Dipartimento di Matematica
Guido Castelnuovo

Giacomo Del Nin (MPI Leipzig)

BMO-type seminorms and local Poincare' constants for BV functions

In 2015 Bourgain, Brezis and Mironescu introduced some BMO-type functionals that measure the oscillation of a function on a family of disjoint $\epsilon$-cubes. These functionals turned out to be related to the total variation of the function, and over the years several authors have addressed the problem of finding an expression for their limit as $\epsilon$ goes to zero. Thanks to the work of many, we now know that for SBV functions the limit exists and coincides with 1/2 times the jump variation plus 1/4 times the absolutely continuous variation. However, for BV functions with a non-trivial Cantor part, the limit might not exist. In this talk I will present a natural relaxation of these functionals that enforces the existence of the limit for any BV function, and I will show that this limit is related to the local Poincare' constant of the function. Based on an ongoing project with Adolfo Arroyo-Rabasa (UCLouvain) and Paolo Bonicatto (University of Trento).

Elia Brué (Università Bocconi)

The fundamental groups of manifolds with nonnegative Ricci curvature

This seminar will focus on Milnor's 1968 conjecture regarding the finitely generated fundamental group of a complete manifold with nonnegative Ricci curvature. We will outline 60 years of progress towards its resolution and present a recent counterexample discovered in collaboration with Aaron Naber and Daniele Semola.

Roberta Bianchini (CNR)

"(In-)stabilities’' of a PDE system for stratified fluids

We will be interested in the analysis of a system of PDEs modeling continuously stratified fluids like the oceans and the atmosphere, under the influence of gravity. I will present some mathematical results related to (in)stability and long-time dynamics. In particular, a version of the classical Beale-Kato-Majda criterion for continuation of classical solutions requires L^1_T L^\infty control of the density gradient. I will focus on a recent result showing that these equations are strongly ill-posed in the class of bounded vorticity and density gradient.

Lucio Boccardo (Sapienza Università di Roma)- Pomeriggio Matematico

Dalla minimizzazione di funzioni di una variabile reale alla minimizzazione in spazi di Banach (cercando di minimizzare la fatica) - h 16:00, aula Picone

In occasione dell'inizio dell'anno accademico si terrà un seminario rivolto a una vasta platea di matematici. Quest'anno l'oratore invitato è Lucio Boccardo, professore emerito del nostro dipartimento.

Ali Hyder (TIFR Bangalore)

Blow-up analysis of stationary solutions to a Liouville-type equation in 3-D

Contrary to the two dimensional case, the Liouville equation in dimension three and higher is supercritical, and in particular it admits singular solutions. We will talk about partial regularity results of stationary weak solutions. Our approach is based on blow-up analysis and a monotonicity formula.

Luca Battaglia (Università Roma 3)

A mean field approach for the double curvature prescription problem

We establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvature on compact surfaces, which is equivalent to a Liouville-type PDE with nonlinear Neumann conditions. We provide three different existence results in the case of positive, zero and negative Euler characteristics. Based on a joint work with Rafael Lopez-Soriano (Universidad de Granada)

Tristan Rivière (ETH Zurich)

Area Variations under pointwise Lagrangian Constraint

Antonio Tribuzio (University of Bonn)

Energy scaling of singular-perturbation models involving higher-order laminates

Motivated by the appearance of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of some N-well problems with surface energy given by a singular higher-order term. In the case of absence of gauge invariances (e.g. with respect to the action of SO(n) or Skew(n)), we provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of the higher order of lamination involved. The upper bound is provided by iterated branching constructions. In the end, we show how a similar approach can be used in the determination of a lower bound for a more realistic model, namely the geometrically linearized cubic-to-tetragonal phase transition, in which a second order lamination is forced by the presence of affine boundary conditions. This is a joint work with Angkana Rüland.

Benedetta Pellacci (Università degli Studi della Campania Luigi Vanvitelli)

Spectral optimization problems arising in logistic models: A singular limit

We will discuss some recent results concerning weighted eigenvalue problems in bounded Lipschitz domains, under Neumann boundary conditions. The optimization of the distribution of resources leads to minimize a principal eigenvalue with respect to the sign-changing weight. Important qualitative properties of the positivity set of the optimal weight, such as being connected, as well as its location, are still not known in general. We will present some asymptotical results in the case of Neumann boundary conditions. Joint works with Dario Mazzoleni (Università di Pavia) and Gianmaria Verzini (Politecnico di Milano).

Martino Fortuna (Sapienza, Università di Roma)

Analisi asintotica di alcuni modelli semi-discreti nello studio delle dislocazioni

Presentiamo l'analisi asintotica, nel senso della Gamma-convergenza, di alcuni modelli utilizzati nello studio della deformazione plastica di corpi cristallini come conseguenza della presenza di dislocazioni. Nella prima parte mostriamo un risultato di omogeneizzazione per energie concentrate su linee rettificabili in tre dimensioni e senza bordo. Come conseguenza della costruzione della recovery sequence otteniamo una dimostrazione alternativa della densità delle correnti uno-rettificabili senza bordo nell'insieme dei campi vettoriali a divergenza nulla. Nella seconda parte presentiamo l'analisi asintotica di un modello bidimensionale per lo studio dei cosiddetti grain-boundaries. Originalmente proposto da Lauteri e Luckhaus, per tale modello mostriamo la Gamma-convergenza ad una energia di interfaccia in accordo con il comportamento logaritmico previsto dalla formula di Read-Shockley.

Mikaela Iacobelli (ETH Zurich)

Stability and singular limits in plasma physics

In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations.

Francois Hamel (Aix-Marseille Université)

Parallel flows in infinite cylinders

In this talk, I will discuss some rigidity results for steady incompressible flows away from stagnation in infinite cylinders with tangential boundary conditions. In two-dimensional strips, Euler flows turn out to be parallel flows. In any dimension, Euler or Navier-Stokes potential flows turn out to be constant. I will also discuss various counter-examples without the main assumptions. The proofs rely on the study of the geometric properties of the streamlines and a combination of ODE and PDE arguments. The talk is based on some joint works with A. Karakhanyan and N. Nadirashvili.

This seminar is funded by the European Union – Next Generation EU.

Nicola Visciglia (Università di Pisa)

Global well posedness for the generalized derivative NLS

The gDNLS equation in the periodic setting has been extensively studied in the literature. By standard compactness argument the associated Cauchy problem is well-known to be locally well posed. Our aim is to show that the local solutions can be globalized for small data, hence answering to a question raised by Ambrose-Simpson. This is a joint work with Masayuki Hayashi.

This seminar is funded by the European Union – Next Generation EU.

Angelo Zanni (Sapienza, Università di Roma)

Decomposizione in profili per operatori astratti: teoria ed applicazione ad un operatore a coefficienti totalmente variabili

In questo seminario vedremo una teoria astratta per una decomposizione in profili di successioni H1 tramite il flusso dell'equazione di Schrödinger. Tale decomposizione verrà poi usata per costruire, sotto opportune ipotesi, una soluzione critica, ossia con flusso precompatto. L'esistenza di tale soluzione critica può essere sfruttata per ottenere lo scattering, come vedremo nel caso di un operatore a coefficienti variabili per l'equazione di Schrödinger defocusing in dimensione 1.

Daniele Bartolucci (Tor Vergata)

Sharp estimates, uniqueness and spikes condensation for superlinear free boundary problems arising in plasma physics

We report about a series of results concerning Grad-Shafranov type equations, which in dimension N=2 describe the equilibrium configurations of a plasma in a Tokamak. In this case we obtain a sharp superlinear generalization of the result of Temam (1977) about the linear case. As a consequence we deduce the first general uniqueness result for superlinear free boundary problems arising in plasma physics. In dimension N\geq 3 the uniqueness result is new but far from sharp, motivating the analysis of a spikes condensation-quantization phenomenon for superlinear and subcritical Grad-Shafranov type free boundary problems, implying among other things a converse of the results about spikes condensation in Flucher-Wei (1998) and Wei (2001). In fact our result plays for these problems essentially the same role as the Brezis-Merle (1993) concentration-compactness theory does for Liouville-type equations. However, because of subcriticality, the Grad-Shafranov spikes condensation phenomenon is due to a sort of infinite mass limit. This is part of a joint research project with A. Jevnikar (Udine

Hardy Chan (University of Basel)

Nonlocal approximation of minimal surfaces: optimal estimates from stability.

For stable nonlocal s-minimal surfaces in R^3 which approximate classical minimal surfaces (as s \to 1^-), we obtain robust C^{2,\alpha} estimates and optimal sheet separation estimates. The interaction between sheets is described by the (local) D'avila--del Pino--Wei system. With a suitable extension of these estimates, we classify stable s-minimal hypersurfaces in R^4 as hyperplanes. Geometric applications are also discussed. This is a joint work with Serena Dipierro, Joaquim Serra and Enrico Valdinoci.

Stephan Luckhaus (University of Leipzig)

Stochastic and kinetic models for the spread of an infectious disease via aerosols -- the example of Covid 19

We discuss models of the so-called Kermak Mc Kendrick type, for human to human infections. The deterministic equations are of Boltzmann type, derived as Kolmogoroff equations for contact processes by Mc Kendrick in 1914 in what is today called a heuristic way (discretization, passage to the large number limit and taking the discretization parameter to zero). For Covid type epidemics one should use a modification. Individuals are infected at meeting points with a probability depending via a dose response curve on the virus exposition. This introduces bistability in the deterministic reaction term, and explains that there is no state independent herd immunity. Joint work with A. Stevens.

Azahara DelaTorre Pedraza (Sapienza, University of Rome)

Uniqueness of least-energy solutions to the fractional Lane-Emden equation in the ball.

In this talk we will show the uniqueness of least-energy solutions for the fractional Lane-Emden equation posed in the ball under homogeneous Dirichlet exterior conditions. This is a non-local semilinear equation with a superlinear and subcritical nonlinearity. Existence of positive solutions follows easily from variational methods, but uniqueness is quite complicated. In the local case, the uniqueness of positive solutions follows from the result of Gidas, Ni and Nirenberg. Indeed, by using the moving plane method, they proved radial symmetry of the solutions which allows the application of ODE techniques. In the non-local case, these arguments don’t seem to work. Our proof makes use of Morse theory, and it is inspired by some results obtained by C. S. Lin in the ’90s. The talk is based on a joint work with Enea Parini.

Alessandro Pigati (Bocconi University, Milan)

Topology of three-dimensional Ricci limits and RCD spaces

Starting from the second half of the nineteenth century, it was understood that curvature, which is infinitesimal (geo)metric information on a space, integrates to give global structure results (in spite of its nonlinear nature), specifically yielding topological rigidity of the space. It was further observed by Gromov that a lower bound on the Ricci curvature is the essential ingredient in order to control the number of degrees of freedom at a metric level as well, allowing to compactify (in a very weak sense) the set of n-dimensional Riemannian manifolds obeying a Ricci lower bound. It was later understood that singular spaces belonging to this compactification (Ricci limits), are special cases of a more general analytic notion (RCD spaces), more stable with respect to certain natural operations, and thus they also inherit a rich analytic structure, allowing to do calculus on them.

In this talk, based on joint work with Elia Bruè and Daniele Semola, we will review some previously known structural results for Ricci limits and RCDs in the non-collapsed case, as well as some instructive examples and counterexamples, and we will see a new, more elementary proof that Ricci limits of dimension three are generalized manifolds, enjoying in particular uniform contractibility. Our tools, together with some results in geometric topology, give an alternative proof that they are in fact topological manifolds. We will also see a new result for tangent cones in higher dimension. If time allows, we will also mention a new structural theorem for RCDs in dimension three.

Xavier Lamy (University of Toulouse)

On the stability of Möbius maps of the n-sphere

A classical theorem of Liouville asserts that if a map from the sphere to itself is conformal, then it must be a Möbius transform: a composition of dilations, rotations, inversions and translations (identifying sphere and euclidean space via stereographic projection). There is a long history of studying stability of this rigidity statement: if a map is nearly conformal, must it be close to a Möbius transform? One can also ask what happens if the image of the map is only nearly spherical. I will present optimal stability estimates obtained with André Guerra and Kostantinos Zemas, which generalize to higher dimensions recent results for the 2-sphere (where, unlike higher dimensions, the problem can be directly linearized).

Antonio De Rosa (Bocconi University, Milan)

Min-max construction of anisotropic CMC surfaces

We prove the existence of nontrivial closed smooth surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. Joint work with G. De Philippis.

Manuel Friedrich (Friedrich Alexander University, Erlangen)

Regularity for minimizers of the Griffith energy

We discuss regularity for the crack set of a minimizer for the Griffith fracture energy, arising in the variational modeling of brittle materials. In the planar setting, we prove an epsilon-regularity result showing that the crack is locally a C^{1,1/2} curve outside of a singular set of zero Hausdorff measure. The main novelty is that, in contrast to previous results, no topological constraints on the crack are required. Joint work with Camille Labourie and Kerrek Stinson.

Roberto Monti (Università di Padova)

Il problema della regolarità delle geodetiche sub-Riemanniane (The regularity problem of sub-Riemannian geodesics)

Presenteremo lo stato dell'arte sul problema della regolarità delle curve minime della lunghezza in varietà sub-Riemanniane, un problema aperto da almeno 40 anni. Il seminario avrà un carattere analitico, informale e introduttivo ma arriveremo a parlare di risultati molto recenti ottenuti con vari collaboratori.

Carlo Mantegazza (Università di Napoli)

Stability for the surface diffusion flow

We study the surface diffusion flow in the flat torus, that is, smooth hypersurfaces moving with the outer normal velocity given by the Laplacian of their mean curvature. This model describes the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential. We show that if the initial set is sufficiently ``close'' to a strictly stable critical set for the Area functional under a volume constraint, then the flow actually exists for all times and asymptotically converges to a ``translated'' of the critical set. This generalizes the analogous result in dimension three, by Acerbi, Fusco, Julin and Morini. Joint work with Antonia Diana e Nicola Fusco.

Maria del Mar Gonzalez (Universidad Autonoma de Madrid)

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