C^1, regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on C¹ domains

We provide a sharp C^1, estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a C¹ domain. To this end, we first obtain a uniform boundary H\"older estimate with the oblique boundary condition in an "almost C¹-flat" domain for the equations which is uniformly elliptic only where the gradient is far from some point, and then we establish a desired C^1, regularity based on perturbation and compactness arguments.