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
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