Nonuniformly elliptic Schauder theory

Schauder estimates are a basic tool in elliptic and parabolic PDE. The idea is to show that solutions are as regular as coefficients allow. They serve as a basic tool in a wide variety of situations: higher regularity of solutions to problems showing any kind of ellipticity, including free boundaries, bootstrap processes, existence theorems and so on.
Their validity in the setting of linear uniformly elliptic problems is classical. First results were obtained by Hopf, Giraud, Caccioppoli and Schauder in the 20/30s of the past century. Extensions were obtained by Agmon, Douglis and Nirenberg. New proofs were achieved over the years by Campanato, Trudinger, Simon (via suitable function spaces, convolution, blow-up, respectively). Nonlinear versions were achieved by Giaquinta and Giusti, DiBenedetto, Manfredi. More recently, nonlocal versions were obtained as well.
As the equations in question are non-differentiable, all these approaches unavoidably rely on perturbation methods, i.e., freezing coefficients and comparing original solutions to solutions with constant coefficients problems. Such approaches, relying on the availability of homogenous estimates for frozen problems, ceases to work in nonuniformly elliptic problems, for which such homogeneity is lost, and for which the validity of Schauder theory has remained an open problem for decades.
We shall present a full solution to the problem of Schauder estimates in the nonlinear, nonuniformly elliptic setting. In particular, we shall present the first direct, non-perturbative approach to pointwise gradient estimates for non-differentiable equations ever.
From recent, joint work with Cristiana De Filippis (University of Parma).