Sobolev space theory of SPDEs with continuous or measurable leading coefficients

We study stochastic partial differential equations with variable coefficients defined on R d , R + d and bounded C 1 domains. For equations with continuous leading coefficients we give existence and uniqueness results in L q ( L p ) -spaces, where it is allowed for the powers of summability with respect to space and time variables to be different. For equations with measurable leading coefficients we give unique solvability in L p -spaces.