14 HABIB AMMARI AND HYEONBAE KANG
In order to derive a representation formula for the solution to (2.25), we need
a periodic Green's function. Let
ei21fn·x
G(x) =-
L
47r21nl2.
nEZd\{0}
Then we get, in the sense of distributions,
~G(x)
=
L
ei21fn•x
=
L
ei21fn·x -
1.
nEZi\{0} neza
It then follows from the Poisson summation formula
L
ei2.,.n·x
=
L
6(x
+
n),
that
(2.26)
neza neZd
~G(x)
=
L
6(x+n) -1.
nEZd
Appearance of the constant 1 in (2.26) may be a little peculiar. It is the volume of
Y and an integration by parts shows that it should be there. In fact,
{
~G(x)dx
= {
8°du,v8
lY laY
and the righthand side is zero because of the periodicity.
For simplicity we only detail the two-dimensional case. The following results
were obtained in
[29].
LEMMA
2.12.
Suppose that d
= 2.
There exists a harmonic function R(x) in
the unit cell Y such that
(2.27)
1
G(x)
=
27r
ln
lxl
+
R(x).
Moreover, the Taylor expansion of R(x) at
0
is given by
1
(2.28)
R(x)
=
R(O)-
4 (x~
+
x~)
+
O(lxl
4
).
The periodic single layer potential of the density function ¢J
E L~(8D)
is defined
by
gD¢J(x)
:= {
G(x- y)¢J(y) du(y), x
E
IR2

laD
Lemma
2.12
shows that
(2.29)
gD¢J(x)
=
SD¢J(x)
+
'RD¢J(x),
where 'RD is a smoothing operator defined by
'RD¢J(x)
:= {
R(x- y)¢J(y) du(y).
laD
Thanks to (2.29), we have
:vgD¢JI± (x)
=
:vSD¢JI± (x)
+
:v'RD¢J(x), x
E
8D.
Thus we can understand tvgD¢JI± as a compact perturbation of tvSD¢JI±· Based
on this natural idea, we obtain the following results from
[29].
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