Korevaar-Schoen energy revisited

Korevaar and Schoen introduced, in a seminal paper in 1993, the notion of `Dirichlet energy’ for a map from a smooth Riemannian manifold to a metric space. They used such concept to extend to metric-valued maps the regularity theory by Eells-Sampson about Lipschitz regularity of harmonic functions, which, in the smooth setting, is based on a lower bound on the Ricci curvature+upper bound on the dimension for the source space and negative sectional curvature+simple connectedness for the target one. Both these assumptions have a natural synthetic counterpart: the RCD(K,N)RCD(K,N) condition on one side and the CAT(0)CAT(0) (or Hadamard) condition on the other. It is therefore natural to try to extend Korevaar-Schoen's theory - and ultimately Eells-Sampson’s one - to this setting. However, already the very definition of energy in the sense of Korevaar-Schoen does not generalize well to non-smooth source spaces: aim of this talk is to discuss this issue and a way to circumvent it.

From a joint work with A. Tyulenev.