  We prove sharp regularity estimates for solutions of highly degenerate fully
nonlinear elliptic equations. These are free boundary models in which a
nonlinear diffusion process drives the system only in the region where the
gradient surpasses a given threshold. Our main result concerns the existence of
a universal modulus of continuity for $Du$, up to the free boundary. Gradient
bounds for the $L^\infty$ norm are proven to be uniform with respect to the
degree of degeneracy. Several new ingredients are needed and among the tools
introduced in this paper is an improvement of regularity lemma designed to
measure the oscillation decay concerning the gradient level-set distance.
Applications of the methods are discussed at the end of the paper.
