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

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