Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2
The leading terms of the asymptotic expansion for the solution to a mixed boundary value problem for the Poisson equation in a thick multi-structure, which is the union of some domain and a large number N of ε-periodically situated thin annular disks with variable thickness of order ε = O(N⁻¹), are...
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irk-123456789-1246062017-09-30T03:04:08Z Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 De Maio, U. Mel'nyk, T.A. The leading terms of the asymptotic expansion for the solution to a mixed boundary value problem for the Poisson equation in a thick multi-structure, which is the union of some domain and a large number N of ε-periodically situated thin annular disks with variable thickness of order ε = O(N⁻¹), are constructed and the corresponding estimates in the Sobolev space H¹ are proved as ε → 0. 2005 Article Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 / U. De Maio, T.A. Mel'nyk // Український математичний вісник. — 2005. — Т. 2, № 4. — С. 463-481. — Бібліогр.: 34 назв. — англ. 1810-3200 2000 MSC. 35B27, 3540, 35J25, 35C20, 35B25. http://dspace.nbuv.gov.ua/handle/123456789/124606 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The leading terms of the asymptotic expansion for the solution to a mixed boundary value problem for the Poisson equation in a thick multi-structure, which is the union of some domain and a large number N of ε-periodically situated thin annular disks with variable thickness of order ε = O(N⁻¹), are constructed and the corresponding estimates in the Sobolev space H¹ are proved as ε → 0. |
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Article |
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De Maio, U. Mel'nyk, T.A. |
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De Maio, U. Mel'nyk, T.A. Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 Український математичний вісник |
| author_facet |
De Maio, U. Mel'nyk, T.A. |
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De Maio, U. |
| title |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 |
| title_short |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 |
| title_full |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 |
| title_fullStr |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 |
| title_full_unstemmed |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 |
| title_sort |
asymptotic solution to a mixed boundary-value problem in a thick multi-structure of type 3:2:2 |
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Інститут прикладної математики і механіки НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/124606 |
| citation_txt |
Asymptotic Solution to a Mixed Boundary-Value Problem in a Thick Multi-Structure of Type 3:2:2 / U. De Maio, T.A. Mel'nyk // Український математичний вісник. — 2005. — Т. 2, № 4. — С. 463-481. — Бібліогр.: 34 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT demaiou asymptoticsolutiontoamixedboundaryvalueprobleminathickmultistructureoftype322 AT melnykta asymptoticsolutiontoamixedboundaryvalueprobleminathickmultistructureoftype322 |
| first_indexed |
2025-07-09T01:42:39Z |
| last_indexed |
2025-07-09T01:42:39Z |
| _version_ |
1837131744469319680 |
| fulltext |
Український математичний вiсник
Том 2 (2005), № 4, 463 – 481
Asymptotic Solution to a Mixed
Boundary-Value Problem in a Thick
Multi-Structure of Type 3:2:2
Umberto De Maio and Taras A. Mel’nyk
(Presented by E. Ya. Khruslov)
Abstract. The leading terms of the asymptotic expansion for the
solution to a mixed boundary value problem for the Poisson equation in
a thick multi-structure, which is the union of some domain and a large
number N of ε-periodically situated thin annular disks with variable
thickness of order ε = O(N−1), are constructed and the corresponding
estimates in the Sobolev space H1 are proved as ε → 0.
2000 MSC. 35B27, 3540, 35J25, 35C20, 35B25.
Key words and phrases. Asymptotic expansions, homogenization,
thick multi-structure.
1. Introduction and Statement of the Problem
It is an interesting problem to study the behaviour of solutions to
boundary-value problems when the domain is perturbed. There are many
kinds of the domain perturbations and we need different asymptotic meth-
ods to study boundary-value problems in perturbed domains. Numerous
monographs and papers (see, e.g., [3,5,9,10,12,16,17,19,31–33] and ref-
erences there) are devoted to asymptotic methods for the investigation of
boundary-value problems in domains with complex dependence on a pa-
rameter of perturbation (perforated domains, partly perforated domains,
lattice frames, junctions of domains with different limit dimensions, etc.).
Boundary-value problems in thick multi-structures (components of
such junctions infinitely increases as the perturbation parameter ε → 0)
have own specific difficulties and such problems deserve special attention.
As it was shown in E. Sanchez-Palencia’s papers [13, 32], such problems
lose the coercitivity as ε → 0 and this creates special difficulties in the
Received 18.11.2004
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
464 Asymptotic Solution...
asymptotic investigation. In [21]– [28], a classification of such thick multi-
structures was given and basic results were obtained for boundary-value
problems in thick junctions of different types. It was shown that qual-
itative properties of solutions essentially depend on the junction type
and on the conditions given on the boundaries of the attached thin do-
mains. There and in [11] a survey of results obtained in this direction
is presented. Here we mention only the pioneer papers [15, 34], where
the asymptotic behaviour of Green’s function of the Neumann problem
for the Helmholtz equation in an unbounded junction body of type 3:2:1
was studied, and the papers [6,7], where the homogenization of nonlinear
problems in thick junctions of types 3:2:1, 2:2:1 was made.
Such thick junctions are prototypes of widely used engineering con-
structions, industrial installations, spaceship grids as well as many other
physical and biological systems with very different characteristic scales.
Despite the enormous growth in computational power, it is often im-
possible to represent a complete system at the finest scale for which the
various constitutive elements may suitably be represented. Increase in
the size of computational domains for thick multi-structures naturally
leads to longer computing times and makes it very difficult to maintain
an acceptable level of accuracy.
Thus, asymptotic analysis of boundary value problems in such do-
mains is an important task for applied mathematics. The aim of the
analysis is to develop rigorous asymptotic methods for boundary value
problems in thick junctions of different types as ε → 0, i.e., when the
number of attached thin domains increases and their thickness decreases.
In this paper we consider a model thick multi-structure (thick junc-
tion) Ωε of type 3:2:2. It consists of a junction’s body
Ω0 =
{
x ∈ R
3 : 0 < x2 < l, r :=
√
x2
1 + x2
3 < a0
}
and a large number N of thin annular disks G(ε) =
⋃N−1
j=0 Gj(ε),
Gj(ε) = {x ∈ R
3 : −ε h−(r) < x2−ε(j+1/2) < εh+(r), a0 ≤ r < a1},
i.e.,
Ωε = Ω0
⋃
G(ε).
Here h− and h+ are piecewise smooth functions on [a0, a1], 0 < h±(r) <
1
2 for r ∈ [a0, a1], and the functions h− and h+ are locally constant and
equal at an enough small neighborhood of the point a0; the number of the
thin disks is equal to a large even integer N, therefore, ε = l/N is a small
parameter, which characterizes the distance between the neighboring thin
U. De Maio, T. A. Mel’nyk 465
disks and their thickness. The type 3:2:2 of the junction refers to the limit
dimensions of the junction’s body, the joint zone (the lateral surface of
Ω0), and each attached thin disk.
In Ωε we consider the mixed boundary value problem
−∆xuε(x) = fε(x), x ∈ Ωε,
uε(x) = 0, x ∈ S(0) ∪ S(l),
∂νuε(x) = 0, x ∈ ∂Ωε \
(
S(0) ∪ S(l)
)
,
(1.1)
where ∂ν = ∂/∂ν is the outward normal derivative; S(0) = {x ∈ ∂Ω0 :
x2 = 0}, S(l) = {x ∈ ∂Ω0 : x2 = l} and the right-hand side fε ∈ L2(Ωε).
The aim of the paper is to construct the leading terms of the asymp-
totic expansion for the solution to problem (1.1) and to prove the cor-
responding asymptotic estimates as ε → 0, i.e., when the number of
attached thin disks increases and their thickness decreases.
There are two ways of investigation of boundary-value problems in
perturbed domains. The first one lies in the proofs of convergence theo-
rems with the help of special extension operators. The second one consists
in the construction the leading terms of the asymptotic expansions for
the solutions and proving the corresponding asymptotic estimates. For
these two ways we need different assumptions for the right-hand sides
and the geometrical structure of the perturbed domains.
To construct the leading terms of asymptotics of solutions to
boundary-value problems in thick junctions, the method of matched
asymptotic expansions and asymptotic methods for thin domains were
used in [21]– [28]. It is turned out that the corresponding limit problem
is derived from limit problems for each domain forming the thick junc-
tion with the help of the solutions to junction-layer problems around the
joint zone. However, the junction-layer solutions behave as powers (or
logarithm) at infinity and do not decrease exponentially. Therefore, they
influence directly the leading terms of the asymptotics.
Regarding the approximations of the solutions it should be mentioned
that in [1,2] the correctors for the solutions to the Laplace equation in a
plane thick junction of type 2:1:1 with the Dirichlet condition were con-
structed outside a layer of width 2ε in a neighborhood of the joint zone.
But for applied problems, it is very important to construct the asymp-
totic expansion for the solution and to prove the asymptotic estimates in
all thick junction, especially in a neighborhood of the joint zone, since
the solution has singularities in the joint zone.
It should be emphasized new moments in the investigation of problem
(1.1). The first one is type 3:2:2; only spectral problems were considered
in such multi-structures (see [4,27]). Secondly, only boundary value prob-
466 Asymptotic Solution...
lems in thick junctions with attached thin domains whose thickness are
unvarying or problems in domains with rapidly oscillating smooth bound-
aries (see [8]) were considered till now. Our thick junction Ωε has the
Lipschitz boundary and the thickness of each thin rings is equal to the
value εh0(r), r ∈ [a0, a1]. Because of this, the coefficients of the corre-
sponding limit problem depend on the function h0 = h− + h+. Finally,
we require for the right-hand side fε of problem (1.1) only the minimal
condition, namely, fε ∈ L2(Ωε). For comparison, in [8] the authors as-
sumed that the right-hand side f does not depend on ε and f ∈ H1(Ωε)
to prove only the convergence theorem for the solution of the Neumann
problem in a plane thick junction of type 2:1:1; in [2] to construct the
leading terms of asymptotics for the solution to the Dirichlet problem in
a plane thick junction of type 2:1:1, the right-hand side f must vanish on
the thin rods and f ∈ H4 in the junction’s body, in [23] fε has special
form like in Corollary 2.
The convergence theorem for the solution to the Neumann problem
for the Ukawa equation in Ωε, when h− = h+, was proved by authors
in [11].
2. Formal Asymptotic Representation for the Solution
In this section, we construct the leading terms of outer expansions
both in the junction’s body and in each thin disks as well as the leading
terms of an inner expansion in neighborhood of the joint zone. Then
using the method of matched asymptotic expansions, the corresponding
limit problem is derived. Also we assume in this section that the right-
hand side in (1.1) is independent of ε, i.e., fε = f0 and f0 is smooth in
Ω1, where Ω1 is the interior of Ω0 ∪D. The domain D = {x : 0 < x2 < l,
a0 < r < a1} is filled up by the thin disks in the limit passage.
2.1. Outer Expansions
We seek the leading terms of the asymptotics for the solution uε,
restricted to Ω0, in the form
uε(x) ≈ v+
0 (x) +
∞∑
k=1
εkv+
k (x, ε), (2.1)
and, restricted to Gj(ε), in the form
uε(x) ≈ v−0 (x) +
∞∑
k=1
εkv−k (x, ξ2 − j), ξ2 = ε−1x2 . (2.2)
U. De Maio, T. A. Mel’nyk 467
The expansions (2.1) and (2.2) are usually called outer expansions.
Substituting the series (2.1) in the equation of problem (1.1) and in
the boundary conditions on the bases S(0), S(l) of the cylinder Ω0 and
collecting the coefficients of the same powers of ε, we get the following
relations for the function v+
0 :
−∆xv
+
0 (x) = f0(x), x ∈ Ω0, v+
0 (x) = 0, x ∈ S(0) ∩ S(l). (2.3)
Now we find limit relations in the domain D. Assuming for the mo-
ment that the functions v−k in (2.2) are smooth, we write their Taylor
series with respect to the x2 at the point x2 = ε(j + 1
2) and pass to the
"fast" variable ξ2 = ε−1x2. Then (2.2) takes the form
uε(x) = v−0
(
x1, ε(j + 1/2), x3
)
+
+∞∑
k=1
εkV j
k (x1, ξ2, x3), x ∈ Gj(ε), (2.4)
where
V j
k = v−k
(
x1, ε(j + 1/2), x3, ξ2
)
+
k∑
m=1
(ξ2 − j − 1
2)m
m!
∂mv−k−m
∂xm2
(
x1, ε(j + 1/2), x3, ξ2
)
. (2.5)
Let us substitute (2.4) into (1.1) instead of uε. Since the Laplace
operator takes the form ∆x = ε−2 ∂2
∂ξ22
+ ∆x̃, where x̃ = (x1, x3), and the
outward normal to the lateral surface of the thin disk Gj(ε) (beside some
set of zero measure) has the form
ν±(x̃, ε) =
1√
1 + ε2|h′±(r)|2
(
−εh′±(r)
x1
r
, ±1, −εh′±(r)
x3
r
)
,
the collection of coefficients of the same power of ε gives us one dimen-
sional Neumann problems with respect to ξ2. Write the first two ones:
∂2
ξ2ξ2V
j
1 (x̃, ξ2) = 0, ξ2 ∈ Ij(r), ∂ξ2V
j
1
(
x̃, j+2−1±h±(r)
)
= 0; (2.6)
− ∂2
ξ2ξ2V
j
2 (x̃, ξ2) = ∆x̃ v
−
0 (x̃, ε(j + 1/2))
+ f0(x̃, ε(j + 1/2)), ξ2 ∈ Ij(r), (2.7)
∂ξ2V
j
2
(
x̃, j + 2−1 ± h±(r)
)
= ±∇x̃ h±(r) · ∇x̃ v
−
0 (x̃, ε(j + 1/2)), (2.8)
where Ij(r) :=
(
j + 1
2 − h−(r) , j + 1
2 + h+(r)
)
, ∂ξ2 = ∂/∂ξ2, ∂
2
ξ2ξ2
=
∂2/∂ξ22 .
468 Asymptotic Solution...
From (2.6), it follows that the function V j
1 does not depend on ξ2.
We restrict ourselves to the leading term of the asymptotics, and thus set
V j
1 = 0. Then, by virtue of (2.5), we have
v−1
(
x1, ε(j+1/2), x3, ξ2
)
= −∂x2v
−
0
(
x1, ε(j+1/2), x3
)(
ξ2−j−1/2
)
, (2.9)
where ∂x2 = ∂/∂x2.
The solvability condition for the problem (2.7)–(2.8) is given by the
differential equation
−divx̃
(
h0(r)∇x̃ v
−
0 (x̃, ε(j + 1/2))
)
= f0(x̃, ε(j + 1/2)), r ∈ (a0, a1),
(2.10)
where h0(r) = h−(r) + h+(r), r ∈ [a0, a1]. Since these planes x2 =
ε(j + 1/2), j = 0, 1, . . . , N − 1, make up the ε-net in D, we can spread
this equation in all domain D. Due to the Neumann conditions for the
solution to problem (1.1) we must require from v−0 to satisfy the condition
∂rv
−
0 (x) = 0, r = a1, x2 ∈ (0, l), (2.11)
where ∂r = ∂/∂r is the derivative with respect to polar radius r which
coincides with the outward normal derivative in this case.
So, it remains to provide the continuity of the asymptotic approxima-
tion and their gradients in the joint zone Γ0 := ∂Ω0 \
(
S(0) ∪ S(l)
)
. It is
doubtless the condition
v+
0 (x) = v−0 (x), x ∈ Γ0. (2.12)
To get the second transmission condition we should use the method of
matched asymptotic expansions for the outer expansions (2.1), (2.2) and
an inner expansion which we will construct in the following section.
2.2. Inner Expansion
In a neighborhood of Γ0 we consider the Laplace operator in the
cylindrical coordinates r, ϕ, x2, where r =
√
x2
1 + x2
3 and tan(ϕ) = x3/x1,
and then pass to the “rapid” coordinates ξ = (ξ1, ξ2), where ξ1 = −ε−1(r−
a0) and ξ2 = ε−1x2. Then Laplace’s operator in the coordinates ξ =
(ξ1, ξ2), ϕ has the following form
ε−2
( ∂2
∂ξ21
+
∂2
∂ξ22
)
− ε−1
a0 − εξ1
∂
∂ξ1
+
1
(a0 − εξ1)2
∂2
∂ϕ2
. (2.13)
We seek the leading terms of the inner expansion in a neighborhood
of Γ0 in the form
uε(x) ≈ v+
0 (x)|r=a0 + ε
(
Z1(ξ)(∂x2v
+
0 (x))|r=a0
+ Z2(ξ)(∂rv
+
0 (x))|r=a0
)
+ . . . . (2.14)
U. De Maio, T. A. Mel’nyk 469
Substituting (2.14) in (2.13) and in the Neumann condition, collecting the
coefficients of the same power of ε, we arrive at junction-layer problems
for the functions Z1 and Z2 :
−∆ξ1ξ2 Zi(ξ) = 0, ξ ∈ Π,
∂ξ1Zi(0, ξ2) = 0, (0, ξ2) ∈ ∂Π+ \ Ih,
∂ξ2Zi(ξ) = −δ1i, ξ ∈ ∂Π− \ Ih,
∂kξ2Zi(ξ1, 0) = ∂kξ2Zi(ξ1, 1), ξ1 > 0, k = 0, 1.
(2.15)
Here Π is the union of semi-infinite strips Π+ = (0,+∞) × (0, 1) and
Π− = (−∞, 0] × Ih, where Ih =
(
(1 − h)/2 , (1 + h)/2
)
, the constant
h is equal to h0(a0). The last periodic condition in (2.15) due to the
periodicity of the thin disks {Gj(ε) : j = 0, . . . , N − 1}.
The same junction-layer problems were investigated in [22]. The main
asymptotic relations for the functions {Zi} can be obtained from general
results about the asymptotic behaviour of solutions to elliptic problems
in domains with different exits to infinity [14, 29]. However, using the
symmetry of the domain Π, we can define more exactly the asymptotic
relations and detect other properties of the junction-layer solutions Z1, Z2
similarly as in the papers [22,23].
Statement 2.1 ([22]). There exist solutions Zi ∈ H1
loc,η2
(Π), i = 1, 2,
of problems (2.15), which have the following differentiable asymptotics
Z1(ξ) =
{
O(exp(−2πξ1)), ξ1 → +∞,
−ξ2 + 1
2 + O(exp(πh−1ξ1)), ξ1 → −∞;
(2.16)
Z2(ξ) =
{
−ξ1 + ch + O(exp(−2πξ1)), ξ1 → +∞,
−h−1ξ1 + O(exp(πh−1ξ1)), ξ1 → −∞.
(2.17)
In addition, the function Z1 is odd in ξ2 and Z2 is even in ξ2 with respect
to 1/2.
Now we verify the matching conditions for the outer expansions (2.1),
(2.2) and the inner expansion (2.14), namely, the leading terms of the
asymptotics of the outer expansions as r → a0 ± 0 must coincide with
the leading terms of the inner expansion as ξ1 → ∓∞ respectively. Near
the point x ∈ Γ0 the function v+
0 has the following asymptotics
v+
0 (x) = v+
0 (x)|r=a0 − ε ξ1 (∂rv
+
0 (x))|r=a0 + O(ε2ξ21), r → a0 − 0.
Taking into account the asymptotics of Z1 and Z2 as ξ1 → +∞, we see
that the matching conditions are satisfied for the expansion (2.1) and
470 Asymptotic Solution...
(2.14). The asymptotics of (2.2) as r → a0 + 0 and the asymptotics of
(2.14) as ξ1 → −∞ are the following
v−0 (x)|r=a0 + ε
(
Y (ξ2)(∂x2v
−
0 (x))|r=a0 − ξ1 (∂rv
−
0 (x))|r=a0
)
+ . . . ,
r → a0 + 0;
v+
0 (x)|r=a0 + ε
(
Y (ξ2)(∂x2v
+
0 (x))|r=a0 − h−1ξ1 (∂rv
+
0 (x))|r=a0
)
+ . . . ,
ξ1 → −∞.
Comparing the main terms of these asymptotics, we get the first trans-
mission condition (2.12) and the second one
∂rv
+
0 (x) = h ∂rv
−
0 (x), x ∈ Γ0. (2.18)
So, the function
v0(x) =
{
v+
0 (x), x ∈ Ω0,
v−0 (x), x ∈ D,
(2.19)
must satisfy the relations (2.3), (2.10)–(2.12), (2.18), which form the limit
problem
−∆xv
+
0 (x) = f0(x), x ∈ Ω0,
−divx̃
(
h0(r)∇x̃ v
−
0 (x)
)
= h0(r) f0(x), x ∈ D,
∂rv
−
0 (x) = 0, r = a1, x2 ∈ (0, l),
v+
0 (x) = 0, x ∈ S(0) ∪ S(l),
v+
0 (x) = v−0 (x), x ∈ Γ0,
∂rv
+
0 (x) = h0(a0) ∂rv
−
0 (x), x ∈ Γ0,
(2.20)
for problem (1.1).
Let us show that there exist a unique weak solution v ∈ H0 to problem
(2.20) if f0 ∈ L2(Ω1). Here
H0 :=
{
ϕ ∈ L2(Ω1) : ϕ ∈ H1(Ω0), ϕ = 0 on S(0) ∪ S(l),
∂x1ϕ ∈ L2(D), ∂x3ϕ ∈ L2(D), ϕ|r=a0−0 = ϕ|r=a0+0 on Γ0
}
is an anisotropic Sobolev space with the scalar product
(
ϕ,ψ
)
H0
=
∫
Ω0
∇xϕ · ∇xψ dx+
∫
D
h0(r)∇x̃ ϕ · ∇x̃ ψ dx.
U. De Maio, T. A. Mel’nyk 471
A function v ∈ H0 is called a weak solution to problem (2.20) if it
satisfies the following integral identity
(
v, ψ
)
H0
=
∫
Ω0
f0(x)ψ(x) dx+
∫
D
h0(r) f0(x)ψ(x) dx, ∀ψ ∈ H0. (2.21)
Next using standard Hilbert space methods and the Lax–Milgram
lemma, it is easy to prove the existence and uniqueness of the weak
solution to problem (2.20).
3. Corrector and Asymptotic Estimates
Let f0 be a function in H3(Ω1). Assume that f0 and ∂x2f0 vanish
on S(0) ∪ S(l). Let v0 ∈ H0 be the unique weak solution to problem
(2.20) with the right-hand side f0. With the help of v0 and the junction-
layer solutions Z1, Z2 (see Statement 2.1), we define the leading terms in
(2.1), (2.2) and (2.14). Then matching these expansions, we construct an
asymptotic approximation Rε belonging to the Hilbert space
Hε := {u ∈ H1(Ωε) : u = 0 on S(0) ∪ S(l)}.
It is equal to
R+
ε (x) := v+
0 (x) + εχ0(r)N+
(
−r − a0
ε
,
x2
ε
, ϕ, x2
)
, x ∈ Ω0; (3.1)
and to
R−
ε (x) := v−0 (x) + ε
(
Y
(x2
ε
)
∂x2v
−
0 (x)
+ χ0(r)N−
(
−r − a0
ε
,
x2
ε
, ϕ, x2
))
, x ∈ D, (3.2)
where Y (ξ2) = −ξ2 + 1
2 + [ξ2]; χ0 ∈ C∞
0 (R) is a cut-off function such
that
χ0(r) =
{
1, |r − a0| ≤ σ/2,
0, |r − a0| ≥ σ,
where σ is an enough small fixed positive number such that the functions
h± are constant and equal on the segment [a0, a0 + σ];
N+(ξ, ϕ, x2) = Z1(ξ) (∂x2v
+
0 (x))|r=a0 +
(
Z2(ξ) + ξ1
)
(∂rv
+
0 (x))|r=a0 ,
N−(ξ, ϕ, x2) =
(
Z1(ξ) − Y (ξ2)
)
(∂x2v
+
0 (x))|r=a0
+
(
Z2(ξ) + h−1ξ1
)
(∂rv
+
0 (x))|r=a0 ,
ξ1 = −ε−1(r − a0), ξ2 = ε−1x2, r =
(
x2
1 + x2
3
)1/2
, tan(ϕ) = x3/x1.
472 Asymptotic Solution...
3.1. Discrepancies in the Domain Ω0
Taking into account the properties of the functions Z1 and Z2, we
conclude that
R+
ε (x) = 0, x ∈ S(0) ∪ S(l),
∂rR
+
ε (x) = −∂ξ1Z1(0, x2/ε)
(
∂x2v
+
0 (x)
)
|r=a0
− ∂ξ1Z2(0, x2/ε)
(
∂rv
+
0 (x)
)
|r=a0 = 0
for any x ∈ ∂Ωε ∩ {r = a0}.
Observing that
[∆x̃ , χ0(r)]Y (x) = ∇x̃ ·
(
Y (x)∇x̃ χ0(r)) + ∇x̃Y (x) · ∇x̃ χ0(r),
where [A,B] = AB − BA is the commutator of two operators A and
B, and taking into account form (2.13) of Laplace’s operator, we get
− ∆xR
+
ε (x) − fε(x) = f0(x) − fε(x)
− χ0(r)
(
∂2
x2ξ2N+(ξ, ϕ, x2) − r−1∂ξ1N+(ξ, ϕ, x2)
)
−∇ξN+(ξ, ϕ, x2) · ∇x̃ χ0(r)
− ε
(
∇x̃ ·
(
N+∇x̃ χ0(r)
)
+ χ0(r)∂
2
x2x2
N+(ξ, ϕ, x2)
+ r−2χ0(r)∂
2
ϕϕN+(ξ, ϕ, x2)
)
,
ξ1 = −r − a0
ε
, ξ2 =
x2
ε
, r =
√
x2
1 + x2
3, tan(ϕ) =
x3
x1
, x ∈ Ω0.
(3.3)
Further, the arguments of functions involved in calculations are in-
dicated only if their absence may cause confusion. We multiply identity
(3.3) by a test function ψ ∈ Hε and integrate by parts in Ω0 :
−
∫
Qε
∂rR
+
ε ψ dSx +
∫
Ω0
∇xR
+
ε · ∇xψ dx−
∫
Ω0
fεψ dx
= I+
1 (ε, ψ) + . . .+ I+
4 (ε, ψ), (3.4)
where Qε = Ωε ∩ {r = a0},
I+
1 (ε, ψ) =
∫
Ω0
(
f0(x) − fε(x)
)
ψ dx,
I+
2 (ε, ψ) =
∫
Ω0
χ0(r)
(
r−1∂ξ1N+ − ∂2
x2ξ2N+
)
ψ dx,
U. De Maio, T. A. Mel’nyk 473
I+
3 (ε, ψ) = ε
∫
Ω0
N+ ∇x̃ χ0(r) · ∇x̃ψ dx−
∫
Ω0
ψ∇ξN+ · ∇x̃ χ0(r) dx,
I+
4 (ε, ψ) = ε
∫
Ω0
χ0(r) (∂x2ψ ∂x2N+ + r−2∂x2ψ ∂ϕN+) dx.
3.2. Discrepancies in the Thin Disks
Denote by S+
j (ε) and S−
j (ε) the right and left lateral surfaces of the
thin disk Gj(ε) respectively;
S+(ε) :=
N−1⋃
j=0
S+
j (ε), S−(ε) :=
N−1⋃
j=0
S−
j (ε).
It is easy to calculate that
(
∂rR
−
ε
)
|r=a1 = 0,
∂rR
−
ε (x) = εY (x2/ε) ∂r(∂x2v
−
0 (x)) + ∂rR
+
ε (x), x ∈ Qε, (3.5)
and
∂νR
−
ε (x) =
1√
1 + ε2|h′±(r)|2
(
−ε∇x̃(h±) · ∇x̃
(
v−0 + εY (ξ2)∂x2v
−
0
)
± ε
(
Y (ξ2)∂
2
x2x2
v−0 (x) + χ0(r)∂x2N−(ξ, ϕ, x2)
))
, x ∈ S±(ε). (3.6)
Putting R−
ε in the differential equation of problem (1.1), we obtain
− ∆xR
−
ε (x) − fε(x) = f0(x) − fε(x) + ∇x̃(lnh0) · ∇x̃v
−
0
+ χ0(r)
(
r−1∂ξ1N− − ∂ξ2N−
)
−∇ξN− · ∇x̃ χ0(r)
− ε∂x2
(
Y
(x2
ε
)
∂2
x2x2
v−0 + χ0(r)
(
∂x2N−
)
|ξ2=x2/ε
)
− ε
(
Y
(x2
ε
)
∆x̃
(
∂x2v
−
0
)
+ ∇x̃ ·
(
N−∇x̃ χ0(r)
)
+ r−2χ0(r) ∂
2
ϕϕN−
)
,
x∈G(ε). (3.7)
Next we will use the following identity
∫
S±
ε
εh±(r)√
1 + ε2|h′±(r)|2
ψ dSx =
∫
Gε
ψ dx− ε
∫
Gε
Y
(x2
ε
)
∂x2ψ dx ∀ψ ∈ Hε.
(3.8)
To prove (3.8) it is enough to integrate by part the last integral.
474 Asymptotic Solution...
Using (3.8) and taking into account the boundary values of ∂νR
−
ε (see
(3.5), (3.6)), we multiply (3.7) by a test function ψ ∈ Hε and integrate
by parts in Gε. This yields
∫
Qε
∂rR
+
ε ψ dSx+
∫
Gε
∇xR
−
ε ·∇xψ dx−
∫
Gε
fεψ dx = I−1 (ε, ψ)+. . .+I−5 (ε, ψ),
(3.9)
where
I−1 (ε, ψ) =
∫
Gε
(
f0(x) − fε(x)
)
ψ dx,
I−2 (ε, ψ) =
∫
Gε
ψ χ0(r)
(
r−1∂ξ1N− − ∂ξ2N−
)
dx,
I−3 (ε, ψ) = ε
∫
Gε
N−∇x̃ χ0(r) · ∇x̃ ψ dx−
∫
Gε
ψ∇ξN− · ∇x̃ χ0(r) dx,
I−4 (ε, ψ) = ε
∫
Gε
χ0(r)
(
∂x2ψ ∂x2N− + r−2∂ϕψ ∂ϕN−
)
dx,
I−5 (ε, ψ) = ε
∫
Gε
Y
(x2
ε
)(
∇x
(
∂x2v
−
0
)
· ∇xψ+ ∂x2
(
ψ∇x̃
(
lnh0
)
· ∇x̃v
−
0
))
dx.
3.3. Asymptotic Estimates
Summing (3.4) and (3.9), we see that the function Rε constructed by
formulas (3.1) and (3.2) satisfies the following integral identity
∫
Ωε
∇xRε · ∇xψ dx−
∫
Ωε
fεψ dx = Fε(ψ), ∀ψ ∈ Hε,
where Fε(ψ) = I±1 (ε, ψ) + . . . + I±4 (ε, ψ) + I−5 (ε, ψ); I±i = I+
i + I−i ,
i = 1, . . . , 4.
Since the weak solution uε ∈ Hε to problem (1.1) satisfies the integral
identity
∫
Ωε
∇xuε · ∇xψ dx−
∫
Ωε
fεψ dx = 0, ∀ψ ∈ Hε,
we have ∫
Ωε
∇x
(
Rε − uε
)
· ∇xψ dx = Fε(ψ), ∀ψ ∈ Hε. (3.10)
Now we are going to estimate the value Fε(ψ).
U. De Maio, T. A. Mel’nyk 475
The sum I±1 (ε, ψ) is a linear bounded functional on Hε. Thus,
|I±1 (ε, ψ)| = ‖fε − f0‖∗ ‖ψ‖H1(Ωε),
where
‖fε − f0‖∗ = sup
ψ∈Hε,
‖ψ‖
H1(Ωε)
=1
|(fε − f0, ψ)L2(Ωε)|.
Obviously, ‖fε − f0‖∗ ≤ C1‖fε − f0‖L2(Ωε). Here and in what follows, all
constants in asymptotic inequalities are independent of the parameter ε.
In order to estimate the terms I+
2 (ε, ψ), I−2 (ε, ψ), we will use the
following lemma.
Lemma 3.1. Let N be an 1-periodic in ξ2 function belonging to the space
L2(Π) and exponentially decreasing at infinity, i.e., there exist positive
constants c,R, γ such that for any |ξ1| ≥ R
∣∣N (ξ)
∣∣ ≤ c exp(−γ|ξ1|).
Then for any δ > 0 there exist positive constants c1, ε0 such that for all
values ε ∈ (0, ε0) the following inequality is valid
∣∣∣∣∣
∫
Ωε
N
(
−r − a0
ε
,
x2
ε
)
ψ(x) dx
∣∣∣∣∣ ≤ c1ε
1−δ‖ψ‖H1(Ωε), ∀ψ ∈ Hε.
Proof. Set Bε,δ = Ωε ∩ {x : |r − a0| ≤ ε1−2δ} for any δ > 0. Then
∣∣∣∣∣
∫
Ωε
N
(
−r − a0
ε
,
x2
ε
)
ψ dx
∣∣∣∣∣ ≤
∣∣∣∣∣
∫
Bε,δ
N
(
−r − a0
ε
,
x2
ε
)
ψ dx
∣∣∣∣∣
+
∣∣∣∣∣
∫
Ωε\Bε,δ
N
(
−r − a0
ε
,
x2
ε
)
ψ dx
∣∣∣∣∣.
The properties of the function N lead us to the conclusion that the
second summand in this inequality decreases exponentially as ε → 0.
Using Lemma 1.5 ([18]), we estimate the first summand:
∣∣∣∣∣
∫
Bε,δ
N
(
−r − a0
ε
,
x2
ε
)
ψ dx
∣∣∣∣∣
≤
( ∫
Bε,δ
N 2
(
−r − a0
ε
,
x2
ε
)
dx
)1/2
‖ψ‖L2(Bε,δ)
≤ c2 ε
1/2‖N‖L2(Π)c3ε
−δ+1/2‖ψ‖H1(Ωε).
The lemma is proved.
476 Asymptotic Solution...
Since the functions ∂ξ1N±, ∂2
x2ξ2
N−, ∂ξ2N+ exponentially decrease
as |ξ1| → +∞, we deduce from Lemma 3.1 that for any fixed δ > 0
|I+
2 (ε, ψ) + I−2 (ε, ψ)| ≤ ε1−δ C2‖ψ‖H1(Ωε). (3.11)
Integrals in I+
3 (ε, ψ), I−3 (ε, ψ) are, in fact, over
supp
(
∇x̃ χ0(r)
)
∩ Ωε = {x : σ/2 < |r − a0| < σ } ∩ Ωε,
where, by virtue of Statement 2.1, the functions N−,∇ξN± are expo-
nentially small and the function N+ is uniformly bounded with respect
to ε. Thus,
|I+
3 (ε, ψ) + I−3 (ε, ψ)| ≤ εC3‖ψ‖H1(Ωε).
Integrals in I+
4 (ε, ψ), I−4 (ε, ψ) are over {x : |r − a0| < σ } ∩ Ωε and
they can be estimated with extracting, if necessary, the exponentially
decreasing part in the corresponding integrand and then with the help of
the Cauchy–Bunyakovsky inequality. Consider for example the integral
∣∣∣∣∣
∫
Ω0
χ0(r) ∂x2ψ ∂x2N+ dx
∣∣∣∣∣ =
∣∣∣∣∣
∫
Ω0
χ0(r) ∂x2ψ
(
Z1 (∂2
x2x2
v+
0 (x))|r=a0
+
(
Z2 − ε−1(r − a0)
)
(∂x2∂rv
+
0 (x))|r=a0
)
dx
∣∣∣∣∣
≤
∫
Ω0
χ0(r) |∂x2ψ| |Z1| |∂2
x2x2
v+
0 (x)|
∣∣
r=a0
dx
+
∫
Ω0
χ0|∂x2ψ|
(
|Z2 − ε−1(r − a0) − ch| |∂x2∂rv
+
0 (x)|
∣∣
r=a0
+ |ch| |∂x2∂rv
+
0 (x)|
∣∣
r=a0
)
dx ≤ c‖ψ‖H1(Ω0)
(√∫
Ω0
χ0 |Z1|2 dx
+
√∫
Ω0
χ0 |Z2 − ε−1(r − a0) − ch|2 dx+ |ch|
√
|Ω0|
)
≤ c‖ψ‖H1(Ω0)
(√
2πa0lε
∫
Π+ Z
2
1 (ξ) dξ
+
√
2πa0lε
∫
Π+ Z
2
2 (ξ + ξ1 − ch) dξ + |ch|
√
|Ω0|
)
≤ c‖ψ‖H1(Ω0)
(√
ε ‖Z1‖L2(Π+) +
√
ε ‖Z2+ξ1−ch‖L2(Π+) + |ch|
√
|Ω0|
)
,
where |Ω0| is the measure of the domain Ω0. On the basis of (2.16) and
(2.17) the value ‖Z1‖L2(Π+) and ‖Z2 + ξ1 − ch‖L2(Π+) are bounded. As a
U. De Maio, T. A. Mel’nyk 477
result, we have
|I+
4 (ε, ψ) + I−4 (ε, ψ)| ≤ εC4‖ψ‖H1(Ωε). (3.12)
Remark 3.1. The constants C2 and C3 in (3.11) and (3.12) respectively
depend on the following quantities
sup
x∈Γ0
∣∣Dα
(
v+
0 (x)
)∣∣, |α| = α1 + α2 + α3 ≤ 2. (3.13)
Applying the odd extension to the limit problem (2.20) with respect to
the planes x2 = 0, x2 = l and taking into account the conditions for the
function f0, we conclude that the function v+
0 and its second derivatives
have no singularities at the points on S(0) ∩ Γ0 and on S(l) ∩ Γ0. Thus,
by virtue of classical results on the smoothness of solutions to boundary
value problems, the quantities (3.13) are bounded.
Since f0 ∈ H3(Ω1), the function ∂x2v
−
0 ∈ H1(Ω1 \ Ω0). Therefore,
|I−5 (ε, ψ)| ≤ εC5 ‖∂x2v
−
0 ‖H1(Ω1\Ω0) ‖ψ‖H1(Ω1\Ω0).
So, with regard to the inequalities obtained, we conclude that for the
right-hand side in (3.10) the following inequality holds
|Fε(ψ)| ≤ c(δ) ε1−δ‖ψ‖H1(Ωε), (3.14)
where δ is an arbitrary fixed positive number. From (3.10) and (3.14) it
follows the following results.
Theorem 3.1. Suppose fε ∈ L2(Ωε), f0 ∈ H3(Ω1) and f0, ∂x2f0 vanish
on S(0) ∪ S(l).
Then for any δ > 0 there exist positive constants c1, ε0 such that
for all values ε ∈ (0, ε0) the difference between the solution uε to prob-
lem (1.1) and the approximation function Rε defined by (3.1) and (3.2)
satisfies the following estimate
‖uε −Rε‖H1(Ωε) ≤ c1
(
ε1−δ + ‖fε − f0‖∗
)
. (3.15)
Corollary 3.1. From (3.15) it follows that
‖uε − v0‖L2(Ωε) ≤ c2
(
ε1−δ + ‖fε − f0‖∗
)
,
where v0 is the weak solution to the limit problem (2.20).
Corollary 3.2. Assume fε(x) = f0(x) + εf1(x, ε), x ∈ Ωε, where the
norm ‖f1(·, ε)‖L2(Ωε) = O(1) as ε → 0. Then for any δ > 0 there exist
positive constants c3, ε0 such that for all values ε ∈ (0, ε0)
‖uε −Rε‖H1(Ωε) ≤ c3 ε
1−δ, ‖uε − v0‖L2(Ωε) ≤ c3 ε
1−δ.
478 Asymptotic Solution...
Example. If the right-hand side f0 of the limit problem (2.20) depends
only on the variable r and x2, then we can find the explicit solution in
the domain D = {x : r ∈ (a0, a1), x2 ∈ (0, l)} and reduce problem
(2.20) to a problem in the junction’s body Ω0.
In this case the solution to the limit problem (2.20) depends only on
the variable r and x2 as well. So, we can rewrite the limit problem in
the following form
−∂r
(
r∂rv
+
0
)
+ r∂2
x2x2
v+
0 = rf0(r, x2), x ∈ Ω0,
−∂r
(
rh0(r)∂rv
−
0
)
= rh0(r)f0(r, x2), x ∈ D,
∂rv
−
0 (a1, x2) = 0, x2 ∈ (0, l),
v+
0 (r, 0) = v+
0 (r, l) = 0, x ∈ S(0) ∪ S(l),
v+
0 (a0, x2) = v−0 (a0, x2), x2 ∈ (0, l),
∂rv
+
0 (a0, x2) = h0(a0) ∂rv
−
0 (a0, x2), x2 ∈ (0, l),
(3.16)
By solving the ordinary equation of problem (3.16) in the domain D with
regard to the Neumann condition at r = a1 and to the first transmission
condition at r = a0 in the joint zone Γ0, we find that
v−0 (r, x2) = v+
0 (a0, x2) +
r∫
a0
1
ρ h0(ρ)
a1∫
ρ
t h0(t) f0(t, x2) dt dρ.
Now, according to the second transmission condition in problem (3.16),
we obtain the classical mixed boundary-value problem
−∂r
(
r∂rv
+
0
)
+ r∂2
x2x2
v+
0 = rf0(r, x2), x ∈ Ω0,
v+
0 (r, 0) = v+
0 (r, l) = 0, x ∈ S(0) ∪ S(l),
∂rv
+
0 (a0, x2) = F̂0(x2), x2 ∈ (0, l),
(3.17)
to find v+
0 . Here
F̂0(x2) = a−1
0
a1∫
a0
t h0(t)f0(t, x2) dt, x2 ∈ (0, l).
Problem (3.17) is called resulting problem for problem (1.1).
U. De Maio, T. A. Mel’nyk 479
Conclusion
We assumed that the functions h− and h+ are locally constant and
equal in an enough small neighborhood of the point a0. This is a technical
condition which allows to avoid additional bulky calculations. Because
of this, the junction-layer solutions are odd or even with respect to 1/2
(see Statement 2.1). As a result, the approximation function Rε sat-
isfies exactly some boundary conditions and we do not need additional
boundary-layer asymptotics.
If the right-hand side has the following form fε =
∑∞
k=0 ε
kfk(x),
we can define the other terms in the asymptotic expansions (2.1), (2.2),
(2.14) and construct an asymptotic approximation to any degree of ac-
curacy.
From results proved in the present paper it follows that for applied
problems in thick junctions we can use the corresponding limit problem
or resulting problem in the junction’s body, which are simpler, instead of
the initial problem with the sufficient plausibility.
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Contact information
Umberto De Maio Department of Applied Mathematics
“R. Caccioppoli”
Federico II University of Naples
Complesso Monte S. Angelo-Edificio “T”
Via Cintia, 80126 Naples,
Italy
E-Mail: udemaio@unina.it
Taras A. Mel’nyk Faculty of Mathematics and Mechanics
Taras Shevchenko University of Kyiv
Volodymyrska str. 64
01033 Kyiv,
Ukraine
E-Mail: melnyk@imath.kiev.ua
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