Elliptic and Parabolic Equations with Measurable Coefficients in $L_p$-spaces with Mixed Norms

The unique solvability results for second order parabolic and elliptic equations in Sobolev spaces with mixed norms are presented. The second order coefficients are measurable in one spatial variable and VMO (vanishing mean oscillation) in the other spatial variables. In the parabolic case, the coefficients (except a 11 ) are further allowed to be only measurable in time. We first prove the solvability results for equations in the whole Euclidean space. Then, using these results as well as some extension techniques, we prove the solvability results for equations on a half space without any boundary estimates. The mixed norms we present here are more general than the usual mixed norm L t q L x p .

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