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

@article{Gigli2019KorevaarSchoensDE,
  title={Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows},
  author={Nicola Gigli and A. Tyulenev},
  journal={arXiv: Functional Analysis},
  year={2019}
}
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… Expand
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