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} } @booklet {2021, title = {A first-order condition for the independence on p of weak gradients}, year = {2021}, author = {Nicola Gigli and Francesco Nobili} } @booklet {2021, title = {On master test plans for the space of BV functions}, year = {2021}, author = {Francesco Nobili and Enrico Pasqualetto and Timo Schultz} } @booklet {2021, title = {Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds}, year = {2021}, abstract = {\

We prove that if\ M\ is a closed\ n-dimensional Riemannian manifold,\ n>=3, with\ Ric>=n-1\ and for which the optimal constant in the critical Sobolev inequality equals the one of the\ n-dimensional sphere\ Sn, then\ M\ is isometric to\ Sn. An almost-rigidity result is also established, saying that if equality is almost achieved, then\ M\ is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the\ RCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.}, author = {Francesco Nobili and Ivan Yuri Violo} }

An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact\ CD\ space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of\ RCD\ spaces and on a Polya-Szego inequality of Euclidean-type in\ CD\ spaces.

As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the\ RCD-setting.