Elliptic Theory for Sets with Higher Co-Dimensional Boundaries

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 ⊂ Rⁿ be an Ahlfors regular set of dimension d < n - 1 (not necessarily integer) and Ω = Rⁿ \ Γ. Let L = - div A∇ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A are bounded from above and below by a multiple of dist(·, Γ)^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 estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to 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 is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L for which the harmonic measure given here is absolutely continuous with respect to the d-Hausdorff measure on Γ and vice versa. It thus extends Dahlberg's theorem to some sets of codimension higher than 1.