Sketches of Nonuniformly Elliptic Schauder Theory

Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting.