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].