Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients

Abstract

We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a C 1 function whose first derivatives are Dini continuous. This improves a recent result in [6]. An extension to fully nonlinear elliptic equations is also presented.2010 Mathematics Subject Classification. Primary 35J25, 35B65; Secondary 35J15.