A $W^n_2$-Theory of Elliptic and Parabolic Partial Differential Systems in $C^1$ domains

Abstract: In this paper second-order elliptic and parabolic partial differential systems are considered on $C^1$ domains. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solutions to blow up near the boundary. The coefficients of the systems are allowed to substantially oscillate or blow up near the boundary.