$C^{1, \alpha}$-regularity for functions in solution classes and its application to parabolic normalized $p$-Laplace equations

We establish the global $C^{1, \alpha}$-regularity for functions in solution classes, whenever ellipticity constants are sufficiently close. As an application, we derive the global regularity result concerning the parabolic normalized $p$-Laplace equations, provided that $p$ is close to 2. Our analysis relies on the compactness argument with the iteration procedure.