The lecture was held within the framework of the Hausdorff Trimester Program: Evolution of Interfaces.

Abstract:
In this talk we discuss recent developments in regularity theory for some classes of nonlinear PDEs. Our arguments relate a problem of interest to another one, for which a richer theory is available. It operates in two distinct layers; first compactness builds upon suitable notions of stability to produce approximation results. Then, a scaling argument localizes the analysis to establish (in some cases, sharp) regularity results. The toy-models we cover include fully nonlinear PDEs, the Isaacs equation, double-divergence problems and degenerate/singular equations. We close the talk with an excursion into the realm of free boundary problems.

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