8 HABIB AMMARI AND HYEONBAE KANG
Here and throughout this paper, we define the space L5(aD) by
L~(aD)
:= {
¢
E
L
2
(aD): hn ¢da
=
0
The key fact in proving Theorem 2.4 is the following: For a function u the
tangential derivative of u along aD is defined to be
au
d-l
au
EJT
:=
L
aT:
Tp,
p=l p
where T1
, ... ,
Td-l
is an orthonormal basis for the tangent plane to aD at x.
LEMMA
2.5. Let D be a bounded Lipschitz domain in
JRd,
d
~
2. Let u be a
function such that either
(i) u is Lipschitz in D and Llu
=
0
in D, or
(ii) u is Lipschitz in JRd \ D, Llu
=
0 in JRd \ D, and
lu(x)l
=
0(1/lxld-2
)
when d
~
3
and
lu(x)l
=
0(1/lxl)
when d
=
2
as
lxl
~
+oo.
Then there exists a positive constant
C
depending only on the Lipschitz character
of D such that
(2.6)
~
II;; L2(8D)
~ II~~
II£2(8D)
~
c
II;; L2(8D).
Lemma 2.5 says that the £
2
norms of the normal and tangential derivatives of
a harmonic function are comparable, and can be proved using the Rellich identity
[132];
see also
[18].
Let us briefly see how Lemma 2.5 leads us to Theorem 2.4.
Let u(x)
=
Snf(x), where f
E L~(aD).
Because of the jump formula (2.3), we
have
and by (2.6)
~~~-
£2(8D)
au
I
a ,
1/
+
£2(8D)
or equivalently
(2.7)
~ 11(~1
+ KiJ)fii£2(8D)
~ 11(~1-
KiJ)fii£2(8D)
~ c 11(~1
+ KiJ)fii£2(8D)·
Since f
=
((1/2)1
+
Ki))f
+
((1/2)1- Ki))f, (2.7) yields that
11(~1
+
Ki))fii£2(8D)
~
Cllfii£2(8D)·
We also have the following estimate from
[16].
LEMMA
2.6. There exists a constant C depending only on the Lipschitz char-
acter of D such that for any
I .AI
~
1/2
II¢11£2(8D)
~ c 1~111(.!-
Ki))
¢11£2(8D)
for all¢
E
£5(aD).
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