BMO solvability and the $A_{\infty}$ condition of the elliptic measure in uniform domains

Abstract: We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $\omega_L\in A_{\infty}(\sigma)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.