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The Dirichlet problem for elliptic operators having a BMO anti-symmetric part
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Journal Article
The Dirichlet problem for elliptic operators having a BMO anti-symmetric part
2022
Overview
The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a
BMO
anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation
div
(
A
∇
u
)
=
0
in the upper half-space
(
x
,
t
)
∈
R
+
n
+
1
is uniquely solvable when
n
≥
2
and the boundary data is in
L
p
(
R
n
,
d
x
)
for some
p
∈
(
1
,
∞
)
. This result is equivalent to saying that the elliptic measure associated to
L
belongs to the
A
∞
class with respect to the Lebesgue measure
dx
, a quantitative version of absolute continuity.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
