L_p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

We prove the H_p,q¹ solvability of second order systems in divergence form with leading coefficients A^αβ only measurable in (t, x¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A¹¹ is measurable in t and has a small BMO semi-norm in the other variables; (ii) A¹¹ is measurable in x¹ and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.