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

Abstract

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