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 $$y_t{\textquoteright}\in -\partial ^-\textsf {E} (y_t)$$one uses in the smooth setting and more precisely that $$y_t{\textquoteright}$$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{\textendash}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.

}, isbn = {1559-002X}, url = {https://doi.org/10.1007/s12220-021-00701-5}, author = {Nicola Gigli and Francesco Nobili} }