The Variations of Yang–Mills Lagrangian

Abstract

We are giving a survey on some of the analysis methods from gauge theory developed in the last decades. We first cover Uhlenbeck’ s compensated compactness theory in critical 4 dimension for the Yang–Mills functional. As an application we present the resolution of minimization processes of Yang– Mills in this critical dimension. In the second part of the survey we present the resolution of similar variational questions in super-critical dimensions and we end up the survey by stating some open problems raised by Tian relative to the regularity of Ω-anti-self-dual instantons in high dimensions.