Regularity in diffusion models with gradient activation

Abstract

We prove 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. The model is mathematically framed through a new notion of viscosity solutions, where test functions with insufficiently large gradients are disregarded. Our main result concerns the existence of a universal modulus of continuity for Du, up to the free boundary. Gradient bounds with respect to the $L^{\infty }$ norm are proven to be uniform with respect to the degree of degeneracy. Several new ingredients are needed and further applications of the methods are discussed at the end of the paper.