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 second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14, 15], 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.