Abstract. We show that for a uniformly elliptic divergence form operator L, defined in an open set Ω with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-A ∞ property) of elliptic-harmonic measure with respect to surface measure on ∂Ω. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of Ω. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that Ω satisfies an interior Corkscrew condition, in the special case that L is the Laplacian.

We prove that the Dirichlet problem for degenerate elliptic equations div(A∇u) = 0 in the upper half-space (x, t) ∈ R n+1 + is solvable when n ≥ 2 and the boundary data is in L p µ (R n ) for some p < ∞. The coefficient matrix A is only assumed to be measurable, real-valued and t-independent with a degenerate bound and ellipticity controlled by an A 2 -weight µ. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in A ∞ with respect to the µ-weighted Lebesgue measure on R n . The Carleson measure estimate allows us to avoid applying the method of ǫ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.

No abstract

No abstract