Nonlinear potential theoretic methods in nonuniformly ellliptic problems

Nonuniform ellipticity is a classical topic in the theory of partial differential equations. While several results in regularity theory have been adding up over decades, many basic issues, as for instance the validity of Schauder theory and sharp dependence of regularity upon data, remained opened for a while. In these notes we give an overview of recent results and techniques about the topic, that, via a novel use of nonlinear potential theoretic methods, allow to answer several of the above questions.