There is a long history of studying boundary value problems for elliptic an
d parabolic differential equations when the equation is posed ill a domain
with sufficiently smooth boundary. In this paper. we prove some pointwise e
stimates: and regularity results for solutions when a directional derivativ
e is prescribed on the boundary, which satisfies an interior cone condition
. we consider very weak hypotheses on the data of the problem. In particula
r, we consider an optimal relation between the interior cent: and the direc
tion of tile prescribed directional derivative. and we assume very little s
moothness of the coefficients in the equation and the boundary condition. (
C) 2001 Academic Press.