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