10 HABIB AMMARI AND HYEONBAE KANG
2.3. Transmission problems.
Let
n
be a bounded domain in
JRd
with a
connected Lipschitz boundary and conductivity equal to 1. Consider a bounded
domain D
cc
n
with a connected Lipschitz boundary and conductivity 0 k
=f
1
+oo.
Let
g
E L~(an),
and consider the following transmission problem:
(2.12)
V' ·
(1
+
(k -1)x(D))V'u
=
0
inn,
au' -
g
av
8!1 - '
{ u(x) da(x) =
0,
lan
where
x(D)
is the characteristic function of
D.
By integrating the equation against
functions with compact supports, we can easily see that the problem (2.12) can be
rephrased as
~u
=
0 in
n
\aD,
ui_- ui+
=
0 on
aD,
kaul -au'
=
0
on
an,
av - av
+
au'
av
8!1
=
g,
{ u(x) da(x)
=
0.
lan
At this point we have all the necessary ingredients to state a decomposition
formula of the steady-state voltage potential
u
into a harmonic part and a refrac-
tion part which will be the main tool for deriving the asymptotic expansion, for
designing efficient reconstruction algorithms, and for calculating effective properties
of composite materials in later sections. This decomposition formula is unique and
inherits geometric properties of the inclusion D.
The following theorem was proved in
[87, 88, 90].
The original formula in
[87]
is rather complicated even if it contains all the idea. The following formula has
appeared in
[90]
with a simpler proof.
THEOREM
2.8. Suppose that D is a domain compactly contained in
n
with a
connected Lipschitz boundary and conductivity 0 k
=f
1
+oo.
Then the solution
u of the Neumann problem (2.12) is represented as
(2.13) u(x)
=
H(x)
+
Sv¢(x),
X
En,
where the harmonic function H is given by
(2.14) H(x)
=
-Sn(g)(x)
+
Vn(f)(x), x
E
!1,
f
:=ulan,
and¢
E L~(aD)
satisfies the integral equation
(2.15)
(.I-Icv)4=aHI onan,
av
8D
(2.16)
k+1
,\=2(k-1)'
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