Mappings minimizing the $L_p$ norm of the gradient

Abstract: Let $M$ and $N$ be two Riemannian manifolds. We will look at the set of $W^{1,p}$ mappings $u:M \to N$ having a fixed trace on boundary of $M$.  Our goal is to study the map that minimizes the $L_p$ norm of the gradient, $\int |D u|^p$, among all such mappings.   Hardt and Lin (1987) proved that a minimizer is locally Holder continuous outside a set of measure zero. In this talk, we will analyze the key steps involved in the proof of the result and do a quick introduction to the generalization of the result in fractional Sobolev spaces. This is based on joint work with Katarzyna Mazowiecka and Armin Schikorra.