W^{2,p}-a priori estimates for the emergent Poincaré Problem

We derive W-2,W-p(Omega)-a priori estimates with arbitrary p epsilon (1, infinity), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field l that is tangent to delta Omega at the points of a non-empty set epsilon subset of delta Omega and is of emergent type on delta Omega.