$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

We prove a global \begin{document}$ W^{2, p} $\end{document} -estimate for the viscosity solution to fully nonlinear elliptic equations \begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{2, \alpha} $\end{document} -domain for every \begin{document}$ \alpha\in (0, 1) $\end{document} . Here, the nonlinearities \begin{document}$ F $\end{document} is assumed to be asymptotically \begin{document}$ \delta $\end{document} -regular to an operator \begin{document}$ G $\end{document} that is \begin{document}$ (\delta, R) $\end{document} -vanishing with respect to \begin{document}$ x $\end{document} . We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global \begin{document}$ W^{2, p} $\end{document} -estimate for the viscosity solution to fully nonlinear parabolic equations \begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{3} $\end{document} -domain.