01404nas a2200157   4500008004100000020001400041245008300055210006900138260001500207300001800222490000700240520091200247100001801159700002201177856004701199       2021                        eng d  a1559-002X00aA Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications0 aDifferential Perspective on Gradient Flows on CATKSpaces and App  c2021/12/01  a11780 - 118180 v313 a<p>We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$-spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$one uses in the smooth setting and more precisely that $$y_t'$$selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$and CAT(0) valued maps: we define the Laplacian of such $$L^2$$map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$, provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$-dense. Basic properties of this Laplacian are then studied.</p>1 aGigli, Nicola1 aNobili, Francesco  uhttps://doi.org/10.1007/s12220-021-00701-5