$W^{2,p}$-A~priori estimates for the neutral Poincaré problem

A degenerate oblique derivative problem is studied for uniformly elliptic operators with low regular coefficients in the framework of Sobolev's classes $W^{2,p}(Ω)$ for {\em arbitrary} $p>1.$ The boundary operator is prescribed in terms of a directional derivative with respect to the vector field $ł$ that becomes tangential to $\partial Ω$ at the points of some non-empty subset $\E\subset \partial Ω$ and is directed outwards $Ω$ on $\partialΩ\setminus\E.$ Under quite general assumptions of the behaviour of $ł,$ we derive {\it a priori} estimates for the $W^{2,p}(Ω)$-strong solutions for any $p\in(1,\infty).$

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