On the W 2 p Estimate for Oblique Derivative Problem in Lipschitz Domains

Abstract

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form 2nd-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [ 16] and [ 17], which work for $C^{1,\alpha }$ domains with $\alpha> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.