OPTiMiSE — Talks & Presentations
2026
- Recent Advances in PDEs and Geometric Analysis, February 17–18, 2026, Politecnico di Milano
Variational Convergence of Metric Gradient FlowsIn this talk we address variational convergence and stability properties of gradient flows in metric spaces. We begin by introducing the notion of metric gradient flow in the sense of Evolution Variational Inequalities (EVI), which provides a robust and intrinsic characterization of dissipative evolutions beyond the Hilbertian setting. Given a sequence of functionals \((\phi_h)\) defined on a metric space and generating corresponding gradient flows, we study the problem of understanding when and how the associated evolutions converge as the functionals vary. In the classical Hilbert space framework, convergence of gradient flows is closely related to Mosco (i.e. \(\Gamma\)-convergence with respect to the strong and weak topology) of convex energies and to the convergence of resolvent operators. In a general metric setting, however, several key tools are missing: resolvents may fail to be well defined or contractive, compactness is often unavailable, and no natural weak topology is present. We present a variational approach to stability that avoids strong coercivity assumptions and relies instead on quantitative error estimates for minimizing movement schemes and their Ekeland relaxations. These estimates provide a robust bridge between discrete variational approximations and continuous EVI flows, and allow one to transfer convergence properties from the energies to the evolutions under suitable variational assumptions. The results apply to a broad class of metric spaces admitting EVI gradient flows and yield general convergence criteria that extend classical Hilbert space results to genuinely metric frameworks, with applications for instance to Wasserstein and RCD spaces.
Joint work with Matteo Muratori.