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,