OPTiMiSE — Talks & Presentations
2026
- Multiscale Stochastics, Patterns, and Analysis of Combinatorial Environments (mSPACE Kickoff Meeting), March 16–19, 2026, Bocconi University, Milano
The De Giorgi Variational Principle for Gradient Flows: A Direct ApproachDe Giorgi's variational principle provides a global-in-time characterization of gradient flows of a functional \(\phi\) as curves minimizing an energy-dissipation functional. While the existence of minimizers is relatively easy to establish, the original formulation does not immediately imply that the minimum value coincides with the initial energy \(\phi(x_0)\). This identification is crucial, since only minimizers attaining this value correspond to genuine solutions of the gradient flow. We present a direct variational approach showing how to prove that the minimum equals \(\phi(x_0)\), without relying on time-discretization or iterative constructions such as the minimizing movements scheme. The key idea is to replace iterative arguments with a global method based on relaxation, convexification, and duality, allowing one to identify the value of the minimum directly at the continuous level. Although the presentation focuses on the simplest case of gradient flows in Hilbert spaces, the same strategy extends to more general frameworks, including doubly nonlinear equations in reflexive Banach spaces, metric gradient flows, and time-dependent energies.
Joint work with Alessandro Pinzi and Filippo Riva. - Horizons in Nonlinear PDEs — Spring School 2026, March 9–13, 2026, Universität Ulm, Germany
Variational Principles for Evolution ProblemsThe course provides an introduction to variational methods for evolution equations in Hilbert and metric spaces. After reviewing the main generation results for gradient flows and contraction semigroups, we will discuss and apply the abstract theory to the paradigmatic example of the Derrida-Lebowitz-Speer-Spohn (DLSS) equation.
- 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.