Nonuniform Ellipticity and Nonlinear Potentials

Nonuniform Ellipticity and Nonlinear Potential Theory are two classical topics in the analysis of Partial Differential Equations. In this talk I show how those themes merge to solve the longstanding open problem of deriving Schauder estimates for minima of functionals (resp. solutions to elliptic equations) featuring polynomial non-uniform ellipticity. I also discuss some delicate borderline regularity results that hold when the polynomial non-uniform ellipticity condition is violated (e.g. if the ellipticity ratio grows exponentially fast) and even apply (almost everywhere) to relaxed minimizers of quasiconvex variational integrals. This is joint work with Giuseppe Mingione (University of Parma).