$C^{1, \alpha}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

In this paper, we establish $C^{1, \alpha}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, \alpha}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary H\"{o}lder estimates for equations which are uniformly elliptic when the gradient is either small or large.