W^{2,p}-Theory of the Poincaré Problem

We present some recent results regarding the W 2,p -theory of a degenerate oblique derivative problem for second order uniformly elliptic operators. The boundary operator is prescribed in terms of directional derivative with respect to a vector field l which is tangent to ¶W at the points of a nonempty set e Ì ¶W : Sufficient conditions are given ensuring existence, uniqueness and regularity of solutions in the L p-Sobolev scales. Moreover, we show that the problem considered is of Fredholm type with index zero.