arXiv subjects
Publications and source records attributed to Francesco Nobili.
We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all $\rm BV$ functions and their total variation measures. In the setting of non-branching ${\sf CD}(K,N)$ spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.
We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on ${\sf CAT}(κ)$-spaces and prove that they can be characterized by the same differential inclusion $y_t'\in-\partial^-{\sf E}(y_t)$ one uses in the smooth setting and more precisely that $y_t'$ selects the element of minimal norm in $-\partial^-{\sf E}(y_t)$. 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 ${\sf CAT}(0)$ valued maps: we define the Laplacian of such $L^2$ map as the element of minimal norm in $-\partial^-{\sf E}(u)$, 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.