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

Abstract

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

This is the definitive version of a lecture delivered at the International Conference on "Recent Advances in PDEs" in memory of Filippo Chiarenza, Messina, December 15--17, 2005; 6 figures