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.
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