GENERALIZED POLARIZATION TENSORS
5
and the solution
u
represents the voltage distribution in n generated by
g.
Here
and throughout this paper aujav
=
'Vu. v and vis the outward normal to an.
We seek the solution to (2.1) in the form
(2.2) u(x)
= {
r(x- y)/J(y)da(y), X
En,
lan
where
r
is a fundamental solution to the Laplacian and given by
{
2_
ln
lxl,
d
=
2,
27f
r(x)
:=
1
2 d
(2 _
d)w)xl - , d ;:::
3.
The function u defined by (2.2) is harmonic inn (also in JRd \ 0). Thus for u to be
the solution to (2.1), it suffices to satisfy the boundary condition. For this purpose,
we have to investigate the boundary behavior of
av
a
r
r(x- y)¢(y)da(y),
1 an
as
X
approaches to an.
2.1. Layer potentials for Laplacian. Given a bounded Lipschitz domain
D
in JRd,
d ;::: 2,
we will denote the single and double layer potentials of a function
¢
E
L
2
(aD) as SD¢ and VD¢, respectively, where
SD¢(x)
:=
r
r(x- y)¢(y) da(y), X
E
JRd,
laD
VD¢(x)
:= {
aa r(x- y)/J(y) da(y), x
E
]Rd
\aD.
laD
Vy
Observe that both SD¢ and VD¢ are harmonic in D and JRd \D. For a function u
defined on JRd \aD, we denote
ui±(x)
:=
lim u(x
±
tvx), x
E
aD,
t-+O+
and
aa
ul (x)
:=
lim ('Vu(x
±
tvx), vx), x
E
aD,
Vx
±
t-+O+
if
the limits exist. Here
Vx
is the outward unit normal to aD at x, and (,) denotes
the scalar product in JRd.
We want to investigate ,tvSDPI±(x) and VD¢I±(x) for x
E
aD. Fix
z
E
aD.
Since
( ( ) )
1 (x-y,vz)
'Vr x -
y ,
Vz
= -
I ld ,
wd
x-y
XED, y
E
aD,
we may expect that
a
a
r
r(x- y)/J(y)da(y)
-+
2_
r
(~-
y,
~:)
/J(y)da(y) as X-+
Z.
V
1
aD
Wd
1
aD
Z -
Y
But this is not the case. The main difficulty is that since
I (z-
y,
Vz)
I :::; Clz-
Yll-d
lz
-yld
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