W^{2,1}_p Solvability for Parabolic Poincare Problem

We study Poincar'e problem for a linear uniformly parabolic operator $\mathrm{\P }$ in a cylinder $Q=\mathrm{\Omega }\times (0,T)\mathrm{.}$ The boundary operator $\text{B}$ is defined by an oblique derivative with respect to a tangential vector field $\text{l}$ defined on the lateral boundary $S\mathrm{.}$ The coefficients of $\mathrm{\P }$ are supposed to be $VMO$ away from the set of tangency $E$ and to possess higher regularity in $x$ near to $E\mathrm{.}$ A unique strong solvability result is obtained in $W_p^{2,1}(Q)$ for all $p\in (1,\mathrm{\infty })\mathrm{.}$