Seminario di Analisi Matematica

SEMINARI A.A. 2021/2022





Nicola Garofalo 
Heat kernels for a class of evolution equations of hybrid type

The aim of my talk is to construct (explicit) heat kernels for some evolution equations which arise in physics, conformal geometry and subelliptic PDEs. A common feature is that the relevant partial differential operator appears in the form L_1 + L_2 - D_t, but the variables of L_1 and L_2 cannot be decoupled. This obstruction results in the fact that the heat kernel of L_1 + L_2 - D_t cannot be written as the product of the heat kernels of the operators L_1  - D_t and L_2 - D_t. One of the highlights will be the construction of the heat kernel for the time-dependent version of an ``extension problem" introduced by Frank, Del Mar Gonzalez, Monticelli and Tan. Such problem plays a pervasive role in conformal CR geometry and, among other things, it serves as a way to define the fractional powers of the conformal sublaplacian on the Heisenberg group. The leitmotif of my talk is emphasising the so far unexplored connection of the relevant hybrid equations with the heat kernel of the generalised operator of Ornstein-Uhlenbeck type in the opening of H\"ormander's groundbreaking 1967 work on hypoellipticity. This is joint wok with Giulio Tralli. 

Università di Padova

Paolo Piccione
Nonplanar minimal spheres in ellipsoids of revolution

In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with R. G. Bettiol (CUNY).

 Universidade de São Paolo 


Cristian Mendico
Asymptotic behavior of solutions to Hamilton-Jacobi-Bellmann equations

The analysis of the ergodic behavior of solutions to Hamilton- Jacobi-Bellmann equations has a long history going back to the seminal paper by Lions-Papanicolaou-Varadhan. Since this work, the subject has grown very fast and when the Hamiltonian is of Tonelli type a large number of results have been proved. However, few results are available if the Hamiltonian fails to be Tonelli, i.e., the Hamiltonian is neither strictly convex
nor coercive with respect to the momentum variable. In particular, such results cover only some specific structure and so, the general problem is still open. In this talk, I will present some recent results obtained in collaboration with Piermarco Cannarsa and Pierre Cardaliaguet concerning the long time-average behavior of solutions to Hamilton-Jacobi-Bellman equations arising from optimal control problems with control of acceleration, first, and then from optimal control problems of sub-Riemannian type. We will show the existence of a critical constant in both cases but the existence of a critical solution only in case of sub-Riemannian geometry. For this latter case, we also show some results on the Aubry set. We conclude presenting open problems and ideas toward the solution of the general case.

Università degli Studi di Padova

Eris Runa
Symmetry breaking and pattern formation for local/nonlocal interaction functionals       

In this talk we will review some recent results we obtained on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimensional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.                                                                                                                                                                                                                   

                                                                        Deutsche Bank

Giulio Ciraolo

Università degli Studi di Milano




Xavier Ros Oton

ICREA & Univerisitat de Barcelona
28/03/2022 Daniela Tonon
Università degli Studi di Padova


SEMINARI A.A. 2020/2021


Luca Rossi 
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.

Università di Roma

Piermarco Cannarsa
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.

Università di Roma
Tor Vergata

Alessandro Carlotto
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
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).                                                                                                                                                                                                                                          

                        Università di Roma

Pieralberto Sicbaldi
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.

Universidad de Granada

Henri Berestycki
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



SEMINARI A.A. 2019/2020

07/10/2019 Andrea Braides
Homogenization of ferromagnetic energies on Poisson Clouds in the plane
Università di Roma
Tor Vergata
14/10/2019 Elena Kosygina
Stochastic homogenization of viscous Hamilton-Jacobi equations with non-convex Hamiltonians: examples and open questions
Baruch College
21/10/2019 Stephan Luckhaus
Federer, Fleming and all that
Universität Leipzig
28/10/2019 Andrei Rodriguez
Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in R^N
Universidad Tecnica Federico Santa Maria
04/11/2019 Daniele Cassani
Bose fluids and positive solutions to weakly coupled elliptic system in the plane
Università dell'Insubria
11/11/2019 Susanna Terracini   
Liouville type theorems and local behaviour of solutions to degenerate or singular problems
Università di Torino
18/11/2019 Nicola Visciglia
NLS: from the completely integrable to the general case
Università di Pisa
25/11/2019 Francesco Clemente

Marco Pozza
A representation formula for viscosity solutions to PDE problems with sublinear operators


Università di Roma
02/12/2019 Gabrielle Saller Nornberg Sapienza
Università di Roma
09/12/2019 Gilles Francfort
Homogenization for a 2D, two-phase, isotropic and periodic mixture in linear elasticity
Université Paris XIII
16/12/2019 Angela Pistoia
Elliptic systems with critical growth 
Università di Roma
20/01/2020 Maria J. Esteban
Domains of singular Dirac operators and how to compute their eigenvalues 
Université Paris Dauphine
27/01/2020 Erwin Topp
Some results for the large time behavior of Hamilton-Jacobi Equations with Caputo Time Derivative
Universidad de Santiago de Chile
03/02/2020 Ning-An Lai  (ANNULLATO)
Strauss exponent for semilinear wave equations with scattering space dependent damping
Lishui University
10/02/2020 Paolo Bonicatto
Uniqueness and non-uniqueness phenomena for the transport equation in R^d
Universität Basel
17/02/2020 Sara Daneri
On the sticky particle solutions to the pressureless Euler system in general dimension
09/03/2020 Andrea Corli   (ANNULLATO)
Traveling waves for degenerate parabolic equations with negative diffusivities

Jean Van Schaftingen   (ANNULLATO)
Ginzburg-Landau functionals for a general compact vacuum manifold on planar domains

Università di Ferrara 


16/03/2020 Xavier Ros-Oton  (ANNULLATO)
Universität Zürich
23/03/2020 Andrea Mondino  (ANNULLATO)
University of Oxford
30/03/2020 Giulio Ciraolo  (ANNULLATO)
Università di Milano

Olivier Ley    (ANNULLATO)

Institut National des Sciences 
Appliquées de Rennes

20/04/2020 Martino Bardi     (ANNULLATO)
Università di Padova

Lorenzo Dello Schiavo   (ANNULLATO)

Universität Bonn

Seminario A.Ma.Ca. 2020
Lucio Boccardo   
El amor a las PDE en los tiempos del coronavirus

Università di Roma

Seminario A.Ma.Ca. 2020
Adriana Garroni
Ruolo delle stime di rigidità in elasticità e plasticità
Eugenio Montefusco
Alcune osservazioni sulle configurazioni limite di sistemi fortemente competitivi


Università di Roma
18/05//2020 Seminario A.Ma.Ca. 2020
Luigi Orsina
Un sistema di equazioni di tipo Kirchhoff-Schrödinger-Maxwell
Marcello Ponsiglione
Limiti di curvature non locali e dei corrispondenti flussi geometrici

Università di Roma
08/06//2020 Seminario A.Ma.Ca. 2020
Luca Di Fazio
Regolarità C^1 nel problema dell'ostacolo sottile per disequazioni variazionali
Andrea Kubin
Potenziali di interazione tra sfere rigide di tipo Riesz

Università di Roma

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