Calculus of Variations and Geometric Measure Theory
home | mail | papers | authors | news | seminars | events | open positions | login

N. Gigli - F. Nobili

A differential perspective on Gradient Flows on ${\sf CAT}(κ)$-spaces and applications

created by nobili on 25 Dec 2020
modified on 06 Oct 2021


Published Paper

Inserted: 25 dec 2020
Last Updated: 6 oct 2021

Journal: J. Geom. Anal.
Year: 2021

ArXiv: 2012.12952 PDF


We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on ${\sf CAT}(\kappa)$-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.

Credits | Cookie policy | HTML 5 | CSS 2.1