$W^{2,p}$-A~priori estimates for the neutral Poincar\'e problem

TL;DR

This paper establishes $W^{2,p}$ a priori estimates for a class of degenerate elliptic boundary value problems with oblique derivatives, extending regularity results to cases with low regular coefficients and tangential boundary conditions.

Contribution

It provides the first comprehensive $W^{2,p}$ a priori estimates for degenerate oblique derivative problems with low regularity coefficients and tangential boundary conditions.

Findings

Derived $W^{2,p}$ estimates for solutions with arbitrary $p>1$.

Handled boundary operators with directional derivatives tangential to the boundary.

Extended regularity theory to degenerate elliptic problems with low regularity coefficients.

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 $\l$ that becomes tangential to $\partial \Omega$ at the points of some non-empty subset $\E\subset \partial \Omega$ and is directed outwards $\Omega$ on $\partial\Omega\setminus\E.$ Under quite general assumptions of the behaviour of $\l,$ we derive {\it a priori} estimates for the $W^{2,p}(\Omega)$-strong solutions for any $p\in(1,\infty).$