Variational convergence of EVI-gradient flows in metric spaces.

It is well known that gradient flows of convex and lower semicontinuous functionals $(\phi^h)_{h\in \mathbb N}$ in a Hilbert space are stable with respect to Mosco-convergence (i.e.~$\Gamma$-convergence with respect to the strong and the weak topology) of the driving functionals $\phi^h$.

In general metric spaces the situation is more delicate, due to lack of a weak topology and of the corresponding compactness properties. Moreover, the existence of a contracting semigroup generated by a (geodesically) convex functional also depends on the properties of the distance function.

We will show that for gradient flows characterized by a family of evolution variational inequalities (EVI-flows), a good stability property is in fact equivalent to a reinforced notion of $\Gamma$-convergence, which does not require any equi-compactness of the sublevels of the functional.