Let
be an
uniformly elliptic matrix with Hölder continuous real coefficients and let
be the fundamental
solution of the PDE
in
. Let
be a compactly
supported
-AD-regular
measure in
and consider the associated operator
We show that if
is bounded in
,
then
is uniformly
-rectifiable. This extends the
solution of the codimension-
David–Semmes problem for the Riesz transform to the gradient of the single-layer potential.
Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given
with finite
Hausdorff measure
,
if
is
bounded in
,
then
is
-rectifiable.
Further, as an application we show that if the elliptic measure associated to
the above PDE is absolutely continuous with respect to surface measure,
then it must be rectifiable, analogously to what happens with harmonic
measure.
Keywords
David–Semmes problem, uniform rectifiability, gradient of
the single-layer potential, elliptic measure,
rectifiability