Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.
To this end, we turn to degenerate elliptic equations. Let
Γ
⊂
R
n
\Gamma \subset \mathbb {R}^n
be an Ahlfors regular set of dimension
d
>
n
−
1
d>n-1
(not necessarily integer) and
Ω
=
R
n
∖
Γ
\Omega = \mathbb {R}^n \setminus \Gamma
. Let
L
=
−
div
A
∇
L = - \operatorname {div} A\nabla
be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix
A
A
are bounded from above and below by a multiple of
dist
(
⋅
,
Γ
)
d
+
1
−
n
\operatorname {dist}(\cdot , \Gamma )^{d+1-n}
. We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or
L
p
L^p
estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to
L
L
, establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions.
In another article to appear, we will prove that when
Γ
\Gamma
is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator
L
L
for which the harmonic measure given here is absolutely continuous with respect to the
d
d
-Hausdorff measure on
Γ
\Gamma
and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.
Noncommutative elliptic theory. Examples
