Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows

Abstract

We develop Korevaar-Schoen's theory of directional energies for metric-valued Sobolev maps in the case of ${\sf RCD}$ source spaces; to do so we crucially rely on Ambrosio's concept of Regular Lagrangian Flow. Our review of Korevaar-Schoen's spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of `differential of a map along a vector field' and about the parallelogram identity for ${\sf CAT}(0)$ targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.