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

Abstract

<p style='text-indent:20px;'>We prove a global <inline-formula><tex-math id="M1">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear elliptic equations <inline-formula><tex-math id="M2">\begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M3">\begin{document}$ C^{2, \alpha} $\end{document}</tex-math></inline-formula>-domain for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\in (0, 1) $\end{document}</tex-math></inline-formula>. Here, the nonlinearities <inline-formula><tex-math id="M5">\begin{document}$ F $\end{document}</tex-math></inline-formula> is assumed to be asymptotically <inline-formula><tex-math id="M6">\begin{document}$ \delta $\end{document}</tex-math></inline-formula>-regular to an operator <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula> that is <inline-formula><tex-math id="M8">\begin{document}$ (\delta, R) $\end{document}</tex-math></inline-formula>-vanishing with respect to <inline-formula><tex-math id="M9">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global <inline-formula><tex-math id="M10">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear parabolic equations <inline-formula><tex-math id="M11">\begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M12">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>-domain.</p>