Neumann problem for non-divergence elliptic and parabolic equations with BMOₓ coefficients in weighted Sobolev spaces

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We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMOₓ coefficients.