Dipartimento di Matematica
Guido Castelnuovo

Luca Rossi - https://meet.google.com/nie-attq-ged 
Are solutions of reaction-diffusion equations asymptotically 1D ?

The symmetry of solutions of elliptic equations is a classical and challenging problem in PDE, connected with stability.
In this talk we are concerned with parabolic equations and we ask whether the 1-dimensional symmetry eventually emerges in the long time,

for solutions which are initially non-symmetric.
We will present a satisfactory answer in the case of the Fisher-KPP equation, together with some counter-examples and open questions.
This topic is the object of a joint work with F. Hamel.

Piermarco Cannarsa - https://meet.google.com/nie-attq-ged
Long time behaviour of solutions to Hamilton-Jacobi equations for sub-Riemannian control systems

Sub-Riemannian systems are an important class of nonlinear control systems with linear dependence on controls. Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields. We will discuss the long-time average behaviour of the value function of optimal control  problems for sub-Riemannian systems, which cannot be addressed by classical weak KAM theory as the Hamiltonian fails to be coercive in the momentum variable.

Alessandro Carlotto - https://meet.google.com/nie-attq-ged
Free boundary minimal surfaces with connected boundary and arbitrary genus

Besides their self-evident geometric significance, which can be traced back at least to Courant, free boundary minimal surfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years thanks to work of Fraser and Schoen, in connection to extremal metrics for Steklov eigenvalues for manifolds with boundary (i. e. for eigenvalues of the corresponding Dirichlet-to-Neumann map).
The theory has been developed in various interesting directions, yet many fundamental questions remain open. One of the most basic ones can be phrased as follows: does the Euclidean unit ball contain free boundary minimal surfaces of any given topological type? In spite of significant advances, the answer to such a question has proven to be very elusive. I will present some joint work with Giada Franz and Mario Schulz where we answer (in the affirmative) the well-known question whether there exist in B^3 (embedded) free boundary minimal surfaces of genus one and one boundary component. In fact, we prove a more general result: for any g there exists in B^3 an embedded free boundary minimal surface of genus g and connected boundary. This result provides a long-awaited analogue of the existence theorem obtained by Lawson in 1970 for closed minimal surfaces in round S^3.
The proof builds on global variational methods, in particular on a suitable equivariant counterpart of the Almgren-Pitts min-max theory, and on a striking application of Simon's lifting lemma.

ETH, Zürich 

Annika Bach - https://meet.google.com/nie-attq-ged
Singularly-perturbed elliptic functionals: Γ-convergence and stochastic homogenisation

In this talk we introduce a general class of singularly-perturbed elliptic functionals Fε and we study their asymptotic behaviour as the perturbation parameter ε > 0 vanishes. Under suitable assumptions, which in particular allow us to bound Fε by the Ambrosio-Tortorelli functionals, we show that the functionals Fε Γ-converge (up to subsequences) to a free-discontinuity functional of brittle type. Moreover, we provide asymptotic formulas for the limiting volume and surface integrands, which show that the volume and surface contributions of Fε decouple in the limit. If time permits, we will discuss the application of the general convergence result to the setting of stochastic homogenisation. This is joint work with R. Marziani and C. I. Zeppieri (Münster).                                                                                                                                                                                                                                          

Pieralberto Sicbaldi - https://meet.google.com/nie-attq-ged
Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold

We consider the problem of finding domains that minimize the first eigenvalue of the Dirichlet Laplacian in a Riemannian manifold under volume constraint (Faber-Krahn minimizers). In the Euclidean setting such domains are balls, and existence and regularity of such domains is trivial. In a non-Euclidean setting very few examples are known. In this talk we will show a general result of existence and regularity of Faber-Krahn minimizers, inspired by the analogous result of existence and regularity of the solutions of the isoperimetric problems in a Riemannian manifold. In particular we will show that Faber-Krahn minimizers are regular in low dimension, and that there
exists a critical dimension after which they can have singularities. Such critical dimension is related to the Alf-Caffarelli cone. This is a joint work with J. Lamboley.

Henri Berestycki - https://meet.google.com/nie-attq-ged
Segregation in predator-prey models and the emergence of territoriality

I report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single prey. We focus on the analysis of stationary states, stability issues, and the asymptotic behavior when the competition parameter becomes unbounded. Existence of solutions is obtained by a bifurcation theory type approach and the segregation analysis rests on a priori estimates and a free boundary problem. We discuss the classification of solutions by using spectral properties of the limiting system. Our results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to have such hostile packs, and on comparing the various territory configurations that arise in this context. These questions lead us to nonstandard optimization problems.

EHESS, Paris