wp,loc
(D2 ) with condition
(1.6) at infinity), which is given by the formulae

(1)

in D2 ,
 V(2) g2
(2)
U (1) = V(1) g1 in D1 ,
U (2) =
V(3) g3
in D2 ,

 π V(2) g = π V(3) g on S ,
S0 −1 2
S0 −1 3
0
where gq , 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
1
(D2 ) with condition (1.6)
of p be fulfilled, U (q) ∈ Wp1 (Dq ), q = 1, 2, (U (2) ∈ Wp,loc
at infinity) be a solution of the boundary-contact problem M1 . Then:
(i)
(2)
(1)
s
s−1
if ϕ1 ∈ Bsr,r (S1 ), ϕ2 ∈ Bs−1
r,r (S2 ), ϕ3 ∈ Br,r (S2 ), fi ∈ Br,r (S0 ), hi ∈
(i)
s+1/r
s+1/r
(q)
∈ Hr
(Dq ), q = 1, 2, (U (2) ∈ Hr,loc (D2 ));
Bs−1
r,r (S0 ), i = 1, 2, we have U
(1)
(2)
(i)
s−1
s
if ϕ1 ∈ Bsr,t (S1 ), ϕ2 ∈ Bs−1
r,t (S2 ), ϕ3 ∈ Br,t (S2 ), fi ∈ Br,t (S0 ), hi ∈
s+1/r
(i)
s+1/r
(q)
∈ Br,t (Dq ), q = 1, 2, (U (2) ∈ Br,t,loc (D2 )).
Bs−1
r,t (S0 ), i = 1, 2, we have U
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
443
4. Asymptotics of Solutions
Now we will write the asymptotics of solutions of the boundary-contact problem M1 . The boundary and contact data are assumed to be sufficiently smooth,
i.e.,
(1)
(2)
),∞
),∞
ϕ1 ∈ Hr(∞,s+2M +1),∞ (S1 ), ϕ2 ∈ H(∞,s+2M
(S2 ), ϕ3 ∈ H(∞,s+2M
(S2 ),
r
r
(i)
(i)
),∞
+1),∞
(S0 ), hi ∈ H(∞,s+2M
(S0 ), i = 1, 2;
fi ∈ H(∞,s+2M
r
r
here the numbers r and s satisfy the conditions of Theorem 3.9.
Let m1 , . . . , m2` be algebraic multiplicities of the eigenvalues λ1 , . . . , λ2` ,
P2`
(j)
j=1 mj = 12, where λj = λR , j = 1, . . . , 2`.
We introduce the notation
¡
¢−1
bR (x1 ) = σR (x1 , 0, +1) σR (x1 , 0, −1).
Let
(1)
b0R (x1 ) = K−1 (x1 )bR (x1 )K(x1 ), x1 ∈ ∂S0 ,
be a quasi-diagonal form, where K is some nondegenerate matrix function
detK(x1 ) 6= 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(x0 , D0 )χ = F,
),∞
),∞
F ∈ H(∞,s+M
(R2+ ) × H(∞,s+M
(R2+ )
r,comp
r,comp
(1)
written in terms of some local coordinate system of the manifold S0 .
Thus we obtain an asymptotic expansion of the solution χ = (χ1 , χ2 )>
´
³
1
1/2+∆(x1 ) 0
log x2,+ K−1 (x1 )c0 (x1 )
χ(x1 , x2,+ ) = K(x1 )x2,+
Bapr −
2πi
M
X
1/2+∆(x1 )+k
+
K(x1 )x2,+
Bk (x1 , log x2,+ ) + χM +1 (x1 , x2,+ ),
(4.1)
k=1
+1),∞
+1),∞
e (∞,s+M
e (∞,s+M
for all sufficiently small x2,+ > 0, χM +1 ∈ H
(R2+ )× H
(R2+ );
r,comp
r,comp
B0apr (t) is defined in [10]; the 12 × 12 matrix function Bk (x1 , t) is a polynomial
of order νk = k(2m0 − 1) + m0 − 1, m0 = max{m1 , . . . , m2` } with respect to the
variable t with 12-dimensional vector coefficients which depend on the variable
x1 and
∆(x1 ) = (∆1 (x1 ), ∆2 (x1 ));
here
(j)
(j)
(j)
(j)
∆j (x1 ) = (δ1 (x1 ), . . . , δ1 (x1 ), . . . , δ` (x1 ), . . . , δ` (x1 )) , j = 1, 2,
|
{z
}
|
{z
}
m1 -times
m` -times
i
1
(1)
arg λk (x1 ) −
|λk (x1 )|,
δk (x1 ) =
2π
2π
1
i
(2)
δk (x1 ) = − arg λk (x1 ) −
|λk (x1 )|, k = 1, . . . , `.
2π
2π
444
O. CHKADUA
Without loss of generality suppose that the matrix B0apr (t) has the form
B0apr (t) = diag{Ba0pr (t), Ba0pr (t)};
here Ba0pr (t) is the upper triangular block-diagonal 6×6-matrix-function defined
in [9].
Hence for the functions χ1 and χ2 we can write an asymptotic expansion.
Indeed, let
µ
¶
K11 (x1 ) K12 (x1 )
K(x1 ) =
K21 (x1 ) K22 (x1 ) 12×12
and
(1)
(2)
K−1 (x1 )c0 (x1 ) = (c0 (x1 ), c0 (x1 ))> ,
(4.2)
(i)
where Kij (x1 ), i, j = 1, 2, are 6 × 6-matrices, c0 , i = 1, 2, are six-dimensional
vector functions. Then
2
³
´
X
1
1/2+∆j (x1 ) 0
(i)
χi (x1 , x2,+ ) =
log x2,+ c0 (x1 )
Kij (x1 )x2,+
Bapr −
2πi
j=1
+
2 X
M
X
1/2+∆j (x1 )+k
Kij (x1 )x2,+
(i)
(i)
Bkj (x1 , log x2,+ ) + χM +1 (x1 , x2,+ ), i = 1, 2, (4.3)
j=1 k=1
(i)
+1)
(i)
e (∞,s+M
(R2+ ) and Bkj (x1 , t) is a polynomial of order νk =
where χM +1 ∈ H
r,comp
k(2m0 − 1) + m0 − 1 with respect to the variable t with six-dimensional vector
coefficients which depend on the variable x1 .
We can also obtain an analogous asymptotic expansion of the solution χ
e=
(e
χ1 , χ
e2 ) of the strongly elliptic equation
),∞
),∞
Q(x0 , D0 )e
χ = Fe, Fe ∈ H(∞,s+M
(R2+ ) × H(∞,s+M
(R2+ )
r,comp
r,comp
(2)
in terms of some local coordinate system on the manifold S0 . Indeed, we have
χ
ei (x1 , x2,+ ) =
2
X
1/2+∆j (x1 )
Kij (x1 )x2,+
j=1
+
M
2 X
X
e j (x1 )+k
1/2+∆
Kij (x1 )x2,+
³
´
1
(i)
Ba0pr −
log x2,+ b0 (x1 )
2πi
(i)
(i)
Bkj (x1 , log x2,+ ) + χ
eM +1 (x1 , x2,+ ), i = 1, 2, (4.4)
j=1 k=1
(i)
+1),∞
e (∞,s+M
e j , j = 1, 2, are defined as ∆j , j = 1, 2, by
where χ
eM +1 ∈ H
(R2+ ), ∆
r,comp
(k)
means of the eigenvalues λQ , k = 1, . . . , 12, of the matrix bQ .
Let us consider the pseudodifferential equation
(3)
e1 and ψe0(2) = −ψe0(3) + E2 ,
πS0 Nψe0 = E
where
(2)
(2)
(2)
(3)
(3)
(3)
N = V−1 (BM )−1 V−1 + V−1 (BM )−1 V−1 .
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
445
The following equalities hold for the principal homogeneous symbols of the
∗ (i)
(i)
operators V−1 and V 0 , i = 2, 3:
σV(2) (x0 , ξ 0 ) = σV(3) (x0 , ξ 0 ) for x0 ∈ S 0 ,
−1
−1
(4.5)
σ ∗(2) (x0 , ξ 0 ) = −σ ∗(3) (x0 , ξ 0 ) for x0 ∈ S 0 .
V0
V0
In view of equality (4.5) we can write the symbol σN (x0 , ξ 0 ) of the ΨDO
operator N as follows:
h³ 1
´¡
¢−1 i−1
σN (x0 , ξ 0 ) =
I + σ ∗(2) (x0 , ξ 0 ) σ−V(2) (x0 , ξ 0 )
V0
−1
2
h³ 1
´¡
¢ i−1
0 0
0 0 −1
∗
+
I − σ (2) (x , ξ ) σ−V(2) (x , ξ )
.
V0
−1
2
Since the symbol σ ∗(2) (x0 , ξ 0 ) is an odd matrix function with respect to ξ 0 , while
V0
the symbol σ−V(2) (x0 , ξ 0 ) is an even matrix function, one can easily ascertain that
−1
the symbol σN (x0 , ξ 0 ) is even with respect to the variable ξ 0 , i.e.,
σN (x0 , −ξ 0 ) = σN (x0 , ξ 0 ),
x0 ∈ S 0 ,
and all eigenvalues of the matrix
¡
¢−1
σN (x1 , 0, +1) σN (x1 , 0, −1) = I,
x1 ∈ ∂S0 ,
(j)
are trivial λN = 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 follow(i)
ing result on asymptotic expansion of the functions ψ0 , i = 2, 3:
(i)
ψ0 (x1 , x2,+ )
i+1
= (−1)
−1/2
c0 (x1 )x2,+ +
M
X
−1/2+k (i)
(i)
dk (x1 )+ψM +1 (x1 , x2,+ ),
x2,+
(4.6)
k=1
(i)
+1),∞
e (∞,s+M
where
∈ C0∞ (R), and the remainder ψM +1 ∈ H
(R2+ ), i = 2, 3,
r,comp
M ∈ N. As we can see from (4.6), due to the properties of the symbol σN (x0 , ξ 0 )
(see [12]) there are no logarithms in the entire asymptotic expansion.
(1)
(1) (2)
(2)
(3)
(3)
Let g = (g1 , g2 , g3 , ϕ0 , ψ0 ϕ0 , ψ0 , ϕ0 , ψ0 ) be a solution of system (3.6),
i.e.,
Ag = Φ,
where
¡ (1) (2) (3)
(1)
(2)
(1)
Φ = Φ0 , Φ0 , Φ0 , f1 − πS (1) Φ0 , h1 + πS (1) Φ0 , f2 − πS (2) Φ0 ,
0
0
0
(3)
(2)
(3) ¢
h2 + πS (2) Φ0 , 0, −(πS0 Φ0 + πS0 Φ0 ) .
(i)
c0 , d k
0
Then
D ◦ Ag = Ψ;
(4.7)
here
¡ (1)
(2) (2)
(3) (3)
(1)
(2)
(1)
Ψ = Φ0 , −V−1 Φ0 , −V−1 Φ0 , f1 − πS (1) Φ0 , h1 + πS (1) Φ0 , f2 − πS (2) Φ0 ,
0
0
0
446
O. CHKADUA
(3)
(2)
(3) ¢
h2 + πS (2) Φ0 , 0, −(πS0 Φ0 + πS0 Φ0 ) .
0
Now adding the expression
©
ª
(2)
(3)
T2M +1 g = diag 0, −(V−1 )2M +1 , −(V−1 )2M +1 , 0, . . . , 0 g
to both parts of system (4.7), we obtain the equality
e
A2M +1 g = Ψ,
(4.8)
where
¡
(2) (2)
(2) 2M +1
(2) (3)
(3)
e = Φ(1)
Ψ
g2 , −V−1 Φ0 − (V−1 )2M +1 g3 , f1 − π
0 , −V−1 Φ0 − (V−1 )
(1)
S0
(2)
(1)
(3)
(2)
(3) ¢
h1 + πS (1) Φ0 , f2 − πS (2) Φ0 , h2 + πS (2) Φ0 , 0, −(πS0 Φ0 + πS0 Φ0 ) .
0
0
0
(2)
(1)
Thus (−L−1
+ ϕ0 , ϕ0 ) satisfies, in some
(1)
fold S0 , the pseudodifferential equation
µ
0
0
R(x , D )
(3)
(1)
Φ0 ,
local coordinate system of the mani-
χ1
χ2
¶
= F,
(1)
and (−L−1
+ ψ0 , ϕ0 ) satisfies, in some local coordinate system of the manifold
(2)
S0 , the pseudodifferential equation
µ
¶
χ
e1
0
0
Q(x , D )
= Fe,
χ
e2
where
F = (L− F1 , F2 ),
(1)
(2)
(2)
(2)
(2)
F1 = f1 − πS (1) Φ0 − πS (1) V−1 (B2M +1 )−1 V−1 Φ0
0
0
(2)
(2)
(2) (2)
−πS (1) V−1 (B2M +1 )−1 V−1 ψ0
(2)
(2)
(2)
− πS (1) V−1 (B2M +1 )−1 (V−1 )2M +1 g2 ,
0
0
³ 1
∗ (1) ´
(2)
(1)
(1)
F2 = h1 + πS (1) Φ0 − πS (1) − I+ V 0 (V−1 )−1 Φ0
0
0
2
³ 1
∗ (1) ´
(1)
(1)
−πS (1) − I+ V 0 (V−1 )−1 ψ0
0
2
and
Fe = (L− Fe1 , Fe2 ),
(1)
(3)
(3)
(3) (3)
Fe1 = f2 − πS (2) Φ0 + πS (2) V−1 (B2M +1 )−1 V−1 Φ0
0
0
(3)
(3)
(3)
(3)
−πS (2) V−1 (B2M +1 )−1 (V−1 )2M +1 ψ0
0
(3)
(3)
(3)
− πS (2) V−1 (B2M +1 )−1 (V−1 )2M +1 g3 ,
0
³ 1
∗ (1) ´
(1)
(1)
(3)
Fe2 = h2 + πS (2) Φ0 − πS (2) − I+ V 0 (V−1 )−1 Φ0
0
0
2
³ 1
∗ (1) ´
(1)
(1)
−πS (2) − I+ V 0 (V−1 )−1 ϕ0 ;
0
2
here
(∞,s+2M ),∞
Fi ∈ Hr,comp
(R2+ ),
),∞
Fei ∈ H(∞,s+2M
(R2+ ), i = 1, 2.
r,comp
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
(2)
447
(3)
Further, we have that (ψ0 , ψ0 ) is a solution of the system
Ã
! µ
¶
(2)
0
ψ0
πS0 C(x, D)
=
.
(2)
(3)
(3)
−(πS0 Φ0 + πS0 Φ0 )
ψ0
This system can be reduced to a pseudodifferential equation with the positivedefinite operator
(3)
(2)
πS0 Nψ0 = E1 ,
where
(2)
(3)
(2)
(3)
ψ0 = −ψ0 − (πS0 Φ0 + πS0 Φ0 ),
(2)
(2)
(3)
(3)
(3)
N = V−1 (B2M +1 )−1 V−1 + V−1 (B2M +1 )−1 V−1
and
(2)
(2)
(2)
(2)
(2)
(2)
(2)
E1 = −πS0 V−1 (B2M +1 )−1 V−1 Φ0 − πS0 V−1 (B2M +1 )−1 (V−1 )2M +1 g2
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(3)
+ πS0 V−1 (B2M +1 )−1 V−1 ϕ0 − πS0 V−1 (B2M +1 )−1 V−1 (πS0 Φ0 + πS0 Φ0 );
here
),∞
E1 ∈ H(∞,s+2M
(R2+ ).
r,comp
Hence we can obtain asymptotic expansions (4.3), (4.4) and (4.6) for the func(2)
(2)
(1)
(3)
(2)
−1 (3)
tions −L−1
+ ϕ0 , ϕ0 , −L+ ψ0 , ϕ0 and ψ0 , ψ0 , respectively.
We now define g1 , g2 and g3 by the first three equations of system (4.8)
(1)
(1)
(1)
(1)
(1)
(1)
g1 = (V−1 )−1 ϕ0 + (V−1 )−1 ψ0 + (V−1 )−1 Φ0 ,
(i)
(i) (i)
gi = −(B2M +1 )−1 V−1 ϕ0 −
(i)
(i)
+ (B2M +1 )−1 Φ0 + Gi ,
(4.9)
(i)
(i) (i)
(B2M +1 )−1 V−1 ψ0
i = 2, 3,
(4.10)
where
(i)
(i)
Gi = (B2M +1 )−1 (−V−1 )2M +1 gi , i = 2, 3,
(∞,s+2M ),∞
G2 ∈ Hr
(1)
(∞,s+2M ),∞
(∂D2 ),
G3 ∈ Hr
(2)
(∂D2 ).
Using expansions (4.9) and (4.10), we obtain the following representation,
i.e., the solutions of the boundary-contact problem M1 are expressed by the
potential-type functions
(1)
(1)
(1)
(1)
U (1) = V(1) (V−1 )−1 ϕ0 + V(1) (V−1 )−1 ψ0 + R1 ,
(4.11)
(2)
(2)
(2)
(2)
(2)
(2)
(3)
(3)
(3)
(3)
(3)
(3)
r1 U (2) = −V(2) (B2M +1 )−1 V−1 ϕ0 − V(2) (B2M +1 )−1 V−1 ψ0 + R2 , (4.12)
r2 U (2) = −V(3) (B2M +1 )−1 V−1 ϕ0 − V(3) (B2M +1 )−1 V−1 ψ0 + R3 , (4.13)
where
(1)
R1 ∈ C M +1 (D1 ), R2 ∈ C M +1 (D2 ),
(1)
(1)
suppϕ0 ⊂ S 0 ,
(2)
suppψ0 ⊂ S 0 ,
(1)
(2)
(3)
(2)
suppψ0 ⊂ S 0 ,
(2)
R3 ∈ C M +1 (D2 ),
(2)
(1)
suppϕ0 ⊂ S 0 ,
(3)
suppϕ0 ⊂ S 0 , suppψ0 ⊂ S 0 .
Thus, taking into account (4.11), (4.12), (4.13), using the asymptotic expan(2)
(3)
(1)
(1)
(2)
−1 (3)
and ψ0 , ψ0 (see (4.3),
sions of the functions −L−1
+ ϕ0 , ϕ0 , −L+ ψ0 , ϕ0
448
O. CHKADUA
(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 M1 in terms of some local
(1)
(2)
coordinate systems of curves ∂S0 , ∂S0 , ∂S0 :
(1)
a) the asymptotic expansion near the contact boundary ∂S0 :
U (q) (x1 , x2 , x3 ) = (u(q) , ω (q) )(x1 , x2 , x3 )
½ nX
`0
2 X
s −1
h
XX
(q)
(q)
m
=
Re
x3 dsjm (x1 , θ)(zs,θ )1/2+∆j (x1 )−m
θ=±1 j=1 s=1
³
m=0
×Ba0pr −
+
M
+2 MX
+2−l
X
´i
1
(q)
(q)
log zs,θ cjm (x1 )
2πi
¾
(q)
(q) 1/2+∆j (x1 )+k+p (q)
(q)
Bskmpj (x1 , log zs,ϑ )
xl2 xm
3 dslmpj (x1 , ϑ)(zs,ϑ )
k,l=0 p+m=0
k+l+p+m6=0
(q)
+UM +1 (x1 , x2 , x3 ) for M >
(q)
(q)
2
− min{[s − 1], 0}, q = 1, 2,
r
(4.14)
(q)
with the coefficients dsjm (·, ±1), cjm , dslmpj (·, ±1) ∈ C0∞ (R) and the remainder
(q)
UM +1 ∈ C0M +1 (R3± ), q = 1, 2, where the signs “+” and “-” refer to the cases
q = 1 and q = 2, respectively. Here
(q)
(q)
(q)
(q)
zs,+1 = (−1)q [x2 + x3 τs,+1 ], zs,−1 = (−1)q+1 [x2 − x3 τs,−1 ],
−π < Argzs,±1 < π, τs,±1 ∈ C0∞ (R),
◦
(q)
0
are all different roots of the polynomial det M (q) ((Jκ> (x1 , 0))−1 ·
{τs,±1 }`s=1
(0, ±1, τ )) of multiplicity ns , s = 1, . . . , `0 , in the complex lower half-plane (ns
(q)
and `0 depend on q). Bskmpj (x1 , t) is a polynomial of order νkmp = νk + p + m
P̀
mj = 6) with respect
(νk = k(2m0 − 1) + m0 − 1, m0 = max{m1 , . . . , m` },
j=1
to the variable t with vector coefficients which depend on the variable x1 .
We write the following relation between the leading (first) coefficients of the
asymptotic expansions (4.14) and (4.3) (see [9, Theorem 2.3]):
1 (s)
1
Gκ (x1 , 0) V −1,m (x1 , 0, +1)σ −1(1) (x1 , 0, +1)K2j (x1 ),
V−1
2π
1 (s)
1
(1)
dsjm (x1 , −1) = − Gκ (x1 , 0) V −1,m (x1 , 0, −1)σ −1(1) (x1 , 0, −1)
V−1
2π
iπ(−1/2−∆j (x1 ))
× K2j (x1 )e
,
(1)
dsjm (x1 , +1) =
2 (s)
(−1)m+1
Gκ (x1 , 0) V −1,m (x1 , 0, +1)σ −1
∗ (2) (x1 , 0, +1)K1j (x1 ),
1
I+V0
2π
2
2 (s)
(−1)m+1
(2)
dsjm (x1 , −1) =
Gκ (x1 , 0) V −1,m (x1 , 0, −1)σ −1
∗ (2) (x1 , 0, −1)
1
I+V0
2π
2
(2)
dsjm (x1 , +1) =
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
449
× K1j (x1 )eiπ(−1/2−∆j (x1 )) ,
j = 1, 2, s = 1, . . . , `0 , m = 0, . . . , ns − 1;
here Gκ is the square root from the Gram determinant of the diffeomorphism κ
and
q (s)
im+1
dns −1−m
(q)
(x
,
0,
±1)
=
(τ − τs,±1 )ns
V −1,m 1
m!(ns − 1 − m)! dτ ns −1−m
£ ◦ (q)
¤−1 ¯¯
>
−1
× M ((Jκ (x1 , 0)) (0, ±1, τ )) ¯ (q) , q = 1, 2.
τ =τs,±1
(q)
The coefficients cjm (x1 ) in (4.14) are defined as follows:
³1
´
(1)
(2)
cjm (x1 ) = ajm (x1 )Ba−pr
+ ∆j (x1 ) c0 (x1 ),
2
j = 1, 2, m = 0, . . . , ns − 1,
³1
´
(2)
(1)
cjm (x1 ) = ajm (x1 )Ba−pr
+ ∆j (x1 ) c0 (x1 ),
2
where
ml
m1
Ba−pr (t) = diag{B−
(t), . . . , B−
)t)},
´
³
´
³
iπ(t+1
1
mr
∂t Γ(t + 1)e 2
,
B−
(t) = B mr −
2πi
r
B mr (t) = kbm
kp (t)kmr ×mr ,
 ³
´
iπ(t+1)
1 ´p−k (−1)p+k dp−k ³

2
Γ(t
+
1)e
, k ≤ p,
r
bm
2πi
(p − k)! dtp−k
kp (t) =

0,
k > p,
p = 0, . . . , mr − 1, r = 1, . . . , `.
Further,
(j)
(j)
ajm (x1 ) = diag{am1 (λ1 ), . . . , am` (λ` )}, j = 1, 2,
3
1
i
λ(1)
argλr (x1 ) +
log |λr (x1 )| + m,
r (x1 ) = − −
2 2π
2π
1
i
3
λ(2)
argλr (x1 ) +
log |λr (x1 )| + m,
r (x1 ) = − +
2 2π
2π
m = 0, 1, . . . , ns − 1;
mr
(j)
amr (λ(j)
r ) = kakp (λr kmr ×mr ,
where

p
(j)
r
X

(−1)p+k (2πi)l−p bm
(µr )

kl

, m = 0, k ≤ p,
 −i
(j)
p−l+1
mr
(j)
(λ
+
1)
r
l=k
akp (λr ) =
(j)
r

m = 1, 2, . . . , ns − 1, k ≤ p,
(−1)p+k bm

kp (λr ),


0,
k > p;
(j)
(j)
(j)
(1)
here λr = −1 + m + µr , 0 < Reµr < 1, j = 1, 2, r = 1, . . . , `, and c0 (x1 ),
(2)
c0 (x1 ) are defined using the first coefficients of the asymptotic expansion of
(1)
(2)
the functions −L−1
+ ϕ0 and ϕ0 , respectively (see (4.3)).
450
O. CHKADUA
(2)
b) the asymptotics of solutions near the contact boundary ∂S0 :
U (q) (x1 , x2 , x3 ) = (u(q) , ω (q) )(x1 , x2 , x3 )
½ nX
`0
2 X
s −1
h
XX
e
(q)
(q)
m
=
Re
x3 dsjm (x1 , θ)(zz,θ )1/2+∆j (x1 )−m
ϑ=±1 j=1 s=1
×Ba0pr
M
+2 MX
+2−l
X
+
³
m=0
´i
1
(q)
(q)
−
log zs,θ cjm (x1 )
2πi
¾
(q)
(q) 1/2+∆j (x1 )+k+p (q)
(q)
Bskmpj (x1 , log zs,ϑ )
xl2 xm
3 dslmpj (x1 , ϑ)(zs,ϑ )
k,l=0 p+m=0
k+l+p+m6=0
(q)
+UM +1 (x1 , x2 , x3 ) for M >
(q)
(q)
2
− min{[s − 1], 0}, q = 1, 2;
r
(q)
(4.15)
(q)
here zs,+1 = (−1)q [x2 + x3 τs,+1 ], zs,−1 = (−1)q+1 [x2 − x3 τs,−1 ]. The coefficients
(q)
(q)
dsjm (·, ±1) are calculated similarly as in a) and the coefficients bjm are defined
(q)
as in a) by using the first coefficients b0 (q = 1, 2) of the asymptotic expansion
(4.4).
c) the asymptotics of solutions near the cuspidal edge ∂S0 :
(ri U (2) )(x1 , x2 , x3 ) = ri (u(2) , ω (2) )(x1 , x2 , x3 )
`0
XX
=
¾
½ nX
+2−l
M
+1 MX
s −1
X
j 1/2−j (i)
l j −1/2+p+k (i)
dslkjp (x1 )
x2 x3 zs,ϑ
Re
x3 zs,θ dsj (x1 , θ) +
ϑ=±1 s=1
k,l=0 j+p=1
l+k+j+p6=1
j=0
(i)
+UM +1 (x1 , x2 , x3 ) for M >
(i)
2
− min{[s], 0}, i = 1, 2,
r
(i)
(4.16)
(i)
with the coefficients dsj (·, ±1), dslkjp ∈ C0∞ (R) and the remainder UM +1 ∈
C0M +1 (R3± ), i = 1, 2; here
(2)
(2)
zs,+1 = −x2 −x3 τs,+1 , zs,−1 = x2 −x3 τs,−1 , −π < Argzs,±1 < π, τs,±1 ∈ C0∞ (R),
◦
(2)
0
{τs,±1 }`s=1
are all different roots of the polynomial det M (2) ((Jκ> (x1 , 0))−1 ·
(0, ±1, τ )) of multiplicity ns , s = 1, . . . , `0 , in the complex lower half-plane.
(i)
The coefficients dsj (x1 , ±1) have the form (see [9, Theorem 2.3]):
(1)
2 (s)
dsj (x1 , +1) = Gκ (x1 , 0) V −1,j (x1 , 0, +1)σ −1
1
∗(2)
I+V0
2
(1)
(x1 , 0, +1)c(j) (x1 ),
2 (s)
dsj (x1 , −1) = −iGκ (x1 , 0) V −1,j (x1 , 0, −1)σ −1
1
∗(2)
I+V0
2
(2)
(x1 , 0, −1)c(j) (x1 )
2 (s)
dsj (x1 , +1) = (−1)j+1 Gκ (x1 , 0) V −1,j (x1 , 0, +1)σ −1
1
∗(3)
I+V0
2
(2)
(x1 , 0, +1)c(j) (x1 ),
2 (s)
dsj (x1 , −1) = (−1)j+1 iGκ (x1 , 0) V −1,j (x1 , 0, −1)σ −1
1
∗(3)
I+V0
2
(x1 , 0, −1)c(j) (x1 ),
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
451
s = 1, . . . , l0 , j = 0, . . . , ns − 1,
where Gκ is the square root from the Gram determinant of the diffeomorphism κ,
2 (s)
ij+1
dns −1−j
(2)
(τ − τs,±1 )ns
n
−1−j
s
j!(ns − 1 − j)! dτ
¡ ◦ (2)
¢−1 ¯¯
× M ((Jκ> (x1 , 0))−1 · (0, ±1, τ )) ¯ (2) ,
V −1,j (x1 , 0, ±1) =
1´
ij ³
(j)
c0 (x1 )
c (x1 ) = √ Γ j −
2
2 π
τ =τs,±1
and c0 (x1 ) 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
M1 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 M1 in the neighbour(1)
hood of the contact boundary ∂S0 . Analogous properties will be valid in the
(2)
neighbourhood of the contact boundary ∂S0 , too.
For the sake of brevity, by γm and δm (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.,
¯
1 ¯¯ 1
1
¯
γm (x1 ) = − ¯ argλm (x1 )¯, δm (x1 ) = − log |λm (x1 )|,
2
2π
2π
(1)
m = 1, . . . , 6, x1 ∈ ∂S0 ,
(m)
where λm (x1 ) = λR (x1 ) m = 1, . . . , 12, are the eigenvalues of the matrix
bR (x1 ) = (σR (x1 , 0, +1))−1 σR (x1 , 0, −1).
Theorem 5.1. The real parts γm , m = 1, . . . , 6, of the exponents of the first
terms of the asymptotic expansion of solutions of the boundary-contact problem
(1)
M1 near the contact boundary ∂S0 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
(2)
aijlk
(1)
aijlk
(2)
→ 0,
then γm → 1/2 (m = 1, . . . , 6);
(b) if for
(2)
aijlk
→ ∞,
(1)
aijlk
bijlk
(1)
bijlk
(2)
→ 0,
(2)
bijlk
(1)
bijlk
cijlk
(1)
cijlk
→ 0,
(2)
→ ∞,
cijlk
(1)
cijlk
→∞
452
O. CHKADUA
the limiting relations
|ζ2 |
|Imhζ1 , ζ2 i|
|ζ1 |
1. M2+
+ M1+
→ 0;
2.
→ c (c > 0),
|ζ2 |
|ζ1 |
|ζ1 | |ζ2 |
are valid, then γm → 0 and δm → 0 (m = 1, . . . , 6),
where ζ = (ζ1 , ζ2 ) is the eigenvector of the matrix bR , while M1+ and M2+
+
+
=
are maximal eigenvalues of the matrices σN
= σN1 (x1 , 0, +1) and σN
2
1
σN2 (x1 , 0, +1).
(q)
(q)
(q)
Here the limits of the coefficients aijlk , bijlk , cijlk are understood in the uniform
sense with respect to the indices i, j, l, k.
(1)
(1)
(1)
Proof. Multiply the coefficients aijlk , bijlk , cijlk by α (α > 0), and the coefficients
(2)
(2)
(2)
aijlk , bijlk , cijlk by β (β > 0); i.e., we consider the differential equations with
(1)
(1)
(1)
(2)
(2)
(2)
the elastic constants αaijlk , αbijlk , αcijlk and βaijlk , βbijlk , βcijlk .
(a) Taking into consideration the estimate obtained in [6] we have
¯1
¯ 1
´
³ √β
1
¯
¯
¯ argλm (x1 )¯ ≤ arctg √ p + + ,
2π
π
α m1 m2
+
+
−1 +
where αm+
m2 are minimal eigenvalues of the matrices σN
and σN
, re1, β
1
2
spectively.
Consequently, as β/α → 0 we get γm → 12 (m = 1, . . . , 6).
(b) Since
1
1
2Imhζ1 , ζ2 i
arg λm (x1 ) =
arctg +
+
2π
2π
hσN2 ζ1 , ζ1 i + hσN
ζ , ζ2 i
1 2
1
2Imhζ1 , ζ2 i
+
arctg −
,
−
2π
hσN2 ζ1 , ζ1 i + hσN
ζ , ζ2 i
1 2
we obtain the estimate
2|Imhζ1 ,ζ2 i|
¯1
¯ 1
|ζ1 | |ζ2 |
¯
¯
.
¯ arg λm (x1 )¯ ≥ arctg
+ |ζ1 |
2|
−1
2π
π
β M2 |ζ2 | + αM1+ |ζ
|ζ1 |
(5.1)
Further, since
|λm (x1 )|2 =
−
−
(hσN
ζ , ζ1 i + hσN
ζ , ζ2 i)2 + (2Imhζ1 , ζ2 i)2
2 1
1 2
,
+
+
(hσN
ζ , ζ1 i + hσN
ζ , ζ2 i)2 + (2Imhζ1 , ζ2 i)2
2 1
1 2
we get
1
≤ |λm (x1 )|2 ≤ b,
b
where
(5.2)
(β −1 M2+ |ζ1 |2 + αM1+ |ζ2 |2 )2 + (2Imhζ1 , ζ2 i)2
b=
;
+
2
2 2
2
(β −1 m+
2 |ζ1 | + αm1 |ζ2 | ) + (2Imhζ1 , ζ2 i)
+
+
here αM1+ and β −1 M2+ are maximal eigenvalues of the matrices σN
and σN
1
2
respectively, while the vectors ζ1 and ζ2 depend in general on α and β.
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
453
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 γm → 0
and δm → 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 M1 γj + iδj (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]):


1
2
(q)
M (q) (∂x ) M (∂x ) 
M(q) (∂x ) =  3
,
4
(q)
(q)
M (∂x ) M (∂x ) 6×6
° k (q)
°
k
(q)
M (∂x ) = ° M ij (∂x )°3×3 , k = 1, 4, q = 1, 2,
1
(q)
M ij (∂x ) = (µq + αq )δij ∆ + (λq + µq − αq )∂i ∂j ,
3
X
2 (q)
3 (q)
εijk ∂k ,
M ij (∂x ) = M ij (∂x ) = −2αq
k=1
£
¤
(q)
M ij (∂x ) = δij (νq + βq )∆ − 4αq + (εq + νq − βq )∂i ∂j ,
4
where δij and εijk are respectively the Kronecker and the Levi-Civita symbol.
The coefficients λq , µq , αq , νq , βq , εq , q = 1, 2, are the elastic constants satisfying
the conditions
µq > 0,
3λq + 2µq > 0,
αq > 0,
νq > 0, 3εq + 2νq > 0, βq > 0.
The stress operator of couple-stress elasticity is written as


1
2
(q)
(q)
N (∂z , n(z)) N (∂z , n(z)) 
,
N (q) (∂z , n(z)) =  3
4
(q)
(q)
(∂
,
n(z))
(∂
,
n(z))
N
N
z
z
6×6
°
°
k
k (q)
(q)
N (∂z , n(z)) = ° N ij (∂z , n(z))°3×3 , k = 1, 4, q = 1, 2,
1 (q)
ij (∂z , n(z))
N
2 (q)
N ij (∂z , n(z))
= λq ni (z)∂j + (µq − αq )nj (z)∂i + (µq + αq )δij
= −2αq
3
X
3 (q)
ij (∂z , n(z))
εijk nk (z), N
∂
,
∂n(z)
= 0,
k=1
4 (q)
N ij (∂z , n(z))
= εq ni (z)∂j + (νq − βq )ni (z)∂i + (νq + βq )δij
∂
.
∂n(z)
±
The matrices σN
= σNq (x1 , 0, ±1), q = 1, 2, have the following expressions
q
à 1
!
à 1
!
±
σ±
σ
0
0
±
±
N1
N2
σN1 =
, σN2 =
,
2
2 ±
σ±
σ
0
0
N1
N
2
6×6
6×6
454
O. CHKADUA
where
1 ±
N1
σ




µ1 + α 1
0
0
ν1 + β1
0
0
2

0
a1 ∓ib1 
0
c1 ∓id1 
=
, σ±
,
N1 =
0
±ib2 a1
0
±id1 c1
3×3
3×3


1
0
0
 µ2 + α2



a
b
1 ±
2
2


0
∓i
σ N2 = 
,

a22 − b22
a22 − b22 



a2
b2
0
±i 2
a2 − b22
a22 − b22
3×3


1
0
0
 ν2 + β2



c2
d2
2 ±


0
∓i
σ N2 = 
,
2
2
2
2 
c2 − d2
c2 − d2 



d2
c2
0
±i 2
c2 − d22
c22 − d22
3×3
here
2(λq + 2µq )(µq + αq )
aq =
,
λq + αq + 3µq
2(µq + αq )2
bq =
,
λq + αq + 3µq
q = 1, 2,
and
2(εq + 2νq )(νq + βq )
2(νq + βq )2
cq =
, dq =
, q = 1, 2.
εq + βq + 3νq
εq + βq + 3νq
Hence in the considered case the eigenvalues λj (j = 1, . . . , 12) of the matrix
¡
¢−1
bR (x1 ) = σR (x1 , 0, +1) σR (x1 , 0, −1)
are calculated in the manner as follows:
λ1,2
λ3,4
λ5,6
λ7,8
λ9,10
p
(µ1 + α1 ) − (µ2 + α2 ) ± 2i (µ1 + α1 )(µ2 + α2 )
=
,
µ1 + α1 + µ2 + α2
( B±√B 2 −AC
for B 2 − AC ≥ 0,
A
=
√
B±i AC−B 2
for B 2 − AC < 0,
A
√
( B± B 2 −AC
for B 2 − AC ≥ 0,
C
=
√
B±i AC−B 2
for B 2 − AC < 0,
C
p
(ν1 + β1 ) − (ν2 + β2 ) ± 2i (ν1 + β1 )(ν2 + β2 )
,
=
ν1 + β1 + ν2 + β2
√

e
e 2 −A
eC
e
B
 B±
e2 − A
eC
e ≥ 0,
for B
e
A
=
√
e
eC−
e B
e2
 B±i
A
e2 − A
eC
e < 0,
for B
e
A
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
λ11,12 =
√


e
B±

e
B±i
e 2 −A
eC
e
B
e
C
e2 − A
eC
e ≥ 0,
for B
eC−
e B
e2
A
e
C
e2 − A
eC
e < 0,
for B
√
455
where
A = a2 a1 + b1 b2 + a22 − b22 + a2 b1 + b2 a1 ,
B = a2 a1 − b1 b2 − a22 + b22 ,
C = a2 a1 + b1 b2 + a22 − b22 − a2 b1 − b2 a1 ,
e = c2 c1 + d1 d2 + c22 − d22 + c2 d1 + d2 c1 ,
A
e = c2 c1 − d1 d2 − c22 + d22 ,
B
e = c2 c1 + d1 d2 + c2 − d2 − c2 d1 − d2 c1 .
C
2
2
Hence we obtain
√
1 1
µ2 + α2
γ1 = − arctg √
, δ1 = 0,
2 π
µ1 + α1
 1
,
B 2 − AC ≥ 0,

2


√
1
1
AC−B 2
−
arctg
, B 2 − AC < 0,
γ2,3 =
2
2π
B

√

 1 arctg AC−B 2
B 2 − AC < 0,
2π
|B|

√
2
1
 − 2π
log B± BA −AC , B 2 − AC ≥ 0,
³ ´
δ2,3 =
 ∓ 1 log A ,
B 2 − AC < 0,
2π
C
√
1 1
ν2 + β2
γ4 = − arctg √
, δ4 = 0,
2 π
ν1 + β1

1
e2 − A
eC
e ≥ 0,

,
B

2


√

ee B
e2
1
1
e2 − A
eC
e < 0,
− 2π arctg AC−
, B
γ5,6 =
e
2
B

√


ee B
e2
 1
e2 − A
eC
e < 0,

arctg A|C−
,
B
e
2π
B|
√

e
e 2 −A
eC
e
B
 − 1 log B±
e2 − A
eC
e ≥ 0,
, B
e
2π
A
³ ´
δ5,6 =
 ∓ 1 log Ae ,
e2 − A
eC
e < 0.
B
e
2π
C
B > 0,
B < 0,
e > 0,
B
e < 0,
B
e2 − A
eC
e 6= 0, then B 0 (t) = I, i.e., the
Remark 5.3. If B 2 − AC 6= 0 and B
apr
first terms of the asymptotic expansion of solutions of the boundary-contact
(i)
problem M1 contain no logarithms near the contact boundary ∂S0 , 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 µ2 + α2 → 0 or µ1 → ∞, then γ1 → 1/2,
γ2 = γ3 = 1/2;
b) if B > 0, B 2 − AC < 0 and µ1 → ∞, then γm → 1/2, m = 1, 2, 3;
c) if B > 0, B 2 − AC < 0 and µ2 + α2 → 0, then γ1 → 1/2, γ2,3 → 1/4;
d) if B < 0 and µ1 + α1 → 0 or µ2 → ∞, then γm → 0, m = 1, 2, 3;
456
O. CHKADUA
e) if |µ1 − µ2 | → 0, |α1 − α2 | → 0, then γm → 1/4, m = 1, 2, 3,
and
e > 0, B
e2 − A
eC
e ≥ 0 and ν2 + β2 → 0 or ν1 → ∞, then γ4 → 1/2,
ã) if B
γ5,6 = 1/2;
e > 0, B
e2 − A
eC
e < 0 and ν1 → ∞, then γm → 1/2, m = 4, 5, 6;
b̃) if B
e > 0, B
e2 − A
eC
e < 0 and ν2 + β2 → 0, then γ4 → 1/2, γ5,6 → 1/4;
c̃) if B
e < 0, ν1 + β1 → 0 or ν2 → ∞, then γm → 0, m = 4, 5, 6;
d̃) if B
ẽ) if |ν1 − ν2 | → 0, |β1 − β2 | → 0, then γm → 1/4, m = 4, 5, 6.
It is not difficult to see that if the conditions
e2 − A
eC
e>0
B 2 − AC > 0 and B
hold, then the exponent of the first term of the asymptotics of solutions of the
boundary-contact problem M1 has the form
n √µ + α √ν + β o
1 1
2
2
2
2
γ = − arctg max √
,√
.
2 π
µ1 + α 1
ν 1 + β1
Since γ < 12 , γ2,3 = γ5,6 = 12 , 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 γj j = 1, . . . , 6,
of exponents of the first terms of the asymptotic expansion are calculated by
simpler formulas.
+
Let αj > 0 and βj > 0, j = 1, . . . , 6, be the eigenvalues of the matrices σN
1
+
and σN
,
respectively.
2
Theorem
If the¶conditions
µ 5.5.
+
σN1 − αj I
1) rank
< 6, j = 1, . . . , 6;
+
σN
− βj I
2
(j)
(j)
2) hζ1 , ζ 1 i 6= 0, j = 1, . . . , 6,
(j)
are satisfied, where ζ1 (j = 1, . . . , 6) is the common eigenvector of the matrices
+
+
σN1 and σN2 , which correspond to the eigenvalues αj and βj , then the oscillation
participanting in the asymptotic expansion of solutions of the boundary-contact
problem M1 vanishes, i.e., δj = 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
1
1 1
, j = 1, . . . , 6.
γj = − arctg p
2 π
αj βj
Proof. From the condition 1) we obtain the existence of the common eigenvec(j)
+
+
tors ζ1 , j = 1, . . . , 6, for the matrices σN
and σN
, i.e.,
1
2
(j)
(j)
+
σN
ζ = αj ζ1 ,
1 1
(j)
(j)
+
σN
ζ = βj ζ1 , j = 1, . . . , 6.
2 1
(5.3)
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
457
−
+
−
Since σ +
N1 = σN1 and σ N2 = σN2 , we have
(j)
(j)
(j)
−
ζ = αj ζ 1 ,
σN
1 1
(j)
−
σN
ζ = βj ζ 1 , j = 1, . . . , 6.
2 1
(5.4)
Further, we look for an eigenvector ζ (j) of the matrix
¡
¢−1
σR (x1 , 0, +1) σR (x1 , 0, −1)
(j)
(j)
in the form ζ (j) = (ζ1 , γ
ej ζ1 ). We obtain
¡
¢−1
σR (x1 , 0, +1) σR (x1 , 0, −1)ζ (j) = λj ζ (j) .
Taking into consideration the expression of the matrices σR (x1 , 0, ±1), we get
(
(j)
(j)
+ (j)
− (j)
ej ζ1 = λj σN
ζ + λj γ
ej ζ1 ,
σN
ζ −γ
2 1
2 1
(5.5)
(j)
(j)
− (j)
+ (j)
ej σN
=
−λ
ζ
+
λ
γ
e
σ
.
ζ1 + γ
ζ
ζ
j
j
j
1
1
1
N1
1
(j)
Now, by the scalar multiplication of both equations (5.5) by the vector ζ
we have
(
(j)
(j)
(j)
(j)
(j)
(j)
+ (j)
− (j)
ej hζ1 , ζ 1 i = λj hσN
ζ , ζ 1 i + λj γ
ej hζ1 , ζ 1 i,
hσN
ζ , ζ1 i − γ
2 1
2 1
(5.6)
(j)
(j)
(j)
(j)
(j)
(j)
− (j)
+ (j)
hζ1 , ζ 1 i + γ
ej hσN
ζ
,
ζ
i
=
−λ
hζ
,
ζ
i
+
λ
γ
e
hσ
ζ
,
ζ
i.
j
j
j
1
1
1
1
1
1
N
1
1
Substituting (5.3) and (5.4) into (5.6), we get

(j)
(j)
(j)
(j)
(j)
(j)
 βj hζ (j) , ζ (j) i − γ
ej hζ1 , ζ 1 i = λj βj hζ1 , ζ 1 i + λj γ
ej hζ1 , ζ 1 i,
1
1
(j)
(j)
(j)
(j)
(j)
(j)
 hζ (j) , ζ (j) i + γ
e α hζ , ζ i = −λ hζ , ζ i + λ γ
e α hζ , ζ i.
1
1
j
j
1
1
j
1
1
(j)
(j)
j j
j
1
1
Taking into account the condition 2), i.e., hζ1 , ζ 1 i 6= 0, we obtain the following
system of equations with respect to λj and γ
ej (j = 1, . . . , 6):
(
βj − γ
ej = λj βj + λj γ
ej ,
1+γ
ej αj = −λj + λj γ
ej αj .
Further,
p
λj = p
αj βj ∓ i
αj βj ± i
s
, γ
ej = ±i
βj
, j = 1, . . . , 6.
αj
Clearly,
|λj | = 1, j = 1, . . . , 6.
Hence
δj = 0, j = 1, . . . , 6.
Calculating γj (j = 1, . . . , 6), we get
γj =
1 1
1
− arctg p
, j = 1, . . . , 6.
2 π
αj βj
¤
458
O. CHKADUA
If under the conditions of Theorem 5.5
αi βi 6= αj βj for all i 6= j, i, j = 1, . . . , 6,
then Ba0pr (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
(1)
(2)
contact surfaces S0 and S0 are parallel to the isotropic plane. We consider
(q)
(q)
(q)
the case, where the coefficients bijlk = 0 and, instead of aijlk and cijlk , we have
(q)
the elastic constants of transversally-isotropic elastic bodies, i.e., instead of aijlk
we have
(q)
(q)
(q)
(q)
(q)
a11 , a33 , a13 , a55 , a66 , q = 1, 2,
(q)
and, instead of cijlk , we have
(q)
(q)
(q)
(q)
(q)
c11 , c33 , c13 , c55 , c66 , q = 1, 2,
which satisfy the following conditions:


(q)
(q)
(q)
(q)
(q)
(q)
(q)
(q)


 a11 − a66 > 0, a55 > 0, a66 > 0,
 c11 − c66 > 0, c55 > 0, c66 > 0,
(q)
(q)
(a13 )2
(c13 )2
(q)
(q)


, q = 1, 2,
, q = 1, 2.
 a33 > (q)
 c33 > (q)
(q)
(q)
a11 − a66
c11 − c66
It is not difficult to see that in this case the eigenvalues αj and βj , j = 1, . . . , 6,
+
+
of the matrices σN
and σN
are calculated explicitly.
1
2
(q)
(q)
(q)
(q)
Let a11 6= a33 and c11 6= c33 (q = 1, 2); then the conditions 1)–2) of Theorem
5.5 have the form
e1 − D
e1
e1
C1 − D1
C
B
B1
=
,
=
,
e2 − C
e2
e2
D2 − C2
B2
D
B
where
¶
µq
µq q
q ¶
(q)
(q)
(q)
(q) (q)
(q)
(q)
(q)
a11 a33 − a13
a11 a55
a2 + a3
a55
1
q q
q q
Bq =
, Cq =
,
2
(q)
(q)
(q)
(q)
(q)
(q)
a55 + a11 a33
a55 + a11 a33
µq
q q
q ¶
(q)
(q) (q)
(q)
(q)
a33 a11 a55
a2 + a3
1
q q
Dq =
, q = 1, 2,
2
(q)
(q)
(q)
a55 + a11 a33
(q)
(q)
a2 and a3 are the roots of the equation (see [26])
£ (q)
(q) (q)
(q) (q)
(q)
(q) (q)
(q) ¤
a11 a55 a2 + (a13 + a55 )2 − a11 a33 − (a55 )2 a + a33 a55 = 0,
and
µq
eq =
B
(q)
c55
¶
q
(q)
c11
(q)
c33
µq
(q)
c13
−
q q
(q)
(q)
(q)
c55 + c11 c33
,
eq = 1
C
2
q = 1, 2,
q ¶
(q)
+ c3
q q
,
(q)
(q)
(q)
c55 + c11 c33
(q) (q)
c11 c55
(q)
c2
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
q
(q)
eq = 1
D
2
(q)
459
µq
q ¶
(q) (q)
(q)
(q)
c11 c55
c2 + c3
q q
, q = 1, 2,
(q)
(q)
(q)
c55 + c11 c33
q
c33
(q)
where c2 and c3 are the roots of the equation
£ (q)
(q) (q)
(q)
(q) (q)
(q) ¤
(q) (q)
c11 c55 c2 + (c13 + c55 )2 − c11 c33 − (c55 )2 c + c33 c55 = 0, q = 1, 2.
(q)
(q)
(q)
(q)
If we assume that a13 = −a55 and c13 = −c55 , q = 1, 2, then the conditions
1)–2) of Theorem 5.5 can be rewritten in a simpler form:
q
q
q
q
q
q
(1)
(1)
(1)
(1)
(1)
(1)
a11 − a33
a55
c11 − c33
c55
q
q
q
=q
and q
=q
.
(2)
(2)
(2)
(2)
(2)
(2)
a33 − a11
a55
c33 − c11
c55
Now let us consider the case, where the domains Dq , 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 Dq , q = 1, 2, are filled with the same material,
then the asymptotic expansion of solutions of the boundary-contact problem M1
(1)
near the contact boundary ∂S0 takes the form
U (q) (x1 , x2 , x3 ) = (u(q) , ω (q) )(x1 , x2 , x3 )
½ nX
l0
2 X
s −1
h
i
XX
1
(q)
(q)
+∆j (x1 )−m (q)
4
Re
xm
d
(x
,
θ)(z
)
=
cjm (x1 )
3
sjm 1
s,theta
ϑ=±1 j=1 s=1
+
+2−l
M
+2 MX
X
m=0
o
(q)
(q) 1 +∆j (x1 )+k+p (q)
(q)
4
xl2 xm
d
(x
,
ϑ)(z
)
B
(x
,
log
z
)
1
1
3 slmpj
s,ϑ
skmpj
s,ϑ
k,l=0 p+m=0
k+l+p+m6=0
(q)
+UM +1 (x1 , x2 , x3 ) for M >
(q)
(q)
2
− min{[s − 1], 1}, q = 1, 2,
r
(5.7)
(q)
with the coefficients dsjm (· , ±1), cjm , dslmpj (· , ±1) ∈ C0∞ (R) and the remainder
(q)
3
UM +1 ∈ C0M +1 (R± ), q = 1, 2,
(q)
(q)
(q)
(q)
(q)
zs,+1 = (−1)q (x2 + x3 τs,+1 ), zs,−1 = (−1)q+1 (x2 − x3 τs,−1 ), −π < Argτs,±1 < π.
In this case the parameters ∆j are calculated by
¡ (j)
¢
(j)
∆j (x1 ) = δ1 (x1 ), . . . , δ6 (x1 ) ,
j = 1, 2,
where
i
1
i
(2)
log |λk (x1 )|, δk (x1 ) = −
log |λk (x1 )|, k = 1, . . . , 6.
2π
2 2π
Proof. When the domains Dq , q = 1, 2, are filled with the same material, we
can show, like in [7], that the eigenvalues λk (x1 ), k = 1, . . . , 12, of the matrix
¡
¢−1
(1)
σR (x1 , 0, +1) σR (x1 , 0, −1), x2 ∈ ∂S0 ,
(1)
δk (x1 ) = −
460
O. CHKADUA
are calculated by means of the eigenvalues βk (x1 ), k = 1, 2, 3, of the matrix σ ∗ ,
V0
i.e.,
 s

1 − 2βk (x1 )


if k = 1, . . . , 6,

 i 1 + 2β (x ) ,
k 1
s
λk (x1 ) =


1 − 2βk−6 (x1 )


, if k = 7, . . . , 12,
 −i
1 + 2βk−6 (x1 )
where −1/2 < βk < 1/2, k = 1, . . . , 6.
Hence the exponent of the first term of the asymptotic expansion of solutions
has the form
1 1
i
∓ −
log |λk (x1 )|, k = 1, . . . , 6.
¤
2 4 2π
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 bound(2)
ary ∂S0 .
6. Solvability and Asymptotics of Solutions of
the Boundary-Contact Wedge-Type Problem M2
We will formulate theorems of the uniqueness and existence of solutions of
problem M2 .
Assume that the following compatibility conditions are fulfilled on the curves
(1)
(2)
∂S0 , ∂S0 ∂S0 :
(i)
(i)
(2)
(1)
(3)
(2)
s−1
s−1
∃Φ0 , i = 1, 2, 3, Φ0 ∈ Bs−1
p,r (∂D1 ), Φ0 ∈ Bp,r (∂D2 ), Φ0 ∈ Bp,r (∂D2 ),
(1)
(2)
e s−1 (S (1) ),
h1 − πS (1) Φ0 + πS (1) Φ0 ∈ B
0
p,r
0
0
(1)
(3)
e s−1 (S (2) )
h2 − πS (2) Φ0 + πS (2) Φ0 ∈ B
0
p,r
0
0
(2)
(3)
s−1
e
πS0 Φ0 + πS0 Φ0 ∈ Bp,r (S0 )
(1)
(6.1)
(2)
s−1
s−1
hold for ϕ1 ∈ Bs−1
p,r (∂D1 ), ϕ2 ∈ Bp,r (∂D2 ), ϕ3 ∈ Bp,r (∂D2 ), 1 < p < ∞,
1 ≤ r ≤ ∞, 1/p − 1/2 < s < 1/p + 1/2.
(i)
Here Φ0 , i = 1, 2, 3, are some fixed extensions of the functions ϕi , i = 1, 2, 3,
(1)
(2)
on ∂D1 , ∂D2 , ∂D2 , respectively.
Theorem 6.1. Let 4/3 < p < 4 and the compatibility conditions (6.1) be
(1)
(2)
fulfilled on ∂S0 , ∂S0 ∂S0 for s = 1 − 1/p. Then the boundary-contact wedgetype problem M2 has solutions of the classes Wp1 (Dq ), q = 1, 2, if and only if
the condition
Z
Z
Z
ϕ3 ([a × z] + b, a) ds
ϕ2 ([a × z] + b, a) ds −
ϕ1 ([a × z] + b, a) ds −
(2)
(1)
S2
S1
S2
Z
Z
h2 ([a × z] + b, a) ds = 0
h1 ([a × z] + b, a) ds +
+
(1)
S0
(2)
S0
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
461
is fulfilled, where a and b are arbitrary three-dimensional constant vectors.
Solutions of the boundary-contact problem M2 are given by the potential-type
functions
(1)
(1)
(1)
(2)
(2)
(2)
(3)
(3)
(3)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(3)
(3)
(3)
(3)
U (1) = V(1) (V−1 )−1 (B2M +1 )−1 ϕ0 + V(1) (V−1 )−1 (B2M +1 )−1 ψ0 + R1 , (6.2)
r1 U (2) = V(2) (V−1 )−1 (B2M +1 )−1 ϕ0 + V−1 (V−1 )−1 (B2M +1 )−1 ψ0 + R2 , (6.3)
r2 U (2) = V(3) (V−1 )−1 (B2M +1 )−1 ϕ0 + V−1 (V−1 )−1 (B2M +1 )−1 ψ0 + R3 , (6.4)
(1)
(2)
with R1 ∈ C M +1 (D1 ), R2 ∈ C M +1 (D2 ), R3 ∈ C M +1 (D2 ), while the functions
(1)
(2)
(1)
(3)
(2)
(3)
(ϕ0 , ϕ0 ), (ψ0 , ϕ0 ) and (ψ0 , ψ0 ) are solutions of the systems
(
(
(
(2)
(1)
(1)
πS0 N3 ψ0 = Ψ3 ,
πS (1) N1 ϕ0 = Ψ1 ,
πS (2) N2 ψ0 = Ψ2 ,
0
0
(2)
ϕ0
=
(1)
ϕ0
+ G1 ,
(3)
ϕ0
(1)
= ψ0 + G2 ,
(3)
(2)
ψ0 = −ψ0 + G3 ,
respectively. Here
(1)
(2)
(1)
(3)
N1 = (B2M +1 )−1 − (B2M +1 )−1 ,
N2 = (B2M +1 )−1 − (B2M +1 )−1 ,
(2)
(3)
N3 = −(B2M +1 )−1 − (B2M +1 )−1 ,
where
³ 1
∗ (1) ´
(1)
(1)
(1)
B2M +1 = −(V−1 )2M +1 + − I+ V 0 (V−1 )−1 ,
2
³1
∗ (i) ´
(i)
(i)
(i)
I+ V 0 (V−1 )−1 , i = 2, 3.
B2M +1 = (V−1 )2M +1 +
2
The operators Nj , j = 1, 2, 3, are positive-definite ΨDOs.
Let
¡ (1)
¢
(1)
(`)
(`)
∆Nj (x1 ) = δj (x1 ), . . . , δj (x1 ), . . . , δj (x1 ), . . . , δj (x1 ) ,
|
{z
}
|
{z
}
m1 -times
m` -times
i
(k)
(k)
δj (x1 ) = −
log |λNj (x1 )|, j = 1, 2, 3, k = 1, . . . , `
2π
(generally speaking, mk and ` depend on j),
(k)
where λNj , j = 1, 2, k = 1, . . . , `, are the eigenvalues of the matrix
¡
¢−1
bNj (x1 ) = σNj (x1 , 0, +1 σNj (x1 , 0, −1)
of multiplicity mk , k = 1, . . . , `,
6
P
k=1
mk = 6; here mk , k = 1, . . . , `, and `
(k)
depend on j, and the eigenvalues of the matrix bN3 = I are trivial, λN3 = 1,
k = 1, . . . , 6.
Note that the boundary data of the problem M2 are assumed to be suf(∞,S+2M ),∞
(∞,S+2M ),∞
(1)
ficiently smooth, i.e., ϕ1 ∈ Hr
(S1 ), ϕ2 ∈ Hr
(S2 ), ϕ3 ∈
(∞,S+2M ),∞
(2)
(∞,S+2M +1),∞
(i)
(∞,S+2M ),∞
(i)
Hr
(S2 ), fi ∈ Hr
(S0 ), hi ∈ Hr
(S0 ), i = 1, 2,
1
− 12 < s < 1r + 21 .
r
462
O. CHKADUA
Recalling that solutions of the problem M2 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 M2 in terms of some local coordinate systems of curves
(1)
(2)
∂S0 , ∂S0 , ∂S0 :
(j)
a) Asymptotic expansion near the contact boundaries ∂S0 , j = 1, 2:
U (q) (x1 , x2 , x3 ) = (u(q) , ω (q) )(x1 , x2 , x3 )
½ nX
`0
s −1
h
i
XX
(q)
(q) 1 −j+i∆(x1 ) (j)
j
2
x3 dsj (x1 , θ)(zs,θ )
=
Re
c (x1 )
ϑ=±1 s=1
+
j=0
M
+1 MX
+2−l
X
¾
− 1 +i∆(x1 )+p+k (q)
(q)
(q)
xl2 xj3 dsljp (x1 , ϑ)zs,ϑ2
Bskjp (x1 , zs,ϑ )
l,k=0 p+j=1
k+l+p+j6=1
(q)
UM +1 (x1 , x2 , x3 ) for M >
2
− min{[s], 0}, q = 1, 2,
r
(6.5)
3
(q)
with UM +1 ∈ C0M +1 (R± ), q = 1, 2. Here ∆ = ∆Nj , j = 1, 2, and
£
£
(q)
(q) ¤
(q)
(q) ¤
zs,+1 = (−1)q x2 + x3 τs,+1 , zs,−1 = (−1)q x2 − x3 τs,−1 , −π < Argzs,±1 < π.
(q)
The first coefficients dsj (·, ±1), c(j) are calculated as in(4.14) (see [9]).
b) Asymptotic expansions near the cuspidal edge ∂S0 (see (4.16)):
ri U (2) (x1 , x2 , x3 ) = ri (u(2) , ω (2) )(x1 , x2 , x3 )
=
`0
XX
Re
½ nX
s −1
ϑ=±1 s=1
1
−j (i)
2
xj3 zs,θ
dsj (x1 , θ)
+
+2−l
M
+1 MX
X
¾
− 1 +p+k (i)
xl2 xj3 zs,ϑ2
dslkjp (x1 )
l,k=0 j+p=1
l+k+j+p6=1
j=0
(i)
+UM +1 (x1 , x2 , x3 ) for M >
2
− min{[s], 0}, i = 1, 2,
r
(6.6)
3
(i)
with UM +1 ∈ C0M +1 (R± ), i = 1, 2. Here
(2)
(2)
zs,+1 = −x2 − x3 τs,+1 , zs,−1 = −x2 − x3 τs,−1 , −π < Argzs,±1 < π.
(i)
The coefficients dsj (x1 , ±1) are calculated as in (4.16).
Remark 6.2. Note that if we consider the boundary-contact problem M2 for
the nonhomogeneous equations
M(q) (∂x )U (q) + F (q) = 0 in Dq , q = 1, 2,
and use the boundary and boundary-contact data
ϕ1 = 0,
ϕ2 = 0, ϕ3 = 0,
fi = hi = 0, i = 1, 2,
(1)
then the compatibility conditions (6.1) are fulfilled automatically on ∂S0 ,
(2)
∂S0 , ∂S0 .
SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS
463
Remark 6.3. It is easy to see that if the matrices bNj , j = 1, 2, are unitary
(b∗Nj = b−1
Nj ), then the oscillation in the asymptotic expansion (42) vanishes, i.e.,
∆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:
e1 D
e1 − B
e2
e1
B1
4C1 D1 − B12
B
4C
1
=
=
.
(6.7)
,
e2
e2 D
e2 − B
e22
B2
4C2 D2 − B22
B
4C
For the centrally symmetric isotropic case condition (6.7) tales the form
λ1 + µ1 − α1 = λ2 + µ2 − α2 ,
ε1 + ν1 − β1 = ε2 + ν2 − β2 .
(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