The Poincaré Problem in L p -Sobolev Spaces II: Full Dimension Degeneracy

We study a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients within the framework of the Sobolev spaces W-2,W-p(Omega) with arbitrary p > 1. The boundary operator is prescribed in terms of a directional derivative with respect to a vector field l that becomes tangent to partial derivative Omega at the points of a non-empty set E subset of partial derivative Omega and is of emergent type on partial derivative Omega. We extend the results from Palagachev (2005) regarding a priori estimates, strong solvability, uniqueness and Fredholmness of the problem under consideration to the general case of arbitrary set of tangency E which may have positive surface measure.