Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a¹¹) are measurable in time and one spatial variable, and have small BMO (bounded mean oscillation) semi-norms as functions of the other spatial variables. The coefficient a¹¹ is measurable in time and have a small BMO semi-norm in the spatial variables. The unique solvability of equations in the whole space is established. Then this result is applied to solving Dirichlet and Neumann boundary value problems for parabolic equations defined on a half-space or on a bounded domain.