Regularity in diffusion models with gradient activation

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.