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