Geometric approach to Boundary Schauder estimates with an application

Abstract: The fundamental role of Schauder estimates (both interior and at the boundary) in the theory of elliptic and parabolic partial differential equations is well-known.  The classical approach to boundary Schauder estimates rely on flattening the boundary  so that reflection techniques can be applied to the constant coefficient problem and    which consequently facilitates the use of interior estimates for the constant coefficient problem and then obtains similar estimates for the original problem via standard perturbative techniques. In this talk, I will present an approach  to boundary Schauder estimates where one doesn't require to flatten the boundary and which   instead  employs  geometric compactness arguments which has its roots in some of the fundamental works of Luis Caffarelli in the early 90's.  If time permits, I will try to indicate  the robustness of this coordinate free approach in the Non-Euclidean setting of Carnot groups where there is a lack of ellipticity  at every point.  This is based on some recent as  well as some   ongoing joint works with Nicola Garofalo and Isidro Munive.