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Abstract: In this talk, we discuss recent advances on up to the boundary gradient estimates for viscosity solutions of free boundary problems governed by fully nonlinear and quasilinear equations with unbounded coecients. We present the new Inhomogeneous Pucci Barriers as new elements for the proof. If time permits, we discuss some of the main steps in the proof, namely, the trace estimate of the solution on the points of the xed boundary that projects nontangentially over the free boundary. These methods are inspired by some ideas of Carlos Kenig in Harmonic Analysis.
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