On $L_p$ estimates of optimal type for the parabolic oblique derivative problem with {VMO} coefficients

Abstract

We generalize recent results by L.G. Softova concerning $ W_p^{2,1} (\Omega_T) $ estimates ($ 1 < p < \infty$) for second order parabolic operators with VMO coefficients and the boundary condition $$ \sum_{i=1}^n b_i(\xi,t) \partial_i u(\xi,t) = g(\xi,t) \ \ \text{ on $ \partial \Omega_T $} $$ in the nondegenerate case (see Remark 2.2 i). While Softova assumed $ [(\xi,t) \mapsto b_i(\xi,t)] \in Lip(\partial \Omega_T) $, we weaken this assumption to $ b_i \in C^{ \alpha, \alpha/2 } (\partial \Omega_T) $ for some $ \alpha > 1 -1/p $.