On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems
A perturbation of the Poisson equation by a biharmonic operator with a small multiplier ε is considered. The asymptotic behavior of the solution of the Dirichlet problem for this equation as ε → 0 is studied. The gradient of the solution is proved to converge to the gradient of the solution to Poiss...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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| Cite this: | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems / O. Anoshchenko, O. Lysenko, E. Khruslov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 115-122. — Бібліогр.: 6 назв. — англ. |
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| citation_txt | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems / O. Anoshchenko, O. Lysenko, E. Khruslov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 115-122. — Бібліогр.: 6 назв. — англ. |
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| description | A perturbation of the Poisson equation by a biharmonic operator with a small multiplier ε is considered. The asymptotic behavior of the solution of the Dirichlet problem for this equation as ε → 0 is studied. The gradient of the solution is proved to converge to the gradient of the solution to Poisson equation in L₁ (Ω) as ε → 0. The di erence of the gradients is also estimated.
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Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 2, pp. 115�122
On Convergence of Solutions of Singularly Perturbed
Boundary-Value Problems
O. Anoshchenko and O. Lysenko
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:anoshchenko@univer.kharkov.ua
lysenko.e@gmail.com
E. Khruslov
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:khruslov@ilt.kharkov.ua
Received January 27, 2009
A perturbation of the Poisson equation by a biharmonic operator with
a small multiplier " is considered. The asymptotic behavior of the solution of
the Dirichlet problem for this equation as "! 0 is studied. The gradient of
the solution is proved to converge to the gradient of the solution to Poisson
equation in L1 (
) as "! 0. The di�erence of the gradients is also estimated.
Key words: singular perturbation, elliptical equations, the Green func-
tions.
Mathematics Subject Classi�cation 2000: 35B25, 35J05, 35J75, 35J40.
1. Problem Statement and Main Result
Let
be a bounded domain in R3 with a su�ciently smooth boundary.
Consider the following boundary-value problem:(
"�2
u" ��u" = F in
;
u" = 0; @u"
@�
= 0 on @
:
(1.1)
Here � is the outer normal to @
at the point x, F 2 Lp (
)
�
p >
6
5
�
, and " > 0
is a small parameter. As known, there exists a unique solution of this problem u" 2
W
4
p (
) (see, e.g., [1]). We are interested in the asymptotic behavior of solution
c
O. Anoshchenko, O. Lysenko, and E. Khruslov, 2009
O. Anoshchenko, O. Lysenko, and E. Khruslov
of this problem when " ! 0. Similar questions for more general equations were
studied by M. Vishik and L. Lyusternik in [2], where the asymptotic expansion
with respect to the powers of " was constructed. The method proposed in the
paper was widely used at that time. However, all the known results appeared
to be not su�cient to our work. We use our result to construct the regularized
solutions of Navier�Stokes�Vlasov�Poisson boundary value problem [3].
To formulate the main result we consider the following boundary-value prob-
lem: (
�u = F in
;
u = 0 on @
;
(1.2)
where F is the same function as in (1.1). There exists a unique solution to this
problem u 2W
2
p (
) (see, e.g., [1]).
The main result of the paper is the following
Theorem 1. Let u" and u be the solutions of problems (1.1), (1.2), respec-
tively. Then
lim
"!0
Z
jru" (x)�ru (x)j dx = 0
uniformly with respect to all functions F such that kFk
Lp(
)
� C.
This theorem is proved in Sections 2 and 3.
2. Estimates of the Green Functions
Let G" (x; y) and G0 (x; y) be the Green functions of problems (1.1) and (1.2),
respectively.
Lemma 1. The following estimates for normal derivatives of the Green func-
tion G0 (x; y) hold: ����@G0
@�
(x; y)
���� � C1
jx� yj2
; y 2
; x 2 @
;
����Dk
�
@G0
@�
(x; y)
���� � C2
(d (y))2+jkj+�
; y 2
; x 2 @
;
where k = (k1; k2) is a multiindex, ki 2 Z, k1 + k2 � 1, jkj = k1 + k2, D
k
� is
a derivative at the point x 2 @
in tangent directions to @
, d (y) is a distance
from the point y 2
to @
, 0 < � < 1, C1 and C2 are constants that depend on
the minimal radius of curvature of @
, k, and � only.
116 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems
P r o o f. As is well known, the Green function G0 (x; y) has the form
G0 (x; y) =
1
4� jx� yj
� g0 (x; y) ; (2.1)
where the regular function g0 (x; y) is a solution of the following boundary-value
problem with respect to the variable x 2
(y 2
is a parameter):(
�g0 = 0 in
;
g0 =
1
4�jx�yj
on @
:
(2.2)
Let us represent g0 (x; y) as a simple layer potential
g0 (x; y) =
1
4�
Z
@
� (�; y)
jx� �j
dS�: (2.3)
The simple layer potential satis�es the Laplace equation in R3n
, tends to
zero when jxj ! 1, and it is a continuous function in x in R3 . Therefore, by
(2.2), it equals to the function 1
4�jx�yj
in R3n�
. Then its normal derivative in
R
3n
is given by
�
@g0
@�
�
e
= @
@�
1
4�jx�yj
.
Hence, taking into account the properties of the simple layer potential, we
obtain the integral equation for the density � (x; y)
1
2
� (x; y)�
1
4�
Z
@
cos � (x; �)
jx� �j2
� (�; y) dS� = �
1
4�
@
@�
1
jx� yj
; (2.4)
where � (x; �) is the angle between the outer normal to @
at the point x 2 @
and the vector x� �.
This equation corresponds to the representation of the solution to the exter-
nal Neumann boundary-value problem in the form of simple layer potential and,
therefore, it has a unique solution in the class C (@
) (see, e.g., [4]). Applying
the iteration method, we obtain the estimate
j� (x; y)j �
C
jx� yj2
: (2.5)
On the other hand, from (2.1) it is clear that�
@G0
@�
�
i
=
1
4�
@
@�x
1
jx� yj
�
�
@g0
@�
�
i
and according to the properties of the simple layer potential
�
@g0
@�
�
i
�
�
@g0
@�
�
e
=
� (x; y). Consequently, @G0
@�
= �� (x; y). So, the �rst estimate of Lemma 1 is
proved.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 117
O. Anoshchenko, O. Lysenko, and E. Khruslov
To establish the second estimate we use the Schauder estimates (see, e.g., [5])
jg0jk+�;
� C
�
j�g0j+ jg0jk+�;@
�
;
where jgj
k+�;@
is the norm of g in C
k+� (@
), k � 2, 0 < � < 1. The constant C
depends on k, �, and @
. The second estimate follows easily from (2.2). Lemma 1
is proved.
Lemma 2. The following estimate holds:
Z
jrG" (x; y)�rG0 (x; y)j dx � C
0@ 4
p
"
d (y)1+�
+
e
�
d(y)p
"
p
"d (y)
1A ; (2.6)
where d(y) is the distance from the point y to @
, 0 < � < 1, and the constant C
depends on
and � only.
P r o o f. It is easy to verify that the function
�" (x; y) =
1
4� jx� yj
�
1� e
�
jx�yjp
"
�
; " > 0 ; (2.7)
is a fundamental solution of the equation (1.1) in R3 .
As is well known, the Green function G" (x; y) can be represented in the form
G" (x; y) = �" (x; y) � g" (x; y) ; where g" (x; y) is a regular function, which is
a solution of the following boundary-value problem:(
"�2
g" ��g" = 0 in
;
g" = �";
@g"
@�
= �" on @
:
(2.8)
According to (2.2) and (2.8),
G" (x; y)�G0 (x; y) = �
e
�
jx�yjp
"
4� jx� yj
� v" (x; y) ; (2.9)
where the function v" (x; y) = g" (x; y)� g0 (x; y) is a solution of8>>>>><>>>>>:
"�2
v" ��v" = 0 in
;
v" = � e
�jx�yjp
"
4�jx�yj
jx=x(s;�) = �
0
" (s; �) on @
;
@v"
@�
=
@G0
@�
� @
@�
e
�jx�yjp
"
4�jx�yj
!!
jx=x(s;�) = �
1
" (s; �) on @
:
(2.10)
118 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems
Here we use a local coordinate system such that (s; �; t) are coordinates in
a neighborhood of x = x (s; �; 0) 2 @
, where t is the distance from the point x
to @
, and (s; �) are coordinates on @
.
Let us introduce a class of functions
fW =
�
w 2W
2
2 (
) : wj@
= �
0
" ;
@w
@�
j@
= �
1
"
�
:
By [6], v" (x; y) minimizes the functional
J" (w) =
Z
n
" (�w)2 + jrwj2
o
dx:
Then
J" (v") � J" (w") 8w" 2 fW: (2.11)
To estimate J" (v") let us construct a representative of the class fW in the form
w" (x) =
�
�
0
" (s; �) + t�
1
" (s; �)
�
'
�
t
p
"
�
; (2.12)
where ' (t) is a smooth function such that ' (t) = 1 for t � 1=2, ' (t) = 0 for
t � 1, ' (t) 2 C
2 (0;1).
Suppose that y 2
Æ �
, with
Æ being a subdomain of
,
Æ = fx 2
: dist (x; @
) > Æg ; (2.13)
where Æ satis�es the condition Æ > r@
�
p
" > 0, and r@
is the minimal radius
of curvature of the surface @
.
Then, using (2.12), the explicit expressions for the functions �0" ; �
1
" (see (2.10)),
and Lemma 1, we obtain the estimate
J" (w") � C
e
�
2dp
"
"d2
+
p
"
d2+�
!
;
where d = d(y) is the distance from y to @
, and the constant C depends on @
and �, 0 < � < 1. Therefore, it follows from (2.11) that
Z
jrv"j2 dx � J" (v") � C
e
�
2dp
"
"d2
+
p
"
d2+2�
!
:
Using this estimate and taking into account (2.9), we obtain (2.6). Lemma 2 is
proved.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 119
O. Anoshchenko, O. Lysenko, and E. Khruslov
3. Proof of Theorem 1
We begin with the following lemma.
Lemma 3. Let
be a bounded domain in R
3 with a boundary of the class
C
2+�, and '" be a solution to the following boundary-value problem:(
"�2
'" ��'" = F in
;
'" = 0; "@'"
@�
= 0 on @
;
(3.1)
where " � 0, F 2 Lp (
)
�
p >
6
5
�
with a support SF �
. ThenZ
jr'"j dx � C kFk
Lp(
)
(mesSF )
5
6
�
1
p ;
where C is a constant that does not depend on ".
P r o o f. The solution of the problem (3.1) minimizes the functional
F̂"('") =
Z
n
" (�'")
2 + jr'"j2 � 2F�F'"
o
dx
in the class of functions '" of
Æ
W
1
2 (
) for " > 0 and of W 1
2 (
) for " = 0. Here by
�F = �F (x) we denote the characteristic function of the set SF .
Since F̂" (0) = 0, then we have F̂" ('") � 0.
This leads to the inequalityZ
n
" (�'")
2 + jr'"j2
o
dx � 2
Z
jF (x)j j�F (x)j j'" (x)j dx:
Applying the H�older inequality with p; q = 6p
5p�6
, and r = 6
�
1
p
+ 1
q
+ 1
r
= 1
�
to
the right-hand side of this bound, we getZ
n
" (�'")
2 + jr'"j2
o
dx � 2 kFk
Lp(
)
kmes�F kLq(x) k'"kL6(
)
� C kFk
Lp(
)
(mesSF )
5
6
�
1
p kr'"kL2(
)
: (3.2)
Here the norm of '" 2
Æ
W
1
2 is estimated according to the embedding of
Æ
W
1
2 (
) in
L6 (
).
120 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems
From (3.2) we conclude that
kr'"kL2(
)
� C kFk
Lp(
)
(mesSF )
5
6
�
1
p :
Now this bound and the Cauchy�Schwarz inequality
Z
jr'"j dx �
0@Z
jr'"j2 dx
1A1=2
j
j1=2
yield the statement of Lemma 3.
We are now in position to complete the proof of Theorem 1.
Denote by
Æ a subdomain of
de�ned in (2.13).
Let us represent the function F (x) as a sum of three components F (x) =
F1 (x) + F2 (x) + F3 (x), where
F1 (x) 2 C1 (
), suppF1 (x) �
Æ, kF1kLp(
) � kFkLp(
);
F2 (x) : kF2kLp(
) < Æ kFk
Lp(
)
;
F3 (x) = F (x)�Æ (x), where �Æ (x) is a characteristic function of the set
n
Æ.
The solutions of problems (1.1) and (1.2) can be represented as u" = u1" +
u2" + u3"; u = u1 + u2 + u3, respectively. Then we haveZ
jru" �ruj dx �
Z
jru1" �ru1j dx+
Z
jru2"j dx
+
Z
jru2j dx+
Z
jru3"j dx+
Z
jru3j dx: (3.3)
Using Lemma 2, we estimate the �rst integral as follows:Z
jru1" �ru1j dx �
Z
jrG" (x; y)�rG0 (x; y)j jF1 (x)j dx
� C
4
p
"
Æ1+�
+
e
�
Æp
"
p
"Æ
!
kF1kLp(
) : (3.4)
To estimate the remaining integrals we use Lemma 3. Thus, we have:Z
jru2"j dx � C kF2kLp(
) j
Æj
5
6
�
1
p � CÆ kFk
Lp(
)
j
Æj
5
6
�
1
p ; (3.5)
Z
jru3"j dx � C kF3kLp(
) j
n
Æj
5
6
�
1
p � C kFk
Lp(
)
Æ
5
6
�
1
p : (3.6)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 121
O. Anoshchenko, O. Lysenko, and E. Khruslov
We set Æ = Æ (") = "
6p
4(11p�6+6�p) . Then, according to (3.3)�(3.6), we obtainZ
jru" �ruj dx � C"
kFk
Lp(
)
;
where
= 5p�6
4(11p�6+6�p)
.
Since kFk
Lp(
)
� C and p >
6
5
, thenZ
jru" �ruj dx! 0
as "! 0.
Theorem 1 is proved.
References
[1] A.I. Koshelev, A Priori Estimates in Lp. � Usp. Mat. Nauk 13 (1958), No. 4 (82),
29�88. (Russian)
[2] M.I. Vishik and L.A. Lyusternik, Regular Degeneration and Boundary Layer for
Linear Di�erential Equations with Small Parameter. � Usp. Mat. Nauk 12 (1957),
No. 5 (77), 3�122. (Russian)
[3] O. Anoshchenko, E. Khruslov, and H. Stephan, Global Weak Solutions of the
Navier�Stokes�Vlasov�Poisson System. � WIAS Prepr. No. 1335, Berlin, 2008.
[4] S.G. Mikhlin, Lectures on Integral Equations. Fizmatgiz, Moscow, 1959. (Russian)
[5] O.A. Ladyzhenskaya and N.N. Uralceva, Linear and Quasilinear Elliptical Equa-
tions. Nauka, Moscow, 1973. (Russian)
[6] S.L. Sobolev, Applications of Functional Analysis in Mathematical Physics. Nauka,
Moscow, 1988. (Russian)
122 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
|
| id | nasplib_isofts_kiev_ua-123456789-106536 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T16:51:26Z |
| publishDate | 2009 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Anoshchenko, O. Lysenko, O. Khruslov, E. 2016-09-30T06:58:38Z 2016-09-30T06:58:38Z 2009 On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems / O. Anoshchenko, O. Lysenko, E. Khruslov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 115-122. — Бібліогр.: 6 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106536 A perturbation of the Poisson equation by a biharmonic operator with a small multiplier ε is considered. The asymptotic behavior of the solution of the Dirichlet problem for this equation as ε → 0 is studied. The gradient of the solution is proved to converge to the gradient of the solution to Poisson equation in L₁ (Ω) as ε → 0. The di erence of the gradients is also estimated. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems Article published earlier |
| spellingShingle | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems Anoshchenko, O. Lysenko, O. Khruslov, E. |
| title | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems |
| title_full | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems |
| title_fullStr | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems |
| title_full_unstemmed | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems |
| title_short | On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems |
| title_sort | on convergence of solutions of singularly perturbed boundary-value problems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106536 |
| work_keys_str_mv | AT anoshchenkoo onconvergenceofsolutionsofsingularlyperturbedboundaryvalueproblems AT lysenkoo onconvergenceofsolutionsofsingularlyperturbedboundaryvalueproblems AT khruslove onconvergenceofsolutionsofsingularlyperturbedboundaryvalueproblems |