The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

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(Rⁿ, dx) 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.