Seminario di Analisi Matematica a.a. 2022/2023



Alessio Porretta

Stime di decadimento per equazioni di Fokker-Planck
La convergenza in tempo lungo per equazioni di Fokker-Planck con drift confinante è un tema classico, affrontato finora sia con metodi variazionali che probabilistici. Nel seminario discuterò un nuovo
approccio per ottenere stime sulla velocità di convergenza, basato su stime di decadimento dell'oscillazione delle soluzioni per equazioni di diffusione-trasporto in regime ergodico. Questo approccio si applica sia ad operatori locali che nonlocali, ed offre una versione interamente analitica di diversi risultati  ottenuti finora con metodi probabilistici di coupling.

Università degli Studi di Roma 2 Tor Vergata



                                                                    SEMINARI PASSATI



Monica Musso

Leapfrogging for Euler equations

We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia). 

University of Bath

Fabricio Macia


Reconstruction of potentials from the Dirichlet-to-Neumann map.

We address the problem of reconstructing a real potential $V$ from the Dirichlet-to-Neumann map of a Schrödinger operator $-\Delta + V$ on the boundary of a domain in Euclidean space (the reconstruction aspect of the Calderón problem). This problem is rather involved in general, from both the analytical and numerical points of view as it enjoys poor stability properties. After reviewing the main tools used in proving that potentials are uniquely determined from the D-t-N map, we introduce an object that is obtained in terms of certain matrix elements of the D-t-N map -- the Born approximation -- which is reminiscent of an approximation for the potential that has been extensively studied in the context of inverse scattering theory. We will show a number of interesting analytical properties of the Born approximation in the case of radial potentials in a ball. Among others, how it can be used to factorize the reconstruction problem into a linear, yet ill-conditioned problem (the Hausdorff moment problem) and a nonlinear step which enjoys Hölder stability. If time permits, we will also present a novel algorithm for numerical reconstruction based on this object. This is based on a series of works in collaboration with  J.A. Barceló, C. Castro, T. Daudé, C. Meroño, F. Nicoleau, D. Sánchez-Mendoza.

Universidad Politécnica de Madrid

Felix Otto

The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows
The fact that the flow of a hypersurface by its mean curvature can be seen as a gradient flow of the surface area has motivated
an influential minimizing movement scheme (Almgren-Taylor-Wang, Luckhaus-Sturzenhecker).
Also Osher's computationally efficient and very popular thresholding scheme for mean curvature flow by Osher et. al.
can be interpreted as a minimizing movement scheme (Esedoglu-O.).

Based on this observation, we use de Giorgi's ideas for minimizing movements in metric spaces (metric slope, variational
interpolation) to give a surprisingly soft -- but conditional -- convergence proof for the thresholding scheme (Laux-0.)

Max-Planck-Institut Leipzig

Andrea Marchese

Generic uniqueness for the Plateau problem

In [Inv. Math., 1978], Morgan proved that almost every curve in R^3 is the boundary of a unique area minimizing surface. I will show how to extend Morgan's result to surfaces of any dimension and codimension. The result follows from the generic existence of boundary points with density 1/2, which exploits a boundary regularity theorem recently proved by De Lellis, De Philippis, Hirsch and Massaccesi. The argument to deduce the generic uniqueness combines a general unique continuation principle with Almgren's celebrated regularity theory, ensuring that the singular set of any area minimizing current has codimension at least two and therefore it cannot disconnect the regular part. I will then explain how to prove a generic uniqueness result for a Plateau-type problem studied in optimal transport, even if the singular set has only codimension one.


Università degli Studi di Trento

Enea Parini

Reverse Faber-Krahn inequality for a truncated laplacian operator

In this talk we will consider a reverse Faber-Krahn inequality for the principal eigenvalue μ_1(Ω) of the fully nonlinear operator P_{+N}u:=λ_N(D2u), where Ω⊂R^N is a bounded, open convex set, and λ_N(D^2u) is the largest eigenvalue of the Hessian matrix of u. The result will be a consequence of the isoperimetric inequality μ_1(Ω)≤π^2diam(Ω)^2. Moreover, we will discuss the minimization of μ1 under various kinds of constraints. The results have been obtained in collaboration with Julio D. Rossi and Ariel Salort (Buenos Aires).

Institut de Mathématique de Marseille

Gian Paolo Leonardi

A vertex-skipping property for perimeter almost-minimizers in convex containers
A crucial issue in capillarity-type problems is understanding the behavior of solutions near singular points in the boundary of the container. In the special case of the relative perimeter functional, we prove that the closure of the internal boundary of a perimeter almost-minimizer in a 3-dimensional convex container Ω cannot contain vertices of Ω. As a byproduct of our study, we also prove a boundary monotonicity formula for perimeter almost-minimizers within a convex container of any dimension, without any extra regularity of the boundary but a suitable closeness to its tangent cones. This is joint work with Giacomo Vianello (Ph.D. UniTN).
Università di Trento

David Ruiz

Sign-changing soluzions to overdetermined elliptic problems in bounded domains
In 1971 J. Serrin proved that, given a smooth bounded domain Ω⊂Rn and u a positive solution of the problem: −Δu=f(u) in Ω, u=0 on ∂Ω, ∂_νu= constant on ∂Ω, then Ω is necessarily a ball and u is radially symmetric. In this paper we prove that the positivity of u is necessary in that symmetry result. In fact we find a sign-changing solution to that problem for a C^2 function f(u) in a bounded domain Ω different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator.
We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. In addition, rather surprisingly and differently from what happens for a scalar equation, the boundedness may fail if the exponents are not comparable.
Universidad de Granada

Boyan Sirakov

Uniform apriori estimates for the Lane-Emden systems in the plane
We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. In addition, rather surprisingly and differently from what happens for a scalar equation, the boundedness may fail if the exponents are not comparable.
Pontifícia Universidade Católica, Rio de Janeiro

Seminari degli  studenti di dottorato

Angelo Ninno 
Variational methods for coarse graining with long range interactions
In this talk we will discuss the use of variational methods for investigating crystallization problems. In such problems the typical functionals considered are associated with pairwise interactions potentials depending on the mutual distances between the particles: among them, short range repulsive/long range attractive potentials are very relevant. The talk is divided in two parts: In the first part of the talk we will adapt the classical crystallization model of Heitmann and Radin in order to predict ordered structures assumed by group of animals; in the second part we will prove that crystallization does not occur for a class of potentials known as Mie potentials.


Paolo De Donato
Struttura dell'insieme singolare di molteplicità 2 per superfici minime
In questo lavoro studiamo la struttura dell’insieme singolare delle funzioni a 2 di classe C^{1,α} su R^n il cui grafico è una varifold stazionaria di codimensione arbitraria in R^{n+k}. Sfruttando le tecniche sviluppate da Almgren e riutilizzate da altri autori sulla funzione frequency dimostreremo che tale insieme singolare risulta essere (n − 2)-localmente rettificabile.

Sapienza Università di Roma

Pierpaolo Esposito

Green functions for a quasilinear operator and applications

We discuss existence, uniqueness and “regularity” issues for the Green function of the quasi-linear operator u \to -\Delta_p u-\lambda |u|^{p-2}u with 1<p \leq N on a bounded N-dimensional domain with Dirichlet boundary condition. Of independent interest, these properties represent a fundamental tool to discuss, as a non-trivial application, existence results for the quasi-linear Brezis-Nirenberg problem in the low-dimensional case, where the problem has a global character which is encoded, as we will see, in the "regular" part of the Green function.

Università degli Studi Roma Tre

Caterina Zeppieri

Stochastic homogenisation of free-discontinuity problems

In this talk we discuss the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

Joint work with F. Cagnetti, G. Dal Maso, and L. Scardia.

Universität Münster

Pierre-Damien Thizy

Large blowup-sets for Q-curvature equations



Université Claude Bernard Lyon 1

Mircea Petrache
Robust Fourier fingerprints for crystals and generalizations


I will start by giving an introductory overview of different notions of generalized crystals (including quasicrystals), unified by the requirement that their Fourier transforms are atomic. Then we study the effect of random perturbations on the Fourier transform of such generalized crystal. The basic result I will present is that under mixing assumptions on the random perturbations, the Fourier transform of a random perturbation is almost surely equal to the Fourier transform of the unperturbed crystal, multiplied by the Fourier transform of the law of the noise. Thus for example for Gaussian i.i.d. perturbations with known law, the perturbed crystal's Fourier transform allows to recover the initial crystal. The case of (perturbations of) lattices is due to Yakir, and we weaken independence hypotheses and extend the theory to quasicrystals, and lattices in finite groups, Heisenberg groups, and other nilpotent groups. This is joint work with Rodolfo Viera from UC Chile.

Pontificia Universidad Católica de Chile 

Giacomo Canevari
Gamma-convergence for the Ginzburg-Landau functional on complex line bundles

The Ginzburg-Landau functional was originally proposed as a model for superconductivity in Euclidean domains. However, invariance with respect to gauge transformations - which is one of the most prominent features of the model - suggests that the functional can be naturally defined in the setting of complex line bundles, where it can be regarded as an Abelian Yang-Mills-Higgs theory. In this talk, we shall consider the Ginzburg-Landau functional on an Hermitian line bundle over a closed Riemannian manifold, in the so-called "non-self dual scaling" (which is closer to the original motivation from superconductivity theory). We shall focus on the variational aspects of the problem; more precisely, we will discuss a Gamma-convergence result for sequences whose energy grows at most logarithmically in the Ginzburg-Landau coupling parameter. As we shall see, the London equation for superconductivity plays a significant role in our analysis. The talk is based on a joint work with Federico Dipasquale (Università Federico II, Napoli) and Giandomenico Orlandi (Verona).

Università degli Studi di Verona


                                                                   A. MA. CA. 2022




  Lucio Boccardo

Lax-Milgram può non funzionare per risolvere problemi di Dirichlet con termini di ordine uno
  Si inizia ricordando come il teorema di Lax-Milgram risulta un semplice e basico strumento per risolvere (in forma debole) il problema di Dirichlet L(u)=f(x), dove L è un operatore differenziale ellittico del secondo ordine in forma di divergenza e f(x) è una funzione in Lm, m=2N/N+2; poi si passa ai risultati dei casi m>2N/N+2 e 1≤m<2N/N+2. Mentre per il problema di Dirichlet lineare [*] L(u)+E(x)∇u=f(x) e per il suo problema duale non sempre è possibile utilizzare Lax-Milgram: nel seminario sono presentati vari approcci all'esistenza (anche in dipendenza da m), i quali poi portano a vari risultati che si sintetizzano nella frase "vale la teoria di Calderon-Zygmund-Stampacchia". Il punto di partenza, che può sorprendere, è l'approssimazione del problema lineare [*] con problemi non lineari. Non ci sarà certamente tempo per discutere la versione non lineare di [*] o del caso parabolico.

Adriana Garroni 

"Grain-boundaried" nei policristalli
In questo seminario mostreremo come si può ottenere un'energia di interfaccia tra grani in un policristallo a partire da energie definite su campi di deformazione elastica incompatibili.

Sapienza Università di Roma


Antonio Siconolfi

Equazioni di Hamilton--Jacobi su networks

Viene presentata una rassegna di recenti risultati ottenuti per equazioni di Hamilton--Jacobi poste su su networks/grafi in collaborazione con Elisabetta Carlini, Marco Pozza e Alfonso Sorrentino. Le equazioni definite su ogni arco, che non sono correlate, sono messe in relazione dalla geometria del network. Si discuteranno risultati di tipo qualitativo, teoremi di esistenza e unicità, formule di rappresentazione e schemi per l'approssimazione numerica.

Marcello Ponsiglione

Movimenti minimizzanti per flussi parabolici frazionari

In questo seminario introdurremo il metodo dei movimenti minimizzanti per flussi parabolici frazionari, geometrici e non. Analizzeremo in particolare il comportamento dei flussi al variare del parametro frazionario, cercando di estendere la teoria a valori inusuali di tale parametro, e studiando alcuni casi particolarmente critici.

Sapienza Università di Roma 



Filomena Pacella

Superfici a curvatura media costante e problemi sopradeterminati in coni

Il seminario verte sulla questione di determinare i domini in coni che ammettono soluzioni di un problema sopradeterminato e, parallelamente, quella di studiare superfici con bordo a curvatura media costante in coni. Una caratterizzazione completa è stata ottenuta nel caso in cui il cono è convesso. Nella seconda parte del seminario discuterò il caso di coni non convessi presentando alcuni risultati recenti che dimostrano l'esistenza di domini o superfici non radiali soddisfacenti le proprietà di cui sopra. Tali risultati sono stati ottenuti studiando l'instabilità delle "soluzioni" radiali come punti stazionari dei corrispondenti funzionali che forniscono una formulazione variazionale dei due problemi.

Nadia Ansini

Teoria variazionale per funzionali di tipo-convoluzione

Studiamo una classe di funzionali integrali  di tipo convoluzione che possono approssimare (via Gamma-convergenza) funzionali locali definiti su spazi di Sobolev. Dopo aver dimostrato un risultato generale di compattezza e rappresentazione integrale mostreremo alcune applicazioni a modelli su `point-clouds', flussi gradiente, omogeneizzazione.

Fabio De Regibus

Sul numero di punti critici di soluzioni di problemi ellittici

Si parlerà di proprietà qualitative di soluzioni di equazioni ellittiche su domini limitati con dato al bordo di Dirichlet. In particolare ci si soffermerà sul numero dei punti critici in relazione alla convessità del dominio. Nel caso di soluzioni positive si discuteranno alcune  stensioni di risultati noti enfatizzando il ruolo del segno della curvatura del bordo del dominio. Infine si affronterà anche il caso di soluzioni nodali concentrandosi sulle autofunzioni del Laplaciano. Lavori in collaborazione con M. Grossi e D. Mukherjee.


Sapienza Università di Roma


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