GENERALIZED POLARIZATION TENSORS
(i)
SD :
L
2
(aD) ___, W[(aD)
bounded.
(ii)
KD :
W[(aD) ___, W[(aD)
bounded.
7
Observe that if D is a two dimensional disk with radius
r,
then, as was observed
in
[87],
(x-
y,
Vx)
lx-yl2
and therefore, for any¢
E
L
2
(aD),
1
2r
'II
x,y
E
aD,x i=- y,
(2.5) K'D¢(x)
=
KD¢(x)
=
4
1
{ ¢(y) da(y),
1fT
laD
for all
x
E
aD.
Ford
2:
3, if
D
denotes a sphere with radius r, then, since
(x-
y,
vx)
1 1
I
ld
I
ld
2
v
x, Y
E
aD, xi=- y,
X-
y 2r
X-
y -
we have, as shown by Lemma 2.3 of
[89],
that for any¢
E
L
2
(aD),
* (2- d)
KD¢(x)
=
KD¢(x)
=
~SD¢(x)
for all
X
E
aD.
In particular, if
Dis
a ball in
JRd,
d
2:
2, then KD is a self-adjoint
operator on
L
2
(aD).
The converse is also true: Let
D
be a Lipschitz domain. If
KD is self-adjoint, then
Dis
a ball. This was proved by Lim in
[108].
There is a conjecture that has not been resolved completely. Observe that
KD(1)
=
1/2,
and hence
K'D(1)
=
1/2
provided that D is a ball. The conjecture
is that if
K'D(1)
=
1/2
and
Dis
a Lipschitz domain, then
Dis
a ball. The conjec-
ture has been proved to be true for some important classes of domains: Piecewise
smooth domains in IR2 by Martensen
[113],
star-shaped C2•"-domains by Payne
and Philippin
[119]
and Philippin
[121],
and C2 •"-domains by Reichel
[122, 123].
Recently Mendez and Reichel proved the conjecture for bounded Lipschitz domains
in IR2 and bounded Lipschitz convex domains in
!Rd (d
2:
3)
[114].
Let us now go back to the problem
(2.1).
If the solution u takes the form Sn¢
for some ¢ in
n,
then by the boundary condition we need to have
:v
Sn¢1_
=
g
on
an.
It then follows from (2.3) that
(-~1+Kv)¢=g
onan.
In other words we need to invert the operator
(-(1/2)1
+
K'D) on £
2
(8!1).
If
Dis
a C1•" domain, then K'D is a compact operator, and hence (
-(1/2)1
+K'D)
is a Fredholm operator of index zero. Therefore, by the Fredholm alternative, the
question of invertibility is reduced to that of injectivity, and injectivity of (
-(1/2)1
+
K'D) can be proved without much effort. See
[70].
But if
aD
is merely Lipschitz, K'D
is not a compact operator and invertibility of (
-(1/2)1
+
K'D) is a serious matter.
The following theorem is due to Verchota
[132]
for
I.AI =
1/2
and Escauriaza et al.
[68]
for
I.AI
1/2.
THEOREM
2.4.
The operator AI- K'D is invertible on
L5(aD)
if
I.AI
2:
1/2,
and for
.A
E]-
oo,
-1/2]U]1/2,
oo[,
AI- K'D is invertible on L
2
(aD).
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