Dipartimento di Matematica
Guido Castelnuovo

Vito Crismale

Quasistatic evolution for some models coupling elastoplasticity and damage

After introducing the main notions of evolution for rate-independent systems and the general abstract approach to prove their existence, I will discuss, in this framework, models coupling linearized elastoplasticity with damage. I will consider both an associative and a non-associative elastoplastic setting, describing responses to suitable loading of crystalline materials/metals and geomaterials/soils, respectively.

Jacopo Ulivelli

The (Self-Similar, Variational) Rolling Stones

The interplay between variational functionals and the Brunn-Minkowski Theory is currently a well-established phenomenon that has been widely investigated in the last thirty years. In this talk, we present a result on the existence of solutions to the even logarithmic Minkowski problems arising from torsional rigidity. The blueprint we lay down works as well for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply self-similar solutions to variational flow problems a la Firey's worn stone problem.

Riccardo Caniato

Variations of Yang-Mills lagrangians in high dimension

In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^p-control of their curvature. We will start from the classical theory in critical dimension (i.e. n=2p) and then move to the super-critical dimension (i.e. n>2p), making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension.

Berardo Ruffini

On some isoperimetric and spectral energies with repulsion

In the talk I will introduce some variational models where an aggregating term, like the perimeter or a Dirichlet-type energy, is in competition with a repulsive one. Examples of such models arise naturally in different fields of physics. It is the case of the Gamow [liquid drop] model and the Hartree energies in quantum mechanics, or the Rayleigh liquid charged drop model in electrowetting theories. I will give an overview of the recent strategies to get well-or-ill posedness of these energies. Then I will focus on a particular case -the reduced Hartree energy of the atom of Helium in a confined potential field- and show a strategy to characterize minimizers for such an energy. The talk is mostly motivated by an ongoing project with Dario Mazzoleni (Pavia) and Cyrill B. Muratov (Pisa).

Albert Fathi

Weak KAM theory and viscosity solutions on metric spaces

Weak KAM theory originally connected Mather theory of Lagrangian Systems with Viscosity Theory of the solutions of the corresponding Hamilton-Jacobi Equation, at least when the Hamiltonian is obtained from a Lagrangian. In such a case the Mañé potential is the minimal action necessary to join two points in arbitrary time. We will show that we can recover just from the Mañé potential concepts like Peierls barrier, Aubry sets, viscosity subsolutions and solutions. This allows the theory to apply in the more general framework of compact metric spaces, opening a way to define solutions of the Hamilton-Jacobi equation on general metric spaces.

Andrea Davini

On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians

We consider the discounted approximation of the critical Hamilton-Jacobi equation set on the real line associated with the Hamiltonian G(x,p):=H(x,p)-V(x), where H is a 1-periodic Tonelli Hamiltonian and V is a continuous and compactly supported potential. The critical constant associated with G is characterized as the unique constant a for which the associated HJ equation G(x,u')=a can have globally bounded solutions. We prove that the solutions of the discounted equation converge to a specific critical solution, which is identified in terms of projected Mather measures for G and of the asymptotic solution of the unperturbed periodic problem. This is joint work with I. Capuzzo-Dolcetta.

Yannick Sire

Harmonic maps with free boundary and beyond

I will introduce a new heat flow for harmonic maps with free boundary. After giving some motivations to study such maps in relation with extremal metrics in spectral geometry, I will construct weak solutions for the flow and derive their partial regularity. The introduction of this new flow is motivated by the so-called half-harmonic maps introduced by Da Lio and Riviere, which provide a new approach to the old topic of harmonic maps with free boundary. I will also state some open problems and possible generalizations.

Tim Laux

Local minimizers of the interface length functional based on a concept of local paired calibrations

Interfacial energy functionals are ubiquitous in nature. However, some of the most basic questions are still open. In this talk, I will address one of these questions and characterize local minimizers of the interface energy. We'll establish that regular flat partitions are locally minimizing the interface energy with respect to L^1 perturbations of the phases. Regular flat partitions are partitions of open sets in the plane whose network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herring angle condition is satisfied. The proof relies on a localized version of the paired calibration method which was introduced by Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a relative energy functional that precisely captures the suboptimality of classical calibration estimates. Vice versa, we show that any stationary point of the length functional (in a sense of metric spaces) must be a regular flat partition. This is joint work with J. Fischer, S. Hensel, and T. Simon.

Luca Martinazzi

Regularity of minimizers of relaxed energies

I will survey on some long-standing open problems and some recent results about the regularity of minimizers of various relaxed energies. I will focus on the model case of harmonic maps from the 3-dimensional ball into the 2-dimensional sphere, as introduced in a celebrated work of Brezis-Coron-Lieb, and which was also widely studied in the context of liquid crystals. I will also mention some related recent results and open problems regarding fractional harmonic maps and Yang-Mills energies.

Carlo Sinestrari

Moto per curvatura a volume costante e disuguaglianze isoperimetriche

Il moto per curvatura media a volume costante è l'evoluzione di una ipersuperficie con velocità data dalla curvatura media, con un termine aggiuntivo non locale tale che il volume racchiuso resti costante. Il termine non locale crea delle difficoltà aggiuntive, ad esempio non è più valido il principio di confronto. D'altra parte, il rapporto isoperimetrico della regione racchiusa decresce nel tempo nel flusso a volume costante, a differenza di quanto accade nel flusso ordinario. La monotonia del rapporto isoperimetrico fornisce un utile strumento per studiare il comportamento per tempi grandi e la convergenza a un profilo sferico delle ipersuperfici convesse. Tale approccio è stato generalizzato recentemente ad altri flussi a volume costante, con velocità date da funzioni più generali delle curvature principali, o dalla curvatura media frazionaria. Questi risultati sono ottenuti in collaborazione con M.C. Bertini, E. Cinti ed E. Valdinoci.

Nicola Soave

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

n this talk we present some results obtained jointly with Matteo Muratori (Politecnico di Milano), focusing on qualitative properties for • Extremals for the Sobolev inequality, • Positive radial solutions of the Lane-Emden equation for the p-Laplacian in the critical and supercritical regimes, • Positive radial solutions of the Lane-Emden system in the critical and supercritical regimes, posed on a Cartan-Hadamard manifold Mn. We are particularly interested in rigidity results, both for the functions themselves, and for the underlying manifold. For instance, we show that if Mn supports an optimal function u for the Sobolev inequality, and the dimension n is less than or equal to 4, then Mn is isometric to Rn, and u is an Aubin-Talenti bubble.

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.

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). 

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.

Felix Otto

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.

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).

Gian Paolo Leonardi

David Ruiz

Boyan Sirakov

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.

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.

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.

Pierre-Damien Thizy

Large blowup-sets for Q-curvature equations

On a bounded domain of the Euclidean space R2m,m>1, Adimurthi, Robert and Struwe pointed out that, even assuming a volume bound ∫e2mudx≤C, some blow-up solutions for prescribed Q-curvature equations (−Δ)mu=Qe2mu without boundary conditions may blow-up not only at points, but also on the zero set of some nonpositive nontrivial polyharmonic function. This is in striking contrast with the two dimensional case (m=1). During this talk, starting from a work in progress with Ali Hyder and Luca Martinazzi, we will discuss the construction of such solutions which involves (possible generalizations of) the Walsh-Lebesgue theorem and some issues about elliptic problems with measure data.

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.

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).

  A.MA.CA.

  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 L^m, 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.

A.MA.CA.

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 

A.MA.CA.

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.