Krylov and Safonov Estimates for Degenerate Quasilinear Elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient. In particular, the given estimate is independent of the regularity of the coefficients. As in the original paper by Krylov and Safonov, the proof relies on a probabilistic interpretation of the equation.