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).