Up to the boundary gradient estimates for viscosity solutions to nonlinear free boundary problems with unbounded measurable ingredients

In this paper, we prove up to the boundary gradient estimates for viscosity solutions to inhomogeneous nonlinear Free Boundary Problems (FBP) governed by fully nonlinear and quasilinear elliptic equations with unbounded measurable ingredients. Here, we build upon our previous results in [9] to construct Inhomogeneous Pucci Barriers (IPB) for the Pucci extremal equations with unbounded coefficients. Using these barriers, we obtain a version of a boundary growth type lemma for inhomogeneous nonlinear equations that may be of independent interest. In a certain way, this lemma detects the expansion of the level sets of supersolutions from the boundary to the interior. The use of this boundary growth type lemma together with the geometry of IPB bridge the interchanging information between the free boundary condition and Dirichlet boundary data on the free and fixed boundary respectively. This produces an estimate on the trace of solutions to FLIP along the fixed boundary. This way, control of such solutions (up to the boundary) by the distance to the negative phase is obtained. Finally, this distance control combined with the PDE boundary gradient estimates render our final result.