GENERALIZED POLARIZATION TENSORS

LEMMA

2.13.

(i) Let¢

E

L~(8D).

The following trace formula holds:

:v9v¢1± (x)

=

(±~I+

B'D)¢(x) on {}D,

where B'D :

L~(8D)-; L~(8D)

is given by

B'D¢(x)

=

p.v. {

8

8

G(x- y)¢(y) de7(y).

lev

Vx

(ii) If¢

E

L~(8D),

then 9v¢ is harmonic in D andY\ D.

(iii) If

I-XI ~ ~'

then the operator AI- B'D is invertible on

L~(8D).

Analogously to Theorem 2.9 the following result holds.

15

THEOREM

2.14.

Let ui be the unique solution to the transmission problem

(2.25).

Then Ui can be expressed as follows

Ui(x) =Xi+ ci

+

9v(AI- B'D)-

1

(vi)(x) in Y,

i

=

1, 2,

where .X is given by

(2.16},

Ci is a constant and vi is the i-component of the outward

unit normal v to {}D.

2.5. Anisotropic transmission problem. Let D be a bounded Lipschitz

domain in JRd,

d

=

2, 3. Let

A

be a positive-definite symmetric matrix and

A*

be the positive-definite symmetric matrix such that A-

1

=

Az. Let rA(x) be the

fundamental solution of the operator 'V · A 'V:

where

IAI

is the determinant of

A,

and 11·11 is the usual norm of the vector in JRd.

The single and double layer potentials associated with A of the density function

¢

on {}Dare defined by

and

S~¢(x)

:=

r

rA(x- y)¢(y)de7(y), X

E

Rd,

lev

V~¢(x)

:=

r

Vy.

A'VrA(x- y)¢(y)de7(y), X

E Rd

\{}D.

lev

The following jump formulae are well-known:

(2.30)

Vx •

A'VS~¢(x)l+-

Vx ·

A'VS~¢(x)l-

=

¢(x) a.e. x

E

8D,

V~¢(x)l+- V~¢(x)l- =

-¢(x) a.e. x

E

8D.

Let

A

be a constant

d

x

d

positive-definite symmetric matrix with

A

=f.

A.

We

always suppose that

A -

A is either positive-definite or negative-definite. We will

useS~ as a notation for the single layer potential associated with the the domain

D

and the matrix

A.

The following result of Escauriaza and Seo

[69]

is of importance to us.