GENERALIZED POLARIZATION TENSORS
It then follows from (2.14) and (2.18) that
m
-u
+
Vn(uian)
=
Sn(g)lan- LSvi¢/ilian inn,
j=l
and hence, by (2.4), we get
1
m
(-21
+
Kn)(ulan)
=
Sn(g)lan- LSvj¢/j)lan on
an.
j=l
We then have from (2.11) that
Av(g)
~ (-~I+
ICn)
-l (
Sn (g)
I
an -
~
Sv,¢0)
loo)
~
Ao(Y) -
t,
Nv,
¢0l,
which is exactly the formula (2.22).
13
We have a similar representation for solutions of the Dirichlet problem. Let
f
E
W{(an),
and let v and
V
be the (variational) solutions of the Dirichlet problems:
2
(2.23)
and
(2.24)
{
V'· (1+(k-1)x(D))V'v=O
v
=
f
on
an,
{
D.V
=
0
inn,
v
= f
on
an.
in
n,
The following representation theorem holds.
THEOREM
2.11. Let v and V be the solution of the Dirichlet problems (2.23)
and (2.24). Then avjav on aD can be represented as
av av a
a
11
(x)
=
av (x)- a
11
Gv¢(x), x
E
an,
where¢ is defined in (2.15) with H given by (2.14) and g
=
avjav on an, and
Gv¢(x)
:= {
G(x, y)¢(y) da(y).
lav
2.4. Periodic transmission problem. We now consider the following peri-
odic transmission problem used in calculating effective properties of dilute compos-
ite materials. Let
Y
=]- 1/2, 1/2[d denote the unit cell and
DC Y.
Consider the
periodic transmission problem:
(2.25)
for
i =
1, ... ,
d.
V' · ( 1 + (k- 1)x(D)) V'ui = 0 in Y,
Ui - Yi periodic with period 1 (in each direction),
i
Ui(y)dy
=
0,
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