GENERALIZED POLARIZATION TENSORS 9

2.2. Neumann and Dirichlet functions.

Let n be a bounded Lipschitz

domain in

~d,

d

2:

2. Let

N(x, z)

be the Neumann function

for~

inn corresponding

to a Dirac mass at

z.

That is,

N

is the solution to

(2.8)

{

~xN(x,

z)

=

-fiz

in

n,

oN 1

f

OVx

I

an=

-lonl

'lan N(x, z) da(x)

=

0 for zEn.

Note that the Neumann function

N(x, z)

is defined as a function of

x

E

0

for

each fixed

z

E n. The operator defined by

N(x, z)

is the solution operator for the

Neumann problem

(2.9)

{

~u

=

o

inn,

au!

- -g

ov an- '

namely, the function U defined by

U(x)

:= {

N(x, z)g(z)da(z)

lan

is the solution to

(2.9)

satisfying

fan

U

da

=

0.

ForD, a subset of n, let

NDf(x)

:=

f

N(x, y)f(y) da(y).

laD

The following lemma from

[15]

relates the fundamental solution with the Neu-

mann function.

LEMMA

2.7.

Forz

En

andx

Eon,

let

rz(x)

:=

r(x-z) andNz(x)

:=

N(x,z).

Then

(2.10)

(-~I+

Kn)

(Nz)(x)

=

r

z(x) modulo constants,

X

E

an,

or, to be more precise, for any simply connected Lipschitz domain D compactly

contained inn and for any g

E

L~(oD),

we have for any x

Eon

laD

(-~I+

Kn)

(Nz)(x)g(z) da(z) =laD rz(x)g(z) da(z).

Observe that we can express

(2.10)

in the following form: for any

g

E

L~(8D)

(2.11) (

1

)-l

NDg(x)=

-2I+Kn

((SDg)lan)(x), xEon.

We have a similar formula for the Dirichlet function. Let

G(x, z)

be the Green's

function for the Dirichlet problem in n, that is, the unique solution to

{

~xG(x,

z)

=

-fiz

in

n,

G(x, z)

=

0 on

an,

and let

Gz(x)

=

G(x, z).

Then for any

X

E

an,

and zEn we can prove that

( ~I

+Kn)-l(arz(Y))(x)

=-

8Gz(x).

2 OVy OVx