Pointwise Estimates for Oblique Derivative Problems in Nonsmooth Domains

There is a long history of studying boundary value problems for elliptic and parabolic differential equations when the equation is posed in a domain with sufficiently smooth boundary. In this paper, we prove some pointwise estimates and regularity results for solutions when a directional derivative is prescribed on the boundary, which satisfies an interior cone condition. We consider very weak hypotheses on the data of the problem. In particular, we consider an optimal relation between the interior cone and the direction of the prescribed directional derivative, and we assume very little smoothness of the coefficients in the equation and the boundary condition.