Nonlinear potentials at the fractional scale: sharp regularity

Nonlinear potential theory and elliptic regularity theory are two classical topics in the modern analysis of partial differential equations. In this talk I show how these themes merge to solve the longstanding open problem, dating back to the seminal contributions of O. A. Ladyzhenskaya & N. N. Ural’tseva, N. S. Trudinger, and L. Simon (1967-1976), of deriving Schauder estimates for minima of functionals (resp. solutions to elliptic equations) featuring polynomial nonuniform ellipticity. The sharp rate of nonuniform ellipticity for the validity of Schauder theory is also disclosed. From recent, joint work with Giuseppe Mingione (Parma).