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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
(2)
(1)
(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)

1 (q)

(∂x ) = k

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)

1 (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)

(q)

N jk (∂z , n(z)) = bijlk ni (z)∂l − aijlk εlrk ni (z),

(q)

N jk (∂z , n(z)) = cijlk ni (z)∂l − blrij εlrk ni (z),

N jk (∂z , n(z)) = aijlk ni (z)∂l ,
3 (q)

,

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
(D2 ). Then r1 U (2) = rD(1) U (2) ∈ Wp1 (D2 )
Let U (1) ∈ Wp1 (D1 ), U (2) ∈ Wp,loc
(2)

2

1
(D2 ), where ri is the restriction operator on
and r2 U (2) = rD(2) U (2) ∈ Wp,loc
(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) }± ∈
′
′
(i)
1/p
1/p
Bp,p (∂D1 ), {ri U (2) }± ∈ Bp,p (∂D2 ), i = 1, 2, p′ = 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) ∈ Wp1′ (D1 )

(1.4)

SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS

and

Z

(i)

D2

431

£ (i) (2)
(i) ¤
V 2 M (∂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 ∈ Wp1′ (D2 ) (V2 ∈ Wp,comp
(D2 )),

where

(1)

E (1) (U (1) , V (1) ) = aijlk ξij (U (1) )ξlk (V
(1)

+ bijlk ξij (V
here U (1) = (u(1) , ω (1) ),

(1)

(1)

(1)

) + bijlk ξij (U (1) )ηlk (V

(1)

(1)

)ηlk (U (1) ) + cijlk ηij (U (1) )ηlk (V

)

(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)

on S2 ,
πS (2) {N (2) (r2 U (2) )}+ = ϕ3


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/p′

1/p′

−1/p

(1)

−1/p

(2)

ϕ1 ∈ Bp,p (S1 ), ϕ2 ∈ Bp,p (S2 ), ϕ3 ∈ Bp,p (S2 ),
(i)

fi ∈ Bp,p (S0 ),

−1/p

(i)

hi ∈ Bp,p (S0 ), i = 1, 2, 1 < p < ∞, p′ = 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/p′

ϕ1 ∈ Bp,p (S1 ),
−1/p

(i)

1/p′

(1)

1/p′

(2)

ϕ2 ∈ Bp,p (S2 ), ϕ3 ∈ Bp,p (S2 ),
1/p′

(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) ′
¢−1 −iτ x3
−1
(q)
H (x) = Fξ′ →x′ ±
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
′
′
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 ′ , 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)
H(1) (x − y)g1 (y)dy S, x ∈
/ ∂D1 ,
V (g1 )(x) =
∂D1
Z
(1)
(2)
H(2) (x − y)g2 (y)dy S, x ∈
/ ∂D2 ,
V (g2 )(x) =
(1)
∂D
Z 2
(2)
H(2) (x − y)g3 (y)dy S, x ∈
/ ∂D2 .
V(3) (g3 )(x) =
(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
V(1) : Bsp,r (∂D1 ) → Bp,r
(D1 ) Bsp,p (∂D1 ) → Hs+1+1/p
(D1 ) ,
p
(1)
(1) ¡
(1)
(1) ¢
s+1+1/p
V(2) : Bsp,r (∂D2 ) → Bp,r
(D2 ) Bsp,p (∂D2 ) → Hs+1+1/p
(D2 ) ,
p
(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)
{V(3) (g3 )(z)}± =
H(2) (z − y)g3 (y)dy S, z ∈ ∂D2 .
(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
(D2 ) with condition (1.6) at infinity).
M1 in the classes W21 (Dq ), q = 1, 2 (W2,loc
Lemma 3.1. A solution of the boundary-contact problem M1 is unique in the
1
(D2 ) and satisfying condition (1.6) at infinity).
classes W21 (Dq ), q = 1, 2 (W2,loc
Proof. Let U (q) , q = 1, 2, be a solution of the homogeneous problem M1 .
(1)
(2)
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 ωi

= 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 ,

1/p′

¤

q = 1, 2.

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 ,
′
′
(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/p
(1)
(2)
e p,p
e −1/p
where Φ0 is some fixed extension of ϕ2 and ϕ0 ∈ B
(S0 ), ψ0 ∈ B
p,p (S0 ).
−1/p
(2)
(3)
Any extension Φ ∈ 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 ,
(3)

(3)

−1/p

(2)

(3)

−1/p

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 .

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 ,
( 12 I+ V 0 )g2 − ϕ0 − ψ0 = Φ0




∗ (3)

(3)
(3)
(3)
(2)
1

(
on ∂D2 ,
I+

V
0 )g3 − ϕ0 − ψ0 = Φ0

2



(1)
(1)
(1)
(2)

on S0 ,
−πS (1) V−1 g2 + ϕ0 = f1 − πS (1) Φ0


0
0

∗ (1)
(2)
(2)
(1)
(3.6)
πS (1) (− 21 I+ V 0 )g1 − ϕ0 = h1 + πS (1) Φ0 on S0 ,

0
0



(1)
(2)
(3)
(1)

on S0 ,
−πS (2) V−1 g3 + ψ0 = f2 − πS (2) Φ0


0
0


∗ (1)

(3)
(2)
(3)

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)

(1)

∃Φ0 ∈ Bs−1
p,r (∂D2 ),
(1)

(3)
(2)
(2)
(3)
e s−1
Φ0 ∈ Bs−1
p,r (∂D2 ) : πS0 Φ0 +πS0 Φ0 ∈ Bp,r (S0 ) (3.7)
(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

(1)

0

p

(1)

(2)

(1)

s−1
s−1
s−1
s
s
s
H p = Hp (∂D1 ) ⊕ Hp (∂D2 ) ⊕ Hp (∂D2 ) ⊕ Hp (S0 ) ⊕ Hp (S0 )
(2)

(1)

(2)

s
s−1
⊕Hsp (S0 ) ⊕ Hs−1
p (S0 ) ⊕ Hp (S0 ) ⊕ Hp (S0 ),

(1)
(2)
s
s−1
s−1
s−1
e s (S (1) ) ⊕ B
e s (S (2) )
B p,r = Bp,r (∂D1 ) ⊕ Bp,r (∂D2 ) ⊕ Bp,r (∂D2 ) ⊕ B
0
0
p,r
p,r
(1)
(2)
e s−1 (S ) ⊕ B
e s−1 (S0 ) ⊕ B
e s−1 (S ) ⊕ B
e s−1 (S0 ),
⊕B
p,r

0

(2)

p,r

p,r

(1)

0

(2)

p,r

(1)

(1)

s−1
s
s
s
s−1
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−1
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 ),

(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


e0 +V−1 ψe0 = Ψ0
I+
)
(
g2 +V−1 ϕ
[(−V
−V
V
0 )]e
−1
−1

2



∗ (3)

(3)
(3) (3)
(3) M
(3) 1
(3) (3)

e0 +V−1 ψe0 = Ψ0
(−V
)
−V
(
g3 +V−1 ϕ
I+

V
0 )e
−1
−1 2




(1)
(2)

e0 = G1
−πS (1) V−1 ge2 + ϕ


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 ( 21 I+ V 0 ), i = 2, 3, are
nonnegative, i.e.,
−1/2

(1)

h−V−1 ϕ, ϕi∂D1 > 0 for all ϕ ∈ H2 (∂D1 ), ϕ 6= 0,
³
D
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)

³1

∗ (3) ´

E

−1/2

(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) 1
(i)
BM = (−V−1 )M − V−1 I+ V 0 , i = 2, 3,
2
are positive operators, i.e.,
− V−1

I+ V 0

ψ, ψ

(2)

−1/2

hBM ϕ, ϕi∂D(1) > 0 for all ψ ∈ H2
2

−1/2

(3)

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
(1)

(1)

(1)

(1)

(1)

(1)

ge1 = (V−1 )−1 ϕ
e0 + (V−1 )−1 ψe0 + (V−1 )−1 Ψ0 ,
(i) (i)
(i) (i)
(i)
(i)
(i)
(i)
e0 − (BM )−1 V−1 ψe0 + (BM )−1 Ψ0 , i = 2, 3.
gei = −(BM )−1 V−1 ϕ

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
0
πS (2) B(x, D)
P=
+ T−∞ ,
0
0
0
πS0 C(x, D) 36×36

where

I



πS (2) V−1 (BM )−1 V−1

I

−I

πS (2) (− 12 I+ V 0 )(V−1 )−1





πS (1) V−1 (BM )−1 V−1



A(x, D) = 
B(x, D) = 

(2)

(2)

(2)

0

πS (1) (− 21 I+
0

−I
(3)

(3)

(3)

0

∗ (1)
(1)
V 0 )(V−1 )−1

∗ (1)

0

(1)

,
,

SOLVABILITY AND ASYMPTOTICS OF SOLUTIONS

C(x, D) =

Ã

(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(x′ , D′ ) and B(x′ , D′ ) be ΨDOs with the symbols σA (x′ , ξ ′ ) and
σB (x′ , ξ ′ ) (ξ ′ = (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
′
′
′
′
R(x , D ) =
◦ A(x , D ) ◦
0 I
0 I
and

′

′

Q(x , D ) =

µ

L− 0
0 I

¶

′

′

◦ B(x , D ) ◦

µ

L+ 0
0 I

¶

,

where L+ = diagΛ+ , L− = diagπ+ Λ− ℓ are 6 × 6 matrix operators, Λ± is a ΨDO
operator with the symbol Λ± (ξ ′ ) = ξ2 ± i ± i|ξ1 |, π+ denotes the operator of
restriction onto R2+ , and ℓ 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(x′ , D′ ) and Q(x′ , D′ ) are
written as
Ã
!
′′
′ ′
′′
′′
(ξ
−i|ξ
|)σ
(x
,
ξ
)(ξ
+i|ξ
|)
(ξ
−i|ξ
|)I
n−1
N
n−1
n−1
2
(1)
σR (x′ , ξ ′ ) =
, x′ ∈ S 0 ,
′′
′ ′
−(ξn−1 + i|ξ |)I
σN1 (x , ξ )
Ã
!
′′
′ ′
′′
′′
(ξ
−i|ξ
|)σ
(x
,
ξ
)(ξ
+i|ξ
|)
(ξ
−i|ξ
|)I
n−1
N
n−1
n−1
3
(2)
, x′ ∈ S 0 ,
σQ (x′ , ξ ′ ) =
′′
′ ′
−(ξn−1 + i|ξ |)I
σN1 (x , ξ )

where σN1 (x′ , ξ ′ ), σN2 (x′ , ξ ′ ) and σN3 (x′ , ξ ′ ) are the principal homogeneous symbols of the ΨDOs
³ 1
∗ (1) ´
(3)
(2)
(3)
(3)
(1)
(2)
(2)
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

σR (x1 , 0, −1) =

µ

σN2 (x1 , 0, −1)
−I
I
σN1 (x1 , 0, −1)

¶

,

440

O. CHKADUA

σR (x1 , 0, +1) =
(k)

µ

σN2 (x1 , 0, +1)
I
−I
σN1 (x1 , 0, +1)

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

σQ (x1 , 0, −1) =
σQ (x1 , 0, +1) =

µ

µ

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
¯
¯
(j)
sup ¯ arg λQ (x1 )¯,
1≤j≤12 2π
(2)

(1)

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(x′ , D′ ) : 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 ) → Hs−1 (R2 ) ⊕ Hs−1 (R2 )
e s (R2 ) ⊕ H
Q(x′ , D′ ) : 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(x′ , D′ ) and Q(x′ , D′ ) 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 G˚
arding 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

are invertible.

¡

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

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 )
π S0 N : H
p
p
¡ s−1
¢
e (S0 ) → Bs (S0 )
B
p,r

and

p,r

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
e p(µ,s−1),k → H(µ,s),k
with weight H
(S0 ) (see [10]).
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 (S (1) )
e s−1 (S (1) ) ⊕ H
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 ) ⊕ B
e s−1 (S0 )
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
is invertible.

¡

(1)

B

s
p,r

(2)

→B

s
p,r

¢

If we take s = 1/p′ in the condition for the operator A to be invertible (see
Theorem 3.7), we conclude that p must satisfy the equality
4
4
0, χM +1 ∈ H
(R2+ )× H
(R2+ );
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 , . . . , m2ℓ } 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 ), . . . , δℓ (x1 ), . . . , δℓ (x1 )) , j = 1, 2,
|
{z
}
|
{z
}
m1 -times
mℓ -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, . . . , ℓ.
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,+ ) =
Kij (x1 )x2,+
Bapr −
log x2,+ c0 (x1 )
2πi
j=1
+

2 X
M
X
j=1 k=1

1/2+∆j (x1 )+k

Kij (x1 )x2,+

(i)

(i)

Bkj (x1 , log x2,+ ) + χM +1 (x1 , x2,+ ), i = 1, 2, (4.3)

+1)
(i)
(i)
e (∞,s+M
where χM +1 ∈ H
(R2+ ) and Bkj (x1 , t) is a polynomial of order νk =
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(x′ , D′ )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
M
X
j=1 k=1

2
X
j=1

1/2+∆j (x1 )

Kij (x1 )x2,+

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)

+1),∞
(i)
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

where

(3)
e1 and ψe0(2) = −ψe0(3) + E2 ,
πS0 Nψe0 = E
(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) (x′ , ξ ′ ) = σV(3) (x′ , ξ ′ ) for x′ ∈ S 0 ,
−1

−1

(4.5)

σ ∗(2) (x′ , ξ ′ ) = −σ ∗(3) (x′ , ξ ′ ) for x′ ∈ S 0 .
V0

V0

In view of equality (4.5) we can write the symbol σN (x′ , ξ ′ ) of the ΨDO
operator N as follows:
h³ 1
´¡
¢−1 i−1
σN (x′ , ξ ′ ) =
I + σ ∗(2) (x′ , ξ ′ ) σ−V(2) (x′ , ξ ′ )
V0
−1
2
h³ 1
´¡
¢ i−1
′ ′ −1
′ ′
∗
+
.
I − σ (2) (x , ξ ) σ−V(2) (x , ξ )
V0
−1
2
Since the symbol σ ∗(2) (x′ , ξ ′ ) is an odd matrix function with respect to ξ ′ , while
V0

the symbol σ−V(2) (x′ , ξ ′ ) is an even matrix function, one can easily ascertain that
−1

the symbol σN (x′ , ξ ′ ) is even with respect to the variable ξ ′ , i.e.,
σN (x′ , −ξ ′ ) = σN (x′ , ξ ′ ),

x′ ∈ S 0 ,

and all eigenvalues of the matrix
¡
¢−1
σN (x1 , 0, +1) σN (x1 , 0, −1) = I,
(j)

x1 ∈ ∂S0 ,

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)
c 0 , dk

(∞,s+M +1),∞
(i)
e r,comp
where
∈ C0∞ (R), and the remainder ψM +1 ∈ H
(R2+ ), i = 2, 3,
M ∈ N. As we can see from (4.6), due to the properties of the symbol σN (x′ , ξ ′ )
(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 ) .
0

Then
here

D ◦ Ag = Ψ;

(4.7)

¡ (1)
(3) (3)
(1)
(2)
(1)
(2) (2)
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

(1)

Φ0 ,

(3) ¢
(2)
(3)
(1)
(2)
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
′

′

R(x , D )
(3)

(1)

µ

local coordinate system of the mani-

χ1
χ2

¶

= F,

and (−L−1
+ ψ0 , ϕ0 ) satisfies, in some local coordinate system of the manifold
(2)
S0 , the pseudodifferential equation
¶
µ
χ
e1
′
′
= Fe,
Q(x , D )
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 ),

(3) (3)
(3)
(3)
(1)
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

here

(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

(∞,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)

where

(2)

(3)

(2)

(3)

ψ0 = −ψ0 − (πS0 Φ0 + πS0 Φ0 ),

πS0 Nψ0 = E1 ,

(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)

here

(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 );

),∞
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 ),∞

G 2 ∈ Hr

(1)

(∞,s+2M ),∞

G3 ∈ Hr

(∂D2 ),

(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)

where

r2 U (2) = −V(3) (B2M +1 )−1 V−1 ϕ0 − V(3) (B2M +1 )−1 V−1 ψ0 + R3 , (4.13)
(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(1)
(3)
(2)
(2)
(1)
−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
ℓ0
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)
xl2 xm
Bskmpj (x1 , log zs,ϑ )
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
{τs,±1 }ℓs=1
are all different roots of the polynomial det M (q) ((Jκ⊤ (x1 , 0))−1 ·
(0, ±1, τ )) of multiplicity ns , s = 1, . . . , ℓ0 , in the complex lower half-plane (ns
(q)
and ℓ0 depend on q). Bskmpj (x1 , t) is a polynomial of order νkmp = νk + p + m
ℓ
P
(νk = k(2m0 − 1) + m0 − 1, m0 = max{m1 , . . . , mℓ },
mj = 6) with respect

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, . . . , ℓ0 , m = 0, . . . , ns − 1;

here Gκ is the square root from the Gram determinant of the diffeomorphism κ
and
q (s)
dns −1−m
im+1
(q)
(τ − τs,±1 )ns
(x
,
0,
±1)
=
V −1,m 1
m!(ns − 1 − m)! dτ ns −1−m
¤−1 ¯¯
£ ◦ (q)
⊤
−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) = B mr −
,
∂t Γ(t + 1)e 2
B−
2πi
r
B mr (t) = kbm
kp (t)kmr ×mr ,
 ³
´
iπ(t+1)
1 ´p−k (−1)p+k dp−k ³

2
, k ≤ p,
Γ(t
+
1)e
r
bm
2πi
(p − k)! dtp−k
kp (t) =

0,
k > p,
p = 0, . . . , mr − 1, r = 1, . . . , ℓ.

Further,

(j)

(j)

ajm (x1 ) = diag{am1 (λ1 ), . . . , amℓ (λℓ )}, j = 1, 2,
1
i
3
argλr (x1 ) +
log |λr (x1 )| + m,
λ(1)
r (x1 ) = − −
2 2π
2π
1
i
3
argλr (x1 ) +
log |λr (x1 )| + m,
λ(2)
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

(µr )
(−1)p+k (2πi)l−p bm

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, . . . , ℓ, and c0 (x1 ),
(2)
c0 (x1 ) are defined using the first coefficients of the asymptotic expansion of
(2)
(1)
the functions −L−1
+ ϕ0 and ϕ0 , respectively (see (4.3)).

450

O. CHKADUA
(2)

b) the asymp