SUB-LAPLACIANS WITH DRIFT
11
DEFINITION.
We define the (first order) correctors ip^n j q (cf. [BLP,
JKO]) as solutions of the problem
(1.5.4) L
M
^ = - C i + Y
ZA^
^') = 0.
1Z7TI
If 1 j n then we just set ^ 0.
Note that (Zibij) = 0, 1 i m and that by the property (6) in section 1.5.1,
(Cj) = 0, n j q. So the correctors are well defined.
Combining (1.5.3) and (1.5.4), we have that there is c 0 independent of /
such that
(1.5-5) L(f+ £ *±.f\
\ \jq
3
J
= - Y (Cii-c*^'+ Y2
fe«w
+ Yl
z*(b*i^))
li,3Q V l£ra l£m /
cta^ dxj
with the function F satisfying \F(x,z)\
c|V3/(x)|,
(x,z)
eRq
x M.
This expression motivates the following definitions:
DEFINITION.
The homogenised sub-Laplacian L# associated to L is defined to
be the operator
with coefficients defined by
Qtj = ( Cij ~
d^J
+ E ^ ^ ^ ) , 1 i,j q-
\ 0£m I
DEFINITION.
We define the (second order) correctors ^ u , 1 hj Q (cf.
[BLP, JKO]) as solutions of the problem
LM^ =
clJ-c^j+
Yl
h^x^3+
Yl Zifri^-Qij*
(rj)
= o-
l£m l£m
Combining the above defionitions with (1.5.5) we have the following expression
which explains the relation between the operators L and L#:
(1.5.6) l(f+ £ V ^ / + £ ^
dxi dxj J
with the function F satisfying \F(x, z)\ c
(|V3/(^)I
+
|V4/(x)|)
,(x,z)eNx M.
3. LH is an elliptic operator on
Rq.
To prove this, it is enough to prove that
if^ = ( 6 , . . . , e n
1
) e R
n i
, ? ^ 0 , t h e n
(1.5.7) Yl
KMJ°-
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