Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure
An asymptotic behavior of solution of the Cauchy problem for the wave equation is studied on the Riemannian manifold Mε depending on a small parameter ε. It is supposed that a topological type of Mε increases as ε → 0. The averaged equation is derived, it describes the asymptotic behavior of the ori...
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| Zitieren: | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure / A.V. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 213-233. — Бібліогр.: 5 назв. — англ. |
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| citation_txt | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure / A.V. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 213-233. — Бібліогр.: 5 назв. — англ. |
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| description | An asymptotic behavior of solution of the Cauchy problem for the wave equation is studied on the Riemannian manifold Mε depending on a small parameter ε. It is supposed that a topological type of Mε increases as ε → 0. The averaged equation is derived, it describes the asymptotic behavior of the original Cauchy problem as ε → 0.
|
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Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 2, pp. 213�233
Klein�Gordon Equation as a Result of Wave Equation
Averaging on the Riemannian Manifold
of Complex Microstructure
A.V. Khrabustovskyi
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:andry9@ukr.net
Received January 4, 2006
An asymptotic behavior of solution of the Cauchy problem for the wave
equation is studied on the Riemannian manifold M" depending on a small
parameter ". It is supposed that a topological type ofM" increases as "! 0.
The averaged equation is derived, it describes the asymptotic behavior of
the original Cauchy problem as "! 0.
Key words: Riemannian manifolds, wave equation, asymptotic behavior,
homogenization.
Mathematics Subject Classi�cation 2000: 35B27, 35K60.
Introduction
We denote byM "
3 the 3-dimensional Riemannian manifold depending on a small
parameter " and described in the following way. Let D"
j
be a union of "holes" in
R
3 � balls of the radius "3 with the centers x"
j
2 R3 and distributed periodically
in R3 with the period ". Let
" = R
3n
[
j
D"
j :
We consider two copies
"
1 and
"
2 of the domain
". Let
"
1 and
"
2 be the
upper and the lower sheets respectively. The boundaries of these sheets consist of
the spheres @D"
1j and @D
"
2j . We join @D"
1j and @D
"
2j by means of 3-dimensional
tubes ("wormholes") G"
j
= S"
j
� [0; 1], where S"
j
is a sphere in R3 of the radius "3.
c
A.V. Khrabustovskyi, 2007
A.V. Khrabustovskyi
Then we obtain the 3-dimensional oriented manifold
M "
3 = (
"
1 [
"
2)
[0
@N(")[
j=1
G"
j
1
A :
We introduce di�erential structure on M "
3 in the standard way (see, e.g., [1]).
This manifold is illustrated in �gure. The points of M "
3 we denote by ~x
�
1
�
�
2
�
�
jG
Figure. Manifold M "
3 .
We de�ne a Riemannian structure on M "
3 by the metric tensor g"
ik
(~x) depend-
ing on ".
By M "
4 we denote the 4-dimensional manifold (space and time)
M "
4 =M "
3 � R
and introduce a pseudo Riemannian metric on M "
4 by the formula
ds2 = [c"(~x)]
2dt2 �
3X
i;k=1
g"ikdxidxk;
where c"(~x) > 0.
The Cauchy problem for the wave equation is considered on M "
4 :
2
"u" � 1
c"(~x)2
@2u"
@t2
� 1
c"(~x)
p
G
"
3X
i;k=1
@
@xi
�
gik" c"(~x)
p
G
" @u"
@xk
�
= 0; (0.1)
u"(~x; 0) = f "; (0.2)
u"t (~x; 0) = g": (0.3)
214 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
(G" = det g"
ik
, gik" , i; k = 1; 2; 3, are the components of the tensor inverse to g"
ik
.)
Suppose that the following conditions hold: on the upper sheet the metric
coincides with the Euclidean metric (outside of some small neighborhoods of the
"holes" D"
j
), and c"(~x) = 1; while on the lower sheet the metric increases or
c" ! 0 (the proper time becomes slower) as "! 0 (in the latter case it is possible
to choose the radiuses of "holes" more than "3).
Then we have the following result. The solution of the problem (0.1)�(0.3)
converges on the upper sheet to the solution of the Cauchy problem for the Klein�
Gordon equation
@2u
@t2
��u" +mu = 0; ~x 2 R3 ; t > 0;
u(x; 0) = f(x);
ut(x; 0) = g(x);
where m de�nitely depends on the characteristics of "wormholes", metric and the
function c"(~x).
This fact admits the following interesting physics interpretation: as a result
of connection with the lower sheet by means of "wormholes" G"
j
a scalar massless
particle gets a mass m as "! 0. In the paper this fact is proved in a more general
statement.
1. The Increasing Metric Case
1.1. The Problem Setting and the Statement of Main Result
Let fD"
j
; j = 1 : : : N(")g be a system of disjoint small domains in R
3 with
a smooth border. Suppose that this system depends on the parameter " > 0 in
such a way that the diameters of the sets D"
j
tend to zero as " ! 0, and their
total number N(") tends to in�nity. Denote
" = R
3n
N(")S
j=1
D"
j
. We consider two
copies
"
1 and
"
2 of the set
". Let
"
1 and
"
2 be called the upper and the lower
sheets, respectively.
Let G"
j
be 3-dimensional manifolds with the boundaries consisting of two dis-
joined components �"1j and �"2j being di�eomorphic to @D"
j
.
By means of these di�eomorphisms and taking account of orientation, we glue
�"1j to the copy of @D"
j
on the upper sheet and �"2j to the copy of @D"
j
on the
lower sheet.
As a result we obtain a di�erentiable manifold M ":
M " = (
"
1 [
"
2)
[0
@N(")[
j=1
G"
j
1
A :
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 215
A.V. Khrabustovskyi
We denote by ~x the points of this manifold. If the point ~x 2
"
k
, then we
assign a pair (x; k) to ~x, where x 2 R3 is a coordinate.
Let B(D"
j
) be the smallest ball containing D"
j
, with the center x"
j
and the
radius d"
j
. We denote
0
"
k
=
8<
:~x = (x; k) 2
"
k
: x 2 R3n
N(")[
j=1
B(D"
j )
9=
; :
On M " we introduce a metric g"
ik
(~x) that coincides with the Euclidean metric
in
0"
1 (g"
ik
= Æik) and increases in
0"
2 as follows:
g"ik =
Æik
�"
; �" > 0 and �" ! 0; "! 0: (1.1)
Consider the following Cauchy problem on M ":
�
"u" � @2u"(~x; t)
@t2
��"u"(~x; t) = 0; ~x 2M "; t > 0; (1.2)
u"(~x; 0) = f "(~x); (1.3)
u"t (~x; 0) = g"(~x); (1.4)
with �" being the Laplace�Beltrami operator on M "
�" =
1p
G
"
3X
i;k=1
@
@xi
�p
G
"
gik"
@
@xk
�
;
where G" = det g"
ik
, gik" , i; k = 1; 2; 3, are the components of the tensor inverse to
g"
ik
, and f ", g" are the smooth functions.
The purpose of this paper is to describe the asymptotic behavior of u"(~x; t)
on the upper sheet as "! 0.
Introduce the notation: r"
j
= dist
B(D"
j
);
S
i6=j
B(D"
i
)
!
,
B"
kj =
�
~x = (x; k) 2
"
k : d
"
j < jx� x"jj < d"j +
r"
j
2
�
;
G
0"
j = G"
j
2[
k=1
�
~x = (x; k) 2
"
k : x 2 B(D"
j)nD"
j
;
~G"
j = G
0"
j
2[
k=1
B"
kj; S"kj = @ ~G"
j \
"
k:
216 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
We consider the following boundary-value problem in the domain ~G"
j
:
�"v = 0; ~x 2 ~G"
j; (1.5)
v = 1; ~x 2 S"1j; (1.6)
v = 0; ~x 2 S"2j: (1.7)
Let v"
j
= v"
j
(~x) be the solution of (1.5)�(1.7). We set
V "
j =
Z
~G"
j
3X
i;k=1
gik"
@v"
j
@xi
@v"
j
@xk
d~x;
where d~x =
p
G"dx1dx2dx3 is the volume element on M ", and introduce the
generalized function
V "(x) =
N(")X
j=1
V "
j Æ(x � x"j):
We introduce the following functional spaces:
L2(M ") is the Hilbert space of real valued functions on M " with the norm
ku"k0" =
8<
:
Z
M"
(u")2d~x
9=
;
1=2
;
H1(M ") is the Hilbert space of real valued functions on M " with the norm
ku"k1" =
8<
:
Z
M"
0
@ 3X
i;k=1
gik"
@u"
@xi
@u"
@xk
+ (u")2
1
A d~x
9=
;
1=2
:
We say that the function f " 2 L2(M ") converges on the upper sheet
to f 2 L2(R3 ), if for any bounded domain
� R
3
lim
"!0
kQ"f " � fkL2(
) = 0; (1.8)
where the operator Q" : L2(M ")! L2(R3) is de�ned by the formula
[Q"f "](x) =
8><
>:
f "(~x); ~x = (x; 1) 2
0"
1 ;
0; x 2
N(")S
j=1
B(D"
j
):
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 217
A.V. Khrabustovskyi
Similarly, we say that u"(~x; t) 2 L2(M "� [0; T ]) converges on the upper sheet
to u(x; t) 2 L2(R3 � [0; T ]), if for any bounded domain
� R
3
lim
"!0
TZ
0
kQ"u"(�; t) � u(�; t)kL2(
)dt = 0: (1.9)
Let us formulate the basic theorem
Theorem 1. Suppose that the following conditions hold:
(i) lim
"!0
max
j
d"
j
= lim
"!0
max
j
r"
j
= 0;
(ii) for any domain G � R
3
lim
"!0
X
x"
j
2G
(d"
j
)2
(r"
j
)3
� C1measG;
and r"
j
> C0d
"
j
(here 0 < C0, C1 <1);
(iii) lim
"!0
max
j
(r"
j
)8
�3"
= 0;
(iv) there exists a limit (in D0(R3))
lim
"!0
V "(x) = V (x);
where V (x) is a measurable bounded nonnegative function;
(v) for any domain G � R
3X
x"
j
2G
measG
0"
j ! 0 ("! 0);
(vi) norms kf "k1" are uniformly bounded with respect to "; when "! 0: f "(~x)
and g"(~x) converge in the sense of (1.8) to the functions f 2 H1(R3 )
and g 2 L2(R3), respectively, andZ
0"
2
S
j
G
0"
j
�
jf "j2 + jg"j2
�
d~x! 0; "! 0:
Then the solution of the problem (1.2)�(1.4) u"(~x; t) converges in the sense of
(1.9) to the solution u(x; t) of the following problem:
@2u
@t2
= �u� V (x)u; ~x 2 R3 ; t > 0; (1.10)
u(x; 0) = f(x); (1.11)
ut(x; 0) = g(x): (1.12)
The proof of the theorem is based on a study of asymptotic behavior of the
operator ��" resolvent as "! 0.
218 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
1.2. Asymptotic Behavior of the Solution of the Stationary Problem
We consider the following problem:
��"u" + �u" = F"; ~x 2M "; (1.13)
u" 2 H1(M "); (1.14)
where � > 0;F" 2 L2(M "):
As it is known, there exists a unique solution u"(x; �) of this problem. The fol-
lowing theorem describes the asymptotic behavior of u"(x; �) on the upper sheet.
Theorem 2. Suppose that conditions (i)�(v) of the Th. 1 hold and suppose
(vi0) F"(x) converges in the sense of (1.8) to the function F(x) 2 L2(R3) as
"! 0 and Z
0"
2
S
j
G
0"
j
(F")2d~x! 0("! 0):
Then the solution of the problem (1.13)�(1.14) converges in the sense of (1.8)
to the solution u(x) of the following problem:
��u+ �u+ V (x)u = F ; x 2 R3 ; (1.15)
u" 2 H1(R3 ): (1.16)
P r o o f. As we know, the solution u"(x; �) of the problem (1.13)�(1.14)
minimizes the functional
J"[u"] =
Z
M"
8<
:
3X
i;k=1
gik"
@u"
@xi
@u"
@xk
+ �(u")2 � 2F"u"
9=
; d~x (1.17)
in the class of functions H1(M "), while the solution u(x; �) of the problem (1.15)�
(1.16) minimizes the functional
J [u] =
Z
R3
�
jruj2 + �u2 + V (x)u� 2Fu
dx (1.18)
in the classH1(R3). The converse assertions are also true. Therefore it is su�cient
to show that the solution of the problem of minimizing (1.17) converges to the
solution of the problem of minimizing (1.18).
Consider an abstract scheme for solving the problem. Let H" be a Hilbert
space depending on the parameter " > 0, (u"; v")"; ku"k" be a scalar product and
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 219
A.V. Khrabustovskyi
a norm in this space, F " be the continuous linear functionals in H" which are
uniformly bounded with respect to ". Let H be a Hilbert space with the scalar
product (u; v) and norm kuk, F be a continuous linear functional in H.
Consider the following two problems of minimization:
ku"k2" + F "[u"]! inf; u" 2 H"; (1.19)
kuk2 + F [u]! inf; u 2 H: (1.20)
We have the following theorem proved in [3].
Theorem 3. Let M be a dense subset of H, and let �" : H" ! H and
P " :M ! H" be the operators satisfying the following conditions:
(a) k�"w"k � Ckw"k;8w" 2 H";
(b1) �
"P "w! w weakly in H as "! 0;8w 2M ;
(b2) lim
"!0
kP "wk" = kwk;8w 2M ;
(b3) for any sequence
" 2 H", such that �"
" !
weakly as "! 0, for any
w 2M one has
lim
"!0
j(P "w;
")"j � Ckwkk
k;
(c) for any sequence
" 2 H", such that �"
" !
weakly as "! 0, we have
lim
"!0
F "[
"] = F [
]:
Then the solution u" of the minimization problem (1.19) converges to
the solution of the minimization problem (1.20) in the following sense:
�"u" !
"!0
u weakly in H:
Note, that Th. 3 holds true if conditions (b3) and (c) hold only for such
sequences
" that the norms k
"k" are uniformly bounded with respect to " (as in
the proof of Th. 3, the conditions (b3) and (c) are used only with these sequences).
Now we consider our abstract scheme. Let H" be the Hilbert space H1(M ")
of the functions on M " with the scalar product
(u"; v")" =
Z
M"
8<
:
3X
i;k=1
gik"
@u"
@xi
@v"
@xk
+ �u"v"
9=
; d~x; (1.21)
and let F " be the linear functional on it de�ned by the formula
F "[u"] =
Z
M"
�2F"u"d~x: (1.22)
220 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
Let H be the Hilbert space H1(R3 ) with the scalar product
(u; v) =
Z
R3
� 3X
i=1
@u
@xi
@v
@xi
+ �uv + V (x)uv
�
dx (1.23)
and F be the linear functional on it de�ned by the formula
F [u] =
Z
R3
�2Fudx: (1.24)
Since jF "[u"]j � 2kF"k0"ku"k0" � 2kF"k0"ku"k" and norms kF"k0" are uni-
formly bounded with respect to ", then the functionals F " are uniformly bounded
with respect to ".
Now we introduce the operators �" and P " satisfying the conditions (a)�(c)
of Th. 3. Let u" 2 H1(M "), u
0" be a contraction of u" on
0"
1 . Then u
0" can
be extended to
N(")S
j=1
B(D"
j
) so that the obtained function ~u
0
" belongs to the space
H1(R3 ) and satis�es the inequality
k~u0
"k � Cku"k";
"
1
; (1.25)
where C does not depend on " [4].
Since this kind of extensions is not unique, we require the norms of the
extended function in the space H1(
S
j
B(D"
j
)) to be minimal. Then we obtain
a unique extension ~u": For this reason we set �"u" = ~u":
It follows from (1.25) that the condition (a) of Th. 3 holds.
We introduce an operator P ". Let '(r) be a twice continuously di�erentiable
non-negative function on the half-line [0;1), which is equal to 1 for r 2 [0; 1=4]
and to 0 for r � 1=2. We set
'"j = '
� jx� x"
j
j � d"
j
r"
j
�
; '"j0 = '
� jx� x"
j
j � d"
j
C0d
"
j
�
:
Let M = C2
0 (R
3 ) be a dense subset of H1(R3) and let w 2 M . De�ne the
operator P " by the equalities
[P "w](~x) =
8>>>>>><
>>>>>>:
w(x) +
N(")P
j=0
(w"
j
� w(x))'"
j0
+
N(")P
j=0
(v"
j
(x)� 1)'"
j
w"
j
; ~x 2
0"
1 ;
v"
j
(~x)w"
j
; ~x 2 G0
"
j
;
N(")P
j=0
v"
j
(x)'"
j
w"
j
; ~x 2
0"
2 ;
(1.26)
where w"
j
= w(x"
j
).
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 221
A.V. Khrabustovskyi
To be sure that the conditions (b1)�(b3) hold we use the following estimates
for the solution v"
j
of the problem (1.5)�(1.7).
Lemma 1. Let ~x = (x; k) 2 B"
kj
and jx� x"
j
j � d"
j
(1 + C0). Then
jD�(v"j (~x)� Æ1k)j � C
d"
j
jx� x"
j
j1+j�j ;
where j�j = 0; 1, and C does not depend on �.
Lemma 2. The following estimates are valid:
Z
B"
kj
jv"j � Æ1kj2d~x �
(
C(d"
j
)3(1 + r"
j
=d"
j
); k = 1;
C(d"
j
)3(1 + r"
j
=d"
j
)�
�3=2
" ; k = 2:
The proofs of these lemmas are carried out in the same way as those of Lems. 1,
2 in [5], by using the inequality 0 � v"
j
� 1, which follows from the maximum
principle.
We verify that the condition (b2) holds. Let w 2M , G = supp(w). Then
kP "u"k2" =
Z
0"
1
(
3X
i=1
�
@w
@xi
�2
+ �w2
)
dx+
X
x"
j
2G
( Z
B"
1j
3X
i=1
�
@v"
j
@xi
�2
w2
jdx
+
Z
G
0"
j
3X
i;k=1
gik"
@v"
j
@xi
@v"
j
@xk
w2
jd~x+
Z
B"
2j
�"
3X
i=1
�
@v"
j
@xi
�2
w2
jd~x
)
+�1(") + �2(") + �G("); (1.27)
where�1(");�2(");�G(") are the remaining integrals over
0"
1 ;
0"
2 ; G
0"
j
, estimated
as follows:
j�1(")j � c1(w)
X
x"
j
2G
(d"j)
3 + c2(w)
(X
x"
j
2G
Z
B1j
3X
i=1
����@vj@xi
����+ jvj � 1j
!
dx
+
X
x"
j
2G
Z
R"
1j
3X
i=1
�
@v"
j
@xi
�2
+ (v"j � 1)2
1
(r"
j
)2
!
dx+
X
x"
j
2G
Z
B"
1j
(v"j � 1)2dx
)
;
j�G(")j � c3(w)
X
x"
j
2G
Z
G
0"
j
(v"j )
2;
222 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
j�2(")j � c4(w)
(X
x"
j
2G
Z
R"
2j
"
3X
i=1
�
@v"
j
@xi
�2
+ (v"j )
2 1
(r"
j
)2
#
�"d~x+
X
x"
j
2G
Z
B"
2j
(v"j )
2d~x
)
;
where R"
kj
=
n
~x = (x; k) 2
0
"
k
: d"
j
+
r"
j
4
< jx� x"
j
j < d"
j
+
r"
j
2
o
.
We estimate the integrals over R"
kj
by means of Lem. 1 and the integrals over
B"
kj
by means of Lem. 2. Taking into account that in
0"
2 : d~x = �
�3=2
" dx1dx2dx3,
we have
�1(") � c(w)
( X
x"
j
2G
(d"
j
)2
(r"
j
)3
max
j
d"j(r
"
j )
3 +
X
x"
j
2G
(r"j )
3
X
x"
j
2G
(d"
j
)2
(r"
j
)3
!1=2
�
�
max r"j +max(r"j)
2
�
+
X
x"
j
2G
(d"
j
)2
r"
j
3
max
j
(r"j )
2
!
+
X
x"
j
2G
(d"
j
)2
(r"
j
)3
max
j
(r"j )
4
!)
;
�2(") � c(w)
( X
x"
j
2G
(d"
j
)2
r"
j
3
max
j
(r"
j
)2
�
1=2
"
+
X
x"
j
2G
(d"
j
)2
(r"
j
)3
max
j
(r"
j
)4
�
3=2
"
)
;
�G(") � c(w)
X
x"
j
2G
measG
0
"
j :
By conditions (i)�(iii) and (v) of the theorem
lim
"!0
(�1(") + �G(") + �2(")) = 0: (1.28)
It follows from (1.27), (1.28) and the condition (iv) that
lim
"!0
kP "wk" = kwk:
Thus the condition (b2) holds.
Now we verify the condition (b3). Let w 2 M , G = supp(w), the sequence
" is such that the norms k
"k" are uniformly bounded with respect to " and
such that �"
" !
(" ! 0) weakly in H. Denote by (u; v)1 the following scalar
product in H1(R3)
(u; v)1 =
Z
R3
"
3X
i=1
@u
@xi
@v
@xi
+ �uv
#
dx:
Integrating by parts, we have
(P "w;
")" = (w;�"
")1 + I"1 + I"2 + I"3 ; (1.29)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 223
A.V. Khrabustovskyi
where
I"1 =�
X
x"
j
2G
w"
j
" Z
R"
1j
2
3X
i=1
@v"
j
@xi
@'"
j
@xi
+ (v"j � 1)�'"j
!
"dx
+
Z
R"
2j
�"
2
3X
i=1
@v"
j
@xi
@'"
j
@xi
+ v"j�'
"
j
!
"d~x
#
;
I"2 = �
X
x"
j
2G
w"
j
" Z
B"
1j
(v"j � 1)'"j
"dx+
Z
B"
2j
v"j'
"
j
"d~x+
Z
G
0"
j
v"j
"d~x
#
;
jI"3 j � C(w)
" X
x"
j
2G
(d"j)
3
#1=2
k�"
"k1 � C(w)
max
j
(d"j)max
j
(r"j)
3
X
x"
j
2G
(d"
j
)2
(r"
j
)3
!1=2
:
Since the norms k�"
"k1 are uniformly bounded with respect to ", it follows from
(i), (ii):
lim
"!0
I"3 = 0: (1.30)
Estimating the integrals over R"
kj
by means of Lem. 1 and the integrals over
B"
kj
by means of Lem. 2, we have the following estimates:
jI"1 j � C
" X
x"
j
2G
(d"
j
)2
(r"
j
)3
jw"
j j2
#1=2
k�"
"k0G + C(w)
" X
x"
j
2G
�1=2"
(d"
j
)2
(r"
j
)3
#1=2
k
"k0";
jI"2 j � C(w)
("
max
j
(r"j)
4
X
x"
j
2G
(d"
j
)2
(r"
j
)3
#1=2
+
"
max
j
(r"
j
)4
�
3=2
"
X
x"
j
2G
(d"
j
)2
(r"
j
)3
#1=2
+
h
meas
[
G
0
"
j
i1=2)
k
"k0";
where k � k0G denotes the norm in L2(G3).
Since �"
" converges weakly in H1(R3) to
, from the embedding theorem
we have
lim
"!0
k�"
"k0G = k
k0G � k
k:
224 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
Since k
"k0" � k
"k" � C, by the conditions (i)�(iii), (v)
lim
"!0
jI1j � Ckwk0k
k0 � Ckwkk
k; (1.31)
lim
"!0
I2 = 0: (1.32)
It follows from (1.29)�(1.32) that the condition (b3) holds.
We verify the condition (b1). Let w 2M . Since
k�"P "wk � ckP "wk ! ckwk;
we have
k�"P "wk � C(w) uniformly with respect to "(" < "0):
Moreover, in the same way as in (b2), it is easy to show that �"P "w ! w strongly
in L2(R3); thus the condition (b1) also holds .
And, �nally, verify that condition (c) holds. Let sequence
" 2 H" be such
that the norms k
"k" are uniformly bounded with respect to " and �"
" !
weakly in H. Then
jF "[
"]� F [
]j � 2
�����
Z
R3
(Q"F" � �"
" � F
)dx
����� + 2
�����
Z
0"
2
S
j
G
0"
j
F"
"d~x
�����: (1.33)
It follows from (1.33) and the condition (vi0) that jF "[
"]� F [
]j ! 0("! 0);
so the condition (c) holds.
Thus all the conditions of Th. 3 hold. Hence �"u" ! u weakly inH. Therefore
by the embedding theorem �"u" ! u strongly in L2(
), where
is a bounded
domain in R3 . Moreover, for any bounded domain G � R
3
X
x"
j
2G
measB(D"
j) = C
X
x"
j
2G
(d"j)
3 � C
X
x"
j
2G
(d"
j
)2
(r"
j
)3
max
j
(r"j )
3max
j
d"j ! 0; "! 0:
(1.34)
Then by (1.34) u" ! u in the sense (1.9). The theorem is proved.
1.3. Asymptotic Behavior of the Solution of the Nonstationary
Problem
We consider the following problem for the complex values of the parameter �:
��"u" + �2u" = �f " + g"; ~x 2M "; (1.35)
u" 2 H1(M "): (1.36)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 225
A.V. Khrabustovskyi
The solution u(~x; �) of this problem is the Laplace transform of the solution
of the problem (1.2)�(1.4). It is proved that for �2 > 0 (that is � 2 Rnf0g)
the solution of the problem (1.35)�(1.36) converges in the sense of (1.8) to the
solution of the problem
��u+ �2u+ V (x)u = �f + g; x 2 R3 ; (1.37)
u 2 H1(R3): (1.38)
Theorem 20. Let � = f� : Re� � Æ > 0g and all conditions of Th. 2 hold.
Then the solution of the problem (1.35)�(1.36) u"(�; �) is a holomorphic function
in �
�
, the following estimate holds:
kQ"u"(�; �)k0 � C; (1.39)
where C does not depend on � and ", and u"(�; �) converges (uniformly on each
bounded subset of �) in the sense of (1.8) to the solution u(�; �) of the problem
(1.37)�(1.38). Besides, u(�; �) is holomorphic in � and satis�es the estimate
(1.39).
P r o o f. Since ��" induces a nonnegative selfadjoint operator in L2(M "),
then for all � 2 �
u" =
1Z
0
dEt(�f
" + g")
t+ �2
;
where Et is a resolution of identity of the operator ��".
Then
ku"k0" �
j�j � kf "k0" + kg"k0"
dist (�2; (�1; 0])
; (1.40)
dist
�
�2; (�1; 0]
�
=
(
jIm�2j; j arg �j > �=4
j�2j; j arg �j � �=4
=
(
j2Im�Re�j; j arg �j > �=4
j�2j; j arg �j � �=4
�
(
2Æj�j sin �=4; j arg �j > �=4
Æj�j; j arg �j � �=4
: (1.41)
It follows from (1.40),(1.41) and the condition (vi) of Th. 1 that (1.39) holds.
Moreover, Q"u"(�; �) and u(�; �) are holomorphic functions for Re� 6= 0 (as
the resolvent is holomorphic outside of the operator spectrum).
Since Q"u"(x; �) converges to u(x; �) in L2(R3 ) for � 2 Rnf0g, then
by the Vitaly theorem, when � 2 �, then Q"u"(x; �) converges (uniformly on
�
As a function of complex variable with values in L2(M").
226 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
each bounded subset of �) in L2(R3) to the function U(x; �). Moreover, U(x; �)
is holomorphic and satis�es the estimate (1.39). Since U(x; �) = u(x; �) for � 2
Rnf0g, then by the uniqueness theorem U(x; �) = u(x; �) for � 2 �. The theorem
is proved.
First, we prove that Q"u�(x; t) converges to u(x; t) weakly in L2(R3 � [0; T ]).
Let D be a set of functions of the form
g(x; t) = '(x) (t);
where '(t) 2 L2(R3), (x) 2 C2
0 [0; T ].
We note, that in view of properties of the Laplace transform, since the solution
u"(~x; �) of the problem (1.13)�(1.14) is the Laplace transform for u"(~x; t), then
u
"(~x;�)
�2
is the Laplace transform for
tR
0
sR
0
u"(~x; �)d�ds, and one has
tZ
0
sZ
0
u"(~x; �)d�ds =
1
2�i
�+i1Z
��i1
u"(~x; �)
�2
e�td�:
Integrating by parts we have
TZ
0
Z
R3
Q"u"(x; t)'(x) (t)dxdt =
Z
R3
'(x)
TZ
0
tZ
0
sZ
0
Q"u"(x; �)d�ds
!
00dtdx
=
Z
R3
'(x)
TZ
0
1
2�i
�+i1Z
��i1
Q"u"(x; �)
�2
e�td�
!
00dtdx
=
1
2�i
�+i1Z
��i1
1
�2
Z
R3
'(x)Q"u"(x; �)dx
TZ
0
e�t 00dt
!
d�: (1.42)
Here the change of order of integration is valid since, in view of the following
inequalities: Z
R3
jQ"u"(x; �)jj'(x)jdx � kQ"u"k0k'k0 � C(');
TZ
0
je�tjj 00jdt � C( ); Re � = �;
the last repeated integral in (1:42) converges absolutely.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 227
A.V. Khrabustovskyi
By Theorem 2, Z
R3
Q"u"(x; �)'(x)dx!
Z
R3
u(x; �)'(x)dx:
Since the norms kQ"u"(�; �)k0" are uniformly bounded with respect to " and
the convergence is uniform on each bounded subset of �, we may pass to the limit
under the integral sign in (1.42).
On the other hand, similarly, we have
TZ
0
Z
R3
u(x; t)'(x) (t)dxdt =
1
2�i
�+i1Z
��i1
0
@ 1
�2
Z
R3
'(x)u(x; �)dx
TZ
0
e�t 00dt
1
A d�:
We will prove later that the set fu"g is uniformly bounded in H1(M "� [0; T ]);
hence, since D is a dense subset in L2(R3 � [0; T ]), then Q"u"(x; t) converges to
u(x; t) weakly in L2(R3 � [0; T ]). In view of (1.34), �"u"(x; t) converges to u(x; t)
weakly in L2(R3 � [0; T ]), where the operator �" : H1(M ")! H1(R3 ) is de�ned
in the proof of Th. 2.
Prove that the set f�"u"g is uniformly bounded in H1(M " � [0; T ]). In view
of (1.2), it is easy to see that the following equation holds
Z
M"
2
4�@u"(~x; t)
@t
�2
+
3X
i;k=1
gik"
@u"(~x; t)
@xi
@u"(~x; t)
@xk
3
5d~x
=
Z
M"
2
4(g"(~x))2 + 3X
i;k=1
gik" (~x)
@f "(~x)
@xi
@f "(~x)
@xk
3
5 d~x: (1.43)
In addition, 8t < T
Z
M"
(u")2d~x =
Z
M"
0
@ tZ
0
@u"(~x; t)
@t
dt+ f "(~x)
1
A
2
d~x
� 2
Z
M"
0
@T
TZ
0
�
@u"(~x; t)
@t
�2
dt+ (f "(~x))2
1
A d~x: (1.44)
By (1.43), (1.44) and the condition (vi) it follows that the set fu"g is uniformly
bounded in H1(M " � [0; T ]), so in view of (1.25), the set f�"u"g is uniformly
bounded in H1(R3 � [0; T ]). Then for any bounded domain
� R
3 the set
f�"u"g is compact in L2(
� [0; T ]). Hence �"u" ! u strongly in L2(
� [0; T ]).
Hence u" converges to u in the sense of (1.9). Theorem 1 is proved.
228 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
1.4. Example
We consider a concrete example of the Riemannian manifold M ". Let D"
j
be
a system of balls with the radius d" = a"3 and the centers at x"
j
periodically
distributed in
� R
3 , that is
D"
j =
n
x 2 R3 : jx� x"j j � d"; x"j = "
3X
i=1
eiz
j
i
o
;
where r" = " is a period, fei; i = 1; 2; 3g is an orthonormal basis in R3 , z
j
i
2 Z
and D"
j
�
; j = 1 : : : N(").
G"
j
is the "pipe", that is G"
j
=
n
('; ; z) : ' 2 [0; 2�]; 2 [��=2; �=2];
z 2 [0; 1]
o
= S3 � [0; 1]:
On M " we introduce a metric g"
ik
(~x) that coincides with the Euclidean metric
in
0"
1 , increases in
0"
2 in the sense of (1.1), and in the "pipe" G
0"
j
we de�ne it by
the following formula for the square of the element of length:
ds2 = q"jdz
2 + (d")2(cos2 d'2 + d 2); (1.45)
where q"
j
> 0.
We introduce the spherical coordinates (r; '; ) in B"
kj8><
>:
x1 = r sin' cos ;
x2 = r cos' cos ;
x3 = r sin ; r 2 [d"; r"=2]; 2 [��=2; �=2]; ' 2 [0; 2�]:
(1.46)
We will state the function v"
j
, supposing that it depends neither on ', nor on .
We denote
v"j =
8><
>:
v1; if ~x 2 B"
1j ;
vg; if ~x 2 G
0"
j
;
v2; if ~x 2 B"
2j :
Then v1; v2; vg satisfy the equations8>><
>>:
1
r2
@
@r
�
r2 @v1
@r
�
= 0; d" < r < r"=2;
@2vg
@z2
= 0; 0 < z < 1;
1
r2
@
@r
�
r2 @v2
@r
�
= 0; d" < r < r"=2;
the boundary conditions (
v1(r
"=2) = 1;
v2(r
"=2) = 0;
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 229
A.V. Khrabustovskyi
and conditions on the boundaries of the upper and the lower sheets
(
v1(d
") = vg(0);
v2(d
") = vg(1);
8<
:
@v1
@r
= � 1p
q
"
j
@vg
@z
;
1p
�
"
@v2
@r
= 1p
q
"
j
@vg
@z
:
Hence we have
v"j =
8><
>:
a1
r
+ b1; ~x 2 B"
1j ;
Az +B; ~x 2 G0
"
j
;
a2
r
+ b2; ~x 2 B"
2j ;
where
a1 = �d"
p
q"
j
d"
+
�
1 +
p
�
"
��
1� 2d"
r"
�!�1
; A = a1
p
q"
j
(d")2
; a2 = �a1
p
�
"
:
If q"
j
can be represented in the form q"
j
= q"l2
j
, lj = l(x"
j
), where l(x) is
a continuous function on
and there exists the limit q = lim
"!0
p
q"
d"
<1, then
V "
j =
Z
~G"
j
3X
i;k=1
gik"
@v"
j
@xi
@v"
j
@xk
d~x =
r
"
=2Z
d"
2�Z
0
�=2Z
��=2
�a1
r2
�2
r2 cos d d'dr
+
1Z
0
2�Z
0
�=2Z
��=2
A2
p
q"
j
(d"j)
2 cos d d'dz +
r"=2Z
d"
2�Z
0
�=2Z
��=2
�"
�1=2
�a2
r2
�2
r2 cos d d'dr
= 4�(a21 + �"
�1=2a22)
�
1
d"
� 2
r"
�
+ 4�
A2(d")2p
q"
j
= 4�d"
�
lj
p
g"
d"
+ 1
��1
(1 + �o(1)):
Since x"
j
are distributed periodically in
and d� = a(r�)3, it follows that
V (x) =
4a�
ql(x) + 1
�
(x) (�
(x) is a characteristic function of
), and the averaged
equation has the form:
@2u
@t2
��u+
4a��
ql(x) + 1
u = 0:
230 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
2. Case of the Delay of Proper Time
We consider the manifold M "
4 =M "
3 �R, where M "
3 is a particular case of the
manifold M " considered in Sect. 1.4 (without assumption that x"
j
are distributed
periodically in
).
Introduce a metric on M "
3 similarly to 1.4 and de�ne a metric in M "
4 by the
following formula:
ds2 = [c"(~x)]
2 dt2 �
3X
i;k=1
g"ikdxidxk;
where
c"(~x) =
8><
>:
1; ~x 2
0
"
1
N(")S
j=1
G
0
"
j
;
~c"; ~x 2
0"
2 ;
~c" ! 0; "! 0;
and g"(~x) = fg"
ik
; i; k = 1; 2; 3g is a metric tensor on M "
3 :
We consider the Cauchy problem on M "
4
2
"u" � 1
c"(~x)2
@2u"
@t2
� 1
c"(~x)
p
G
"
3X
i;k=1
@
@xi
�
gik" c"(~x)
p
G
" @u"
@xk
�
= 0; (2.1)
u"(~x; 0) = f "; (2.2)
u"t (~x; 0) = g": (2.3)
(G" = det g"
ik
, gik" , i; k = 1; 2; 3 are the components of the tensor inverse to g"
ik
.)
Denote
V "
j =
Z
~G"
j
3X
i;k=1
c"(~x)g
ik
"
@v"
j
@xi
@v"
j
@xk
d~x; (2.4)
where v"
j
is the solution of the problem
3X
i;k=1
1p
G
"
@
@xi
�
gik"
p
G
"
c"(~x)
@v
@xk
�
= 0; ~x 2 ~G"
j; (2.5)
v = 1; ~x 2 S"1j; (2.6)
v = 0; ~x 2 S"2j: (2.7)
It follows from the form of metric in G
0
"
j
(1.45), that
V "
j = 4�(a21 + ~c"a
2
2)
1
d"
j
� 1
�"
j
!
+ 4�
A2(d"
j
)2
p
q"
j
;
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 231
A.V. Khrabustovskyi
where �"
j
= d"
j
+ r"
j
=2;
a1 = �~c"d"j
~c"
p
q"
j
d"
j
+
�
~c" + 1
��
1�
d"
j
�"
j
�!�1
; A = a1
p
q"
j
(d"
j
)2
; a2 = �a1
1
~c"
:
Introduce the following generalized function
V "(x) =
N(")X
j=1
V "
j Æ(x � x"j) (2.8)
and let the following limit exist in D0(R3)
lim
"!0
V "(x) = V (x); (2.9)
where V (x) is a measurable bounded nonnegative function.
Theorem 4. Suppose the conditions (i),(iv) of Th. 1 hold and:
(ii0) for any domain G � R
3
lim
"!0
X
x"
j
2G
~c2"
(d"
j
)2
(r"
j
)3
� C1measG;
and r"
j
> C0d
"
j
(here 0 < C0; C1 <1);
(iii0) lim
"!0
max
j
(r"
j
)4
~c3"
= 0;
(vi0) the norms kf "k1" are uniformly bounded with respect to "; when " ! 0,
f "(~x) and g"(~x) converge in the sense of(1.8) to the functions f 2 H1(R3 ) and
g 2 L2(R3 ), respectively, andZ
0"
2
S
j
G
0"
j
jf "j2 + jg"j2
c"(~x)
d~x! 0; "! 0:
Then the solution of the problem (2.1)�(2.3) u"(~x; t) converges in the sense of
(1.9) to the solution u(x; t) of the problem (1.10)�(1.12), where V (x) is de�ned
by (2.8)�(2.9).
P r o o f. The theorem is proved in the same way as Th. 1. But here, for
the solution v"
j
of the problem (2.5)�(2.7), a stronger estimate than in Lem. 1 is
used. Namely, as it appears from an explicit form of the function v"
j
, the following
inequality holds:
jD�(v"j (~x)� 1)j � C
~c"d
"
j
jx� x"
j
j1+j�j ; ~x = (x; 1) 2 B"
1j ; (2.10)
where j�j = 0; 1, and C does not depend on ".
232 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Klein�Gordon Equation as a Result of Wave Equation...
As for the rest, the theorem is proved in the same way as Th. 1.
R e m a r k. This result is obtained for the discontinuous metric tensor g"(~x).
However, g"(~x) can be approximated by the smooth tensor g"
Æ
(~x) that coincides
with g"(~x) outside of the small neighborhood of @D"
kj
. We introduce the local
coordinates (x1; x2; x3) in the neighborhood of the point ~x 2 @D"
kj
, such that
r = d"
j
+ jx1j(x1 � 0), z = x1(x1 � 0), x2 = ', x3 = and set
ds2 = q"jÆ(x1)dx
2
1 +
�
(d"j)
2(1� '"jÆ(x1)) + (d"j + jx1j)2'"jÆ(x1)
�
(cos2 x3dx
2
2 + dx23);
where q"
jÆ
(x1) is a smooth nonnegative function equal to 1 for x1 � 0 and q"
j
for
x1 � Æ > 0, '"
jÆ
(x1) = (q"
jÆ
(x1)� q"j)=(1� q"j ). Then g"(~x) = g"
Æ
(~x) for x1 62 (0; Æ).
In the same way we can approximate c"(~x) by the function c"Æ(~x). Similarly,
we introduce the local coordinates (x1; x2; x3) in the neighborhood of the point
~x 2 @D"
2j and set
c"Æ(~x) = 1� '"jÆ(x1) + ~c"'
"
jÆ
(x1):
We suppose Æ = �o(d"
j
). Then the function V (x), calculated by the formulae
(2.4),(2.8),(2.9), but with a tensor g"
Æ
(~x) and a function c"Æ(~x), will be equal to
the function V (x) calculated with a tensor g"(~x) and a function c"(~x):
The Author thanks deeply Prof. E.Ya. Khruslov for the statement of problem
and the attention he paid to this work.
References
[1] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry. Nauka,
Moscow, 1986. (Russian)
[2] J.M. Ziman, Elements of Advanced Quantum Theory. Mir, Moscow, 1970.
(Russian)
[3] L. Boutet de Monvel and E.Ya. Khruslov, Averaging of the Di�usion Equation on
Riemannian Manifolds of Complex Microstructure. � Tr. Mosk. Mat. Obshch. 58
(1997), 137�161 (Russian); (Transl. Moscow Math. Soc. 1997).
[4] E.Ya. Khruslov, Asymptotic Behavior of the Solution of the Second Boundary-Value
Problem Under Fragmentation of the Boundary of the Domain. � Mat. Sb. 106
(148) (1978), 603�621; (Russian) (Engl. transl.: Math. USSR Sb. 35 (1979).)
[5] E.Ya. Khruslov, The Method of Orthogonal Projections and the Dirichlet Boundary-
Value Problem in Domains with Fine-Grained Boundaries. � Mat. Sb. 88 (130)
(1972), 38�60; (Russian) (Engl. transl.: Math. USSR Sb. 17 (1972).)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 233
|
| id | nasplib_isofts_kiev_ua-123456789-106446 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T18:25:21Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Khrabustovskyi, A.V. 2016-09-28T19:03:32Z 2016-09-28T19:03:32Z 2007 Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure / A.V. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 213-233. — Бібліогр.: 5 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106446 An asymptotic behavior of solution of the Cauchy problem for the wave equation is studied on the Riemannian manifold Mε depending on a small parameter ε. It is supposed that a topological type of Mε increases as ε → 0. The averaged equation is derived, it describes the asymptotic behavior of the original Cauchy problem as ε → 0. The Author thanks deeply Prof. E.Ya. Khruslov for the statement of problem and the attention he paid to this work. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure Article published earlier |
| spellingShingle | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure Khrabustovskyi, A.V. |
| title | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure |
| title_full | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure |
| title_fullStr | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure |
| title_full_unstemmed | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure |
| title_short | Klein-Gordon Equation as a Result of Wave Equation Averaging on the Riemannian Manifold of Complex Microstructure |
| title_sort | klein-gordon equation as a result of wave equation averaging on the riemannian manifold of complex microstructure |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106446 |
| work_keys_str_mv | AT khrabustovskyiav kleingordonequationasaresultofwaveequationaveragingontheriemannianmanifoldofcomplexmicrostructure |