Seminari di Analisi Matematica Bruno Pini

It is well known that solutions to elliptic systems may be unbounded. Nevertheless, for some special classes of systems, it can be proved that solutions are bounded. We mention a recent result of this kind and we discuss some examples suggested by double phase functionals.

After recalling some integrability by compensation results related to conformally invariant Lagrangians in dimension 2, we will present a recent result concerning the upper-semi-continuity of the Morse index plus the nullity of critical points to such variational problems under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse Indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. This is a joint work with Matilde Gianocca and Tristan Rivière

L'abstract del seminario è contenuto nel file allegato.

In questa presentazione verranno mostrati risultati sull'esistenza e la localizzazione delle soluzioni per problemi differenziali non locali in spazi astratti. In particolare, verrà illustrato un procedimento basato su teoremi di punto fisso associati alle cosiddette condizioni di trasversalità. Una particolare attenzione verrà dedicata ad alcune tecniche che permettono di indebolire le ipotesi di compattezza classiche spesso presenti in letteratura per lo studio di equazioni differenziali in spazi astratti con metodi topologici. Questo approccio fornisce un metodo unificante per lo studio di modelli che descrivono processi di diffusione in diversi contesti. Permette di considerare condizioni periodiche e condizioni iniziali non locali più generali come ad esempio condizioni multipoint oppure condizioni iniziali di tipo integrale, e di gestire nonlinearità con crescite superlineari, ad esempio polinomi cubici o mappe che dipendono dall'integrale della soluzione, includendo così comportamenti di diffusione non locale.

By using a family of harmonic functions introduced by Ulku Kuran in 1972, we define a new harmonic invariant that measures the gap between the perimeter of a domain D and the perimeter of the biggest ball contained in D and centered at a fixed point x_0 of D. From the properties of this harmonic invariant we get new proofs, generalizations and partial improvements of several rigidity and stability Theorems by Lewis and Vogel, by Preiss and Toro, by Fichera and by Aharonov, Schiffer and Zalcman. The complete proofs of all these new results will appear in a paper in collaboration with Giovanni Cupini.

Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize some results proved by Ambrosio, Serra Cassano, Vittone valid in Heisenberg groups to the more general setting of Carnot groups

The functional analytic setting of various variational models in Fracture Mechanics requires the use of classes of functions with set of discontinuities of codimension one. The difficulty of finding good discretization for such classes of functions makes the direct numerical simulation of those variational problems challenging and highly problematic. For this reason, numerous regularizations have been proposed, the most successful of which are phase-field functionals. These elliptic regularizations were first introduced and analyzed in the work of Ambrosio and Tortorelli for the Mumford-Shah energy in image segmentation, inspired by a now classical example in phase transition by Modica and Mortola. Ambrosio and Tortorelli type approximations have become very popular both in the communities of Calculus of Variations and of Computational Mechanics to address a number of problems in applied sciences, especially in brittle fracture. In the talk, we will comment on some of those phase-field models, starting with Ambrosio and Tortorelli's, which eventually led to a useful variant for approximating cohesive energies in Fracture Mechanics.

In this talk, we will explore the uniqueness of solutions to variable coefficient Schrödinger equations. We will show that, given appropriate decay assumptions on the coefficients and on the solution at two different times, the corresponding solution must be identically zero. Based on a joint work with Zongyuan Li and Xueying Yu.

The Gauss-Green and integration by parts formulas are of significant relevance in many areas of mathematical analysis and physics, and such applications motivated several investigations to extend these formulas to less regular integration domains and vector fields. These endeavours naturally led to the definition of the divergence-measure fields, which are L^p-summable vector fields whose divergence is a Radon measure. By applying a Leibniz rule between functions of bounded variation and essentially bounded divergence-measure fields, we will prove Gauss--Green formulas for these fields on sets with finite perimeter. It is also of interest to consider as integration domains sets with possibly fractal boundary, such as sets with finite fractional perimeter. To this purpose, we will present a distributional approach to fractional Sobolev spaces and fractional variation, which exploits the notions of fractional Riesz gradient and divergence. This will allow us introduce the fractional divergence-measure fields, which, in perfect analogy with the integer case, are L^p-summable vector fields whose fractional divergence is a Radon measure. Finally, we will provide Leibniz rules involving such fields and suitably regular scalar functions, leading to the fractional version of the Gauss-Green formula. The talk is mainly based on joint works with Kevin R. Payne and Giorgio Stefani.

We present some recent results about a way of defining suitable fractional powers of the sub-Laplacian on an arbitrary Carnot group through an analytic continuation approach introduced by Landkof in Euclidean spaces. Furthermore, we present a stronger outcome in the setting of the Heisenberg group, which is the simplest non-commutative stratified group. Eventually, in this context we propose a geometrical application of our result: we compute the value of suitable momenta with respect to the heat kernel. This is joint work with Fausto Ferrari.

We discuss the characterization of gauge-symmetric functions in the Heisenberg group via various geometric prescriptions on their level sets. In this talk we mainly focus on a family of overdetermined boundary value problems of Serrin type which exhibit both similarities and differences with respect to the classical symmetry result for the torsion function. We show uniqueness results for gauge balls under suitable partial symmetry assumptions for the class of competitor sets. The main technical tool is a new Bochner-type identity for functions with toric/cylindrical invariances. This is a joint project with V. Martino.

The fractional p-Laplacian is a nonlinear, nonlocal operator with fractional order and homogeneity exponent p>1, arising in game theory and extending (in some sense) both the classical p-Laplacian and the linear fractional Laplacian. While behaving similarly to its local counterpart from the point of view of variational and topological methods, this operator requires an "ad hoc" approach in regularity theory. We will give an account on some regularity results for elliptic equations driven the fractional p-Laplacian, either free or coupled with nonlocal Dirichlet conditions: in particular we will discuss interior and boundary Hölder continuity, a special form of weighted Hölder regularity, and a recent local clustering lemma. Finally, we will rapidly hint at some applications such as comparison principles, Hopf type lemmas, Harnack inequalities, and an equivalence principle between Sobolev and Hölder minimizers of the associated energy functional. The talk is mainly based on some very recent collaborations with F.G. Düzgün, S. Mosconi, and V. Vespri.

In this talk we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. More precisely, we develop an existence theory in the cases of linear, superlinear and singular perturbations; in the particular case of linear perturbations, we also investigate an associated mixed Sobolev inequality, detecting the optimal constant, which we show that is never achieved. The results discussed in this seminar are obtained in collaboration with S. Dipierro, E. Valdinoci and E. Vecchi.

We present a pseudo-differential Weyl calculus on graded nilpotent Lie groups, especially the Heisenberg group, which extends the celebrated Weyl calculus on R^n. This Weyl calculus is a particular instance of a general symbolic calculus we develop for a large class of quantization schemes that are defined via the (operator-valued) group Fourier transform. The symbol classes we consider are the Hörmander-type classes introduced by Fischer and Ruzhansky, which on R^n coincide with the classical ones. As a by-product, we also recover the classical Kohn-Nirenberg calculus on R^n and Fischer and Ruzhansky's KN-calculus on general graded groups. A few immediate applications of our theory are the expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\aa rding inequality for elliptic operators, and a generalized Poisson bracket on stratified groups. We also discuss two simple algebraic criteria which determine the Weyl quantization uniquely at least on R^n and the Heisenberg group. The talk is based on joint-work with Serena Federico and Michael Ruzhansky.

Let n ∈ N, n > 2 and let Ω ⊆ R^n be open. Let H, G : R^n → R^{n×n} be of appropriate regularity. We discuss the existence of an immersion u : Ω → R^n of appropriate regularity, satisfying (∇u)^tH(u)(∇u) = G in Ω. (1) We consider both local and global problems. Equation (1) comes up in diverse contexts. When H (and hence G) is symmetric and positive definite, Equation (1) is connected to the problem of equivalence of Riemannian metrics. The symmetric case is also important in the non-linear elasticity theory because of its connection with the Cauchy-Green deformation tensor. When H (and hence G) is skew-symmetric, Equation (1) comes up in the context of the problem of equivalence of differential two-forms. The aim of the talk is to present a survey of recent progress and advances in the context of Equation (1). We also discuss the general case when H, G are neither symmetric nor skew-symmetric. The talk is based on joint works with Bernard Dacorogna, Vladimir Matveev and Marc Troyanov.

In this talk, I will discuss the behavior of the spectrum of the Laplacian on bounded domains, subject to varying mixed boundary conditions. More precisely, let us assume the boundary of the domain to be split into two parts, on which homogeneous Neumann and Dirichlet boundary conditions are respectively prescribed; let us then assume that, alternately, one of these regions “disappears” and the other one tends to cover the whole boundary. In this framework, I will first describe under which conditions the eigenvalues of the mixed problem converge to the ones of the limit problem (where a single kind of boundary condition is imposed); then, I will sharply quantify the rate of this convergence by providing an explicit first-order asymptotic expansion of the “perturbed” eigenvalues. These results have been obtained in collaboration with L. Abatangelo, V. Felli and B. Noris.

The aim of this talk is to discuss the most recent developments of the De Giorgi-Nash-Moser weak regularity theory for kinetic operators. The analysis will be presented in the model case of the Fokker-Planck equation with measurable coefficients in divergence form and the focus will be on exploring qualitative and quantitative methods to prove an invariant Harnack inequality for non-negative solutions. Finally, some applications of this inequality to various kinetic models will be discussed.

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).

In this talk we discuss the problem of the real analytic regularity for the solutions of sums of squares of vector fields. While the problem of the C^\infty hypoellipticity has been settled from the very beginning by Hörmander, the problem of the analytic hypoellipticity is still open and seems much more involved. Treves conjecture states that a “sum of squares”-type operator is analytic hypoelliptic if and only if all the Poisson strata of its characteristic set are symplectic. We show that this conjecture, as stated, does not hold. However, we briefly discuss some model examples which would suggest that the analytic regularity still depends on a suitable stratification of the characteristic variety of the operator.

In questo seminario trattiamo lo studio di una condizione necessaria e sufficiente per l'esistenza di soluzioni con una determinata regolarita` per un sistema di equazioni lineari a coefficienti di regolarita` data. Iniziamo esponendo un metodo risolutivo per la determinazione di soluzioni continue di equazioni lineari a coefficienti continui dovuto a C. Fefferman. Passiamo poi a presentare il risultato di C. Fefferman e G. Luli di soluzioni di classe C^m nel caso di coefficienti polinomiali. Consideriamo infine un nostro risultato per determinare una condizione necessaria e sufficiente per l'esistenza di soluzioni semialgebriche continue nel caso di un sistema di equazioni lineari con coefficienti semialgebrici continui.

My aim is to give , in this talk , some topics on the question of regularity of Analytic-Gevrey vectors of partial differential operators (p.d.o.) with analytic-Gevrey coefficients . Since the results obtained in the sixties on elliptic p.d.o's , which are both hypoelliptic (Cˆ{\infty} setting) , analytic-Gevrey hypoelliptic (analytic-Gevrey setting) and satisfy the so-called Kotake-Narasimhan property , a lot of works and articles were devoted to these problems in case of non elliptic p.d.o's under suitable hypotheses (for example on the degeneracy of ellipticity). I will consider the third problem on analytic-Gevrey vectors in the three cases of global (on compact manifolds ), local ( near a point in the base-space ), microlocal (near a point in the cotangent space ) , situations , and say few words on the main two methods used in order to obtain positive (or negative) results . Finally I will focus on some new microlocal results on degenerate elliptic (also called sub-elliptic ) p.d.o's of second order , obtained in a common work with Gregorio Chinni .

In this seminar we revisit several previous works concerning anisotropic differential equations, i.e. evolution equations in which the diffusion takes a different form within different space directions. Recently, in a joint collaboration with S. Ciani, we went back to these anisotropic PDEs, focusing on the singular ones, and started to work on some of their open problems. We’ll briefly discuss this ongoing project pointing out some of the difficulties one needs to address and overcome.

We will present results concerning existence and uniqueness of solutions to a nonlinear equation driven by an operator arising from the superposition of a Laplacian and a signed power of a fractional Laplacian. Of particular interest are the boundary conditions naturally associated to the equation. We will also go over a couple of strongly related problems. This is a joint work with M. Cozzi (UniMi).

In this talk, I will present a recent result which establishes optimal regularity for isoperimetric sets with densities, under mild H\¨older regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach the optimal regularity class $C^{1, \frac{α}{2−α}}$ in any dimension. This is a joint work with L. Beck and C. Seis.

The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature, and can be interpreted as the gradient flow of the length. In this talk I will consider the same flow for networks (finite unions of sufficiently smooth curves whose endpoints meet at junctions). I will explain how to define the flow in a classical PDE framework, and then I will list some examples of singularity formation, both at finite and infinite time, and explain the resolution of such singularities obtained by geometric microlocal analysis techniques. I will describe a stability result based on Lojasiewicz–Simon gradient inequalities and give a rough estimate on the basin of attraction of critical points. Furthermore, I will motivate the coarsening-type behavior clearly visible in numerical simulations. This seminar is mainly based on recent papers in collaboration with Jorge Lira (Uni- versidade Federal do Ceará), Rafe Mazzeo (Stanford University), Mariel Saez (P. Universidad Catolica de Chile) and Marco Pozzetta (Università di Napoli Federico II).

In this talk I present some recent results of a joint work with Paolo Salani and Tadeusz Kulczycki concerning the fractional Bernoulli problem, where I will focus on the so called spectral half Laplacian. After a short introduction, I will explain how to construct a solution to this problem using the Beurling method on the extended problem. Moreover, we discuss geometric properties of the solution and, if there is time, give some remarks on the problem with the usual half Laplacian.

Starting from motivations coming from Physics, we will introduce the Dirac-Einstein equations on a spin manifold. After reviewing some known background material, we will show some recent results, in particular: the compactness of the variational solutions, the classification of the Palais-Smale sequences for the related conformal problem, and finally, an existence result of Aubin type.

We prove a parabolic version of the standard Poincaré inequality, and we show that the elliptic version of the Moser argument can be applied even in the parabolic and Kolmogorov setting to deduce the Hölder regularity of the solutions. The price to pay is the lack of uniformity, in the constants. The proof is elementary and unifies in a natural way several results in the literature on Kolmogorov equations, subelliptic ones and some of their variations. This result is a joint work with M. Manfredini and Y. Sire.

Abstract: Recently, in a joint work with Bruno Franchi and Pierre Pansu, we have proved some new interior Poincaré and Sobolev inequalities for the Rumin complex in the Heisenberg group in the endpoint situation p = Q (the homogeneous group dimension) or p = Q/2, depending on the degree of the forms. We refer to these inequalities as (\infty, Q)-Poincaré or Sobolev inequalities (or (\infty, Q/2)-Poincaré or Sobolev inequalities respectively). These results complement and complete the program on Poincaré-Sobolev inequalities that we developed in a series of previous papers for $1\le p< Q$ (or p<Q/2). In this talk I'll present a further improvement of the global (\infty, Q)-Poincaré inequality, still obtained in collaboration with Franchi and Pansu, showing that it possible to upgrade bounded primitives to bounded and continuous primitives in the case (\infty, Q) (or (\infty, Q/2), depending on the degree of the forms). The argument we use relies on our previous results and duality (i.e. Hahn-Banach) and generalizes to differential forms a Bourgain-Brezis's duality argument for a Poincaré inequality for periodic functions (Bourgain-Brezis, JAMS 2003).

Abstract (joint research with P. Ciatti and Y. Sire): In this talk we derive several properties of the fundamental solution for the heat equation associated with the Rumin's complex on Heisenberg groups. As an application, we use the heat kernel for Rumin's differential forms to construct a Calder\'on reproducing formula for Rumin’s differential forms. Titolo: Sul nucleo del calore per il complesso di Rumin e la reproducing formula di Calder\’on ( ricerca in collaborazione con P. Ciatti e Y. Sire). Riassunto: In questo seminario otteniamo varie proprie\`a della soluzione fondamentale dell’equazione del calore associata al complesso di Rumin nei gruppi di Heisenberg. Come applicazione, usiamo il nucleo del calore per le forme differenziali di Rumin per costruire la \it{reproducing formula} di Calder\’on per le forme differenziali di Rumin.

Let D be a bounded open set of R^n with \sigma(\partial D)< \infty and let x_0 be a point of D. Assume that u(x_0) equals the average of u on \partial D for every harmonic function u in D continuous up to the boundary. In this case one says that D is a harmonic pseudosphere centered at x_0. In general, harmonic pseudospheres are not spheres as a two-dimensional example due to Keldysch and Lavrentiev (1937) shows. As a consequence, the following problem naturally arose: when a pseudosphere is a sphere? Or, roughly speaking: is it possible to characterize the Euclidean spheres via the Gauss mean value property for harmonic function? The answer is yes. The most general result in this direction was obtained by Lewis and Vogel in 2002: they proved that a harmonic pseudosphere \partial D is a sphere if D is Dirichlet regular and the surface measure on \partial D has at most an Euclidean growth. Preiss and Toro, in 2007, proved the stability of Lewis and Vogel's result. Namely: a bounded domain D satisfying the Lewis and Vogel’s regularity assumptions, has the boundary geometrically close to a sphere centered at x_0 if the Poisson kernel of D with pole at x_0 is close to a constant. In collaboration with Giovanni Cupini we proved that the previous rigidity and stability results hold true if the domain D has the boundary with finite area and only satisfies the following property: the boundary of D is Lyapunov-Dini regular in at least one point of \partial D closest to x_0. Our approach to the rigidity ad stability properties of the Surface Mean Value Theorem for harmonic functions is quite elementary in spirit: it does not uses the profound harmonic analysis and free boundary techniques instead used by Lewis and Vogel and by Preiss and Toro, but it relies on careful estimates of the Poisson kernel of the biggest ball centered at x_0 and contained in D.

We study a class of second order strongly degenerate kinetic operators L in the framework of special relativity. More precisely, the operator L we consider here is a possible suitable relativistic generalization of the kinetic Fokker-Planck operator. We first describe L as a Hormander operator which is invariant with respect to Lorentz trans- formations. We then prove a Lorentz-invariant Harnack type inequality, and we derive accurate asymptotic lower bounds for positive solutions to L f = 0. As a consequence, we obtain lower bounds for the density of the relativistic stochastic process associated to L . This is a joint work with Francesca Anceschi (Università Politecnica delle Marche) and Sergio Polidoro (Università degli Studi di Modena e Reggio Emilia).

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap. This improves on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the absence of a spectral gap, as occurs when the vanishing profile fails to be isolated (and may belong to a continuum of such profiles). Joint work with Beomjun Choi and Robert J. McCann.

In this talk, we show recent Lp-Lq estimates, for general (p,q), obtained in collaboration with M.R. Ebert, for the solution to the Cauchy problem for the visco-elastic damped wave equation. These results complete the picture that was started by Y. Shibata (MMAS, 2000), who obtained the endpoint estimates (p,q)=(1,1) and (p,q)=(1,∞). We show that optimal Lp-Lq estimates in the general case cannot be derived by a simple interpolation due to the “noneffective” nature of the damping term. We show the analogous but different result which holds for other strongly damped p-evolution models, as the plate equation. We mention a straightforward application of the obtained estimates to derive global existence of small data solutions for problems with critical or supercritical power nonlinearities.

In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation -\Delta u + u = a(x)|u|^{p-2}u set in an annulus of R^N. Here p>2 is allowed to be supercritical in the sense of Sobolev embeddings, and a(x) is a positive weight with additional symmetry and monotonicity properties, which are shared by the solution that we construct. For this problem, we find a new type of positive, axially symmetric solutions. Moreover, in the case where the weight a(x) is constant, we detect a condition, depending only on the exponent p and on the inner radius of the annulus, that ensures that the solution is nonradial. In this setting, the major difficulty to overcome is the lack of compactness in a nonradial framework. The proofs rely on a combination of variational methods and dynamical system techniques. This is joint work with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), and Tobias Weth (Goethe-Universitat Frankfurt).

We will investigate the effects of the lack of compactness of the critical Folland-Stein-Sobolev embedding by proving that a famous conjecture of Brezis and Peletier (Progr. Nonlinear Differential Equations Appl. 1989) still holds in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point which can be localized via the Green function associated to the involved domain. In order to achieve the aforementioned result we will combine several new estimates and specific tools to attack the related CR Yamabe equation (Jerison-Lee, J. Diff. Geom. 1987) with new feasible results in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as a De Giorgi's Gamma-convergence approach to provide fine energy approximations in very general (possible non-smooth) domains; Caccioppoli-type boundedness estimates depending on the datum for the solutions to even more general subelliptic equations; the asymptotic control of the optimal functions via the Jerison&Lee estremals realizing the equality in the critical Sobolev inequality (J. Amer. Math. Soc. 1988); the celebrated Global Compactness result which we will extend in the Heisenberg framework via a completely different approach with respect to the original one by Struwe (Math. Z. 1984). Il seminario si basa su un lavoro in collaborazione con Mirco Piccinini (Univ. Parma) e Letizia Temperini (Indam - Univ. Firenze).

We present a classical result by Malmheden based on a simple algorithm to determine the harmonic function in a ball, and we discuss the fractional counterpart, from which one obtains a representation formula for s-harmonic functions as a linear superposition of weighted classical harmonic functions and a new simple proof of the fractional Harnack inequality. This is based on a joint work with Giovanni Giacomin and Enrico Valdinoci.

Simple models for nutrient-oriented bacterial migration are compared. In particular, the potential of certain cross-degenerate diffusion mechanisms to adequately describe experimentally observed phenomena related to emergence and stabilization of structures are discussed. Resulting mathematical challenges are described and possible approaches outlined, both at levels of basic existence theories and the stage of qualitative analysis.

In this talk, we discuss recent advances on up to the boundary gradient estimates for viscosity solutions of free boundary problems governed by fully nonlinear and quasilinear equations with unbounded coefficients. We present the new Inhomogeneous Pucci Barriers as new elements for the proof. If time permits, we discuss some of the main steps in the proof, namely, the trace estimate of the solution on the points of the fixed boundary that projects nontangentially over the free boundary. These methods are inspired by some ideas of Carlos Kenig in Harmonic Analysis.

I will discuss some recent results on the family of fractional Poisson problems $(-\Delta)^s u =f$ in $\Omega$, $u=0$ on $\Omega^c$ of order $2s$ and its connection to the logarithmic Laplacian operator. This connection allows, in particular, to characterize the $s$-dependence of solutions to this family. Special attention will be given to the case $f\equiv 1$, i.e., the fractional torsion problem. This is joint work with Sven Jarohs and Alberto Saldana.

Carnot groups provide the simplest instance of metric spaces that are non-Riemannian but are still endowed with a rich structure of dilations and translations, making possible to develop a non-Riemannian Geometric Measure Theory. The first step of this program consists in the search of a good (i.e natural) notion of regular submanifolds and in the study of their properties. In this talk we present few chapters of this program along the guidelines of a joint monograph with Raul Serapioni and Francesco Serra Cassano, Some topics of Geometric Measure Theory in Carnot Groups (in preparation). - I gruppi di Carnot forniscono l'esempio più semplice di spazi metrici che non sono riemanniani ma che sono comunque dotati di una ricca struttura di dilatazioni e traslazioni che permettono di sviluppare una Teoria geometrica della misura non riemanniana. Il primo passo di questo programma consiste nella ricerca di una buona (cioè naturale) nozione. di sottovarietà regolari e nello studio delle loro proprietà. In questo seminario presentiamo alcuni capitoli di questo programma secondo le linee di una monografia scritta in collaborazione con Raul Serapioni e Francesco Serra Cassano, Some topics of Geometric Measure Theory in Carnot Groups (in preparazione).

The Integration by Parts Formula, which is equivalent with the Divergence Theorem, is one of the most basic tools in Analysis. Originating in the works of Gauss, Ostrogradsky, and Stokes, the search for an optimal version of this fundamental result continues through this day and these efforts have been the driving force in shaping up entire subbranches of mathematics, like Geometric Measure Theory. In this talk I will review some of these developments (starting from elementary considerations to more sophisticated versions) and I will discuss recents result regarding a sharp divergence theorem with non-tangential traces. This is joint work with D. Mitrea and M. Mitrea.

We prove the existence of extremal functions for second order Adams inequalities with Navier boundary conditions on balls in R^n in any dimension n>3. The proof is based on a symmetrization argument and the ideas introduced by Carleson and Chang to prove the existence of extremal functions in the first order case, i.e. extremal functions for the Trudinger-Moser inequality on balls. We also derive a supercritical version of this result for spherically symmetric functions.

Una delle maggiori difficoltà tecniche, nella risoluzione col metodo di Perron del ''Problema di Dirichlet'' per l'equazione del calore, è la costruzione di una base di aperti della topologia euclidea sui quali quel problema è risolubile. Nel seminario verrà dimostrato che la difficoltà si può superare in modo elementare, utilizzando un argomento tratto dalla teoria dei polinomi calorici, il principio del massimo e un semplice risultato di algebra lineare.

In this talk, we consider the wave equation where the Laplacian is replaced by a sub-Laplacian (also called ``Hörmander sum of square''), which is an hypoelliptic operator. We handle the problem of describing the propagation of singularities in such equations : the main new phenomenon that we describe is that singularities can propagate along abnormal curves at any speed between 0 and 1. This general result extends an idea due to R. Melrose, and we then illustrate it on an example, the Martinet case, following a joint work with Y. Colin de Verdière. Our statements are part of a classical/quantum correspondance between sub-Riemannian geometry (on the classical side) and the hypoelliptic operator (on the quantum side), which is also helpful to interpret results in control theory and spectral theory of hypoelliptic operators.

We consider viscosity solutions to a one-phase free boundary problem for a nonlinear elliptic PDE with non-zero right hand side. We obtain regularity results for solutions and their free boundaries. The operator under consideration is a model case in the class of partial differential equations with non-standard growth. This type of operators have been used in the modelling of non-Newtonian fluids, such as electrorheological or thermorheological fluids, also in non-linear elasticity and image reconstruction. We also obtain some new results for the governing operator that are of independent interest. This is joint work with Fausto Ferrari (University of Bologna, Italy)

We consider the generalized inviscid surface-quasigeostrophic equations (gSQG) and analyse the existence of a smooth compactly supported solution to the (gSQG) which is concentrated around N moving vortices. The result we discuss could be understood as the extension to the case of the (gSQG) of the seminal result of Marchioro and Pulvirenti concerning the bi-dimensional incompressible Euler equations. However, the information about the dynamic behaviour and the shape of the constructed solution that we obtain is much more precise than the obtained by Marchioro and Pulvirenti. The talk is based on a joint work with Manuel del Pino.

We consider the generalized inviscid surface-quasigeostrophic equations (gSQG) and analyse the existence of a smooth compactly supported solution to the (gSQG) which is concentrated around N moving vortices. The result we discuss could be understood as the extension to the case of the (gSQG) of the seminal result of Marchioro and Pulvirenti concerning the bi-dimensional incompressible Euler equations. However, the information about the dynamic behaviour and the shape of the constructed solution that we obtain is much more precise than the obtained by Marchioro and Pulvirenti. The talk is based on a joint work with Manuel del Pino.

We present recent results about radial solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. We show that a new critical exponent appears, related to the existence or nonexistence of sign-changing solutions. Furthermore we analyze the new concentration phenomena occurring as the exponents approach the critical values. Based on joint works with A. Iacopetti, G. Galise and F. Pacella.

We present some results concerning the local and global regularity of the solutions for some classes/models of sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type. Moreover we also illustrate a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj.

We shall prove that in the first Heisenberg group with a sub-Finsler structure, a complete, stable, Euclidean Lipschitz surface without singular points is a vertical plane. This is joint work with Manuel Ritoré.

We shall present a characterization of simply interpolating sequences in the Dirichlet space. The same characterization is conjectured to hold in all complete Nevanlinna Pick spaces but the problem remains open despite recent progress. Finally, we are going to discuss some variants of the classical interpolation problem, such as random interpolation. This is a field where numerous questions remain open. Extended abstract: https://site.unibo.it/complex-analysis-lab/en/news-1/piniseminarabstract.pdf/@@download/file/PiniSeminarAbstract.pdf

We show a link between weighted Hilbert-Schmidt norms of Wick operators on Bargmann space and $L^2$-norm of Wick symbols with respect to a class of measures on the complex phase space. As an application, we derive the flow of discrete NLS equations by the mean field asymptotics of a many body quantum model for $N$ interacting particles as $N$ becomes large.

Dimostriamo formule di media, di superficie e di volume, per soluzioni classiche di equazioni paraboliche in forma di divergenza sotto ipotesi naturali sulla regolarità dei coefficienti. La dimostrazione si basa sulle proprietà usuali della soluzione fondamentale delle equazioni paraboliche, su un teorema di divergenza generalizzato e su un preciso risultato dovuto a Dubovickii sulla regolarità degli insiemi di livello delle funzioni C^1. Discuteremo infine la generalizzazione di queste formule di media al contesto degli operatori subellittici nei gruppi di Carnot. I risultati di questo seminario sono stati ottenuti in collaborazione con Diego Pallara ed Emanuele Malagoli.

In this talk, we shall exhibit a mini-max characterization of the second eigenvalue of the p-Laplacian operator on p-quasi-open sets, using a construction based on minimizing movements on non-linear gradient flows. The following outline shall be presented: the notion of non-linear eigenvalues and their properties, the statement of the characterization, the notion of Quasi-open sets, and a sketch of the proof of the theorem. This is based on a joint work with Nicola Fusco and Yi Zhang.

The talk will introduce the main concepts and tools of Optimal Transport between probability measures and its recent extensions to the unbalanced case, involving entropic regularizations. A few applications will also be discussed.

I will start the talk by recalling the notion of gradient flow in its easiest occurrence: the evolution equation x'(t)=-grad F(x(t)) in the Euclidean space. In particular, the focus will be on the implicit Euler scheme as a sequence of iterated minimization problems. I will then move to a more involved setting, where the point x is replaced by a probability density ρ evolving in the space of probabilities endowed with the so-called Wasserstein distance, induced by optimal transport. For suitable choices of the functional F one can recover linear diffusion PDEs (heat and Fokker-Planck equations) as well as non-linear ones (porous medium, fast diffusion, models for crowd motion). The iterated minimization scheme is called in this case JKO scheme (from Jordan-Kinderlehrer-Otto). After explaining why this scheme heuristically provide the desired equation at the limit, I will show how its optimality conditions can be exploited to prove estimates on its solutions, in particular BV, Sobolev and Lipschitz bounds. Lipschitz estimates can also be interpreted as bounds on the maximal displacement of each particle in the optimal transport map, and have a numerical interest, which I will discuss in two examples, where a potential drift is coupled either with linear diffusion or with a pressure effect due to density constrained in crowd motion.

In this talk, I will discuss the development of semi-classical analysis for sub-elliptic operators such as sub-Laplacians. For an elliptic operator, this is well understood as the tools and methods to study e.g. quantum ergodicity or Schrödinger equations have become well established over the past fifty years. They rely on the pseudo-differential theory, and in the elliptic case the space of principal symbols is commutative. The aim of this talk is to present my approach to define similar tools for sub-Laplacians, leading to more non-commutative concepts.

Some new ideas in Calculus of Variation and Geometric Measure Theory allowed, in the last decade, to revisit from a precise mathematical point of view some physical models. Instances of such models are the Lord Rayleigh model of charged liquid drops in electrowetting, the liquid drop model by Gamow to describe nuclear fissions, the Hartree equations in atomic physics. In the seminar I will give a brief overview on such results. Later I will focus on recent results about some of those topics. The topic of the talk is partially based upon works in collaboration with M. Goldman, D. Mazzoleni, C.B. Muratov and M. Novaga.

I will introduce the nonlocal curvature flows, discussing existence, uniqueness and stability of solutions. In the particular case of the fractional mean curvature flow, I will also describe the long time behaviour of graphical solutions and some issues related to the formation of singularities.

There is currently a great deal of interest in the scientific community in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease. I will present a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations for the two proteins. In the model neurons are treated as a continuous medium and structured by their degree of mal- functioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of misfolded tau protein and iii) neuron-to- neuron prion-like transmission of the disease. These processes are modelled by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. I will explain in detail the structure of the model and give a hint of the main results obtained and the techniques used for the purpose. Eventually I will also show the output of some numerical simulations, of some significance even if performed in an over-simplified 2D geometry.

In this talk we present some recent results concerning the regularity of the unique weak solution vanishing at infinity of the prescribed mean curvature equation in the Lorentz-Minkowski space for spacelike hypersurfaces, when the mean curvature belongs to $L^p(R^N)$, with $p>N$. This equation is also known as the ``Born-Infeld'' equation, as it comes from the nonlinear model of electromagnetism introduced by M. Born and L. Infeld, but it also plays a crucial role in Relativity. In the first part of the talk we will show a new gradient estimate for smooth solutions of the prescribed mean curvature equation and prove that, under our assumptions, the unique minimizer of the Born-Infeld energy, which is a priori only Lipschitz continuous, is actually a strictly spacelike weak solution of class $W^{2,p}$. In the second part the we will discuss some other related results concerning the existence of spacelike radial graphs of prescribed mean curvature and some open problems. These results are collected in a series of joint works with Prof. D. Bonheure (Université Libre de Bruxelles).

We will present a series of results regarding the behaviour of solutions to boundary value problems driven by non-integer powers of the Laplacian operator. Special attention will be paid to the failure of maximum principles and its consequences.

We consider a class of non-local ultraparabolic Kolmogorov operators and we study suitable fractional Holder spaces that take into account the intrinsic sub-riemannian geometry induced by the operator. We prove a characterization relating the regularity along the vector fields to the existence of appropriate instrinsic Taylor formulas which extends in the non-local context the characterization given in the diffusive setting.

TBA

Totally positive functions play an important role in approximation theory and statistics. I will discuss some recent applications of totally positive functions in sampling theory and time-frequency analysis. At this time totally positive functions are the only functions for which optimal results for sampling in shift-invariant spaces and for Gabor frames have been proved.

In this talk we illustrate results concerning radial positive solutions of semi-linear elliptic equations such as $$\Delta u + k(|x|) u^{q-1}=0 \qquad \qquad \qquad (1)$$ where $x \in \mathbb{R}^n$, $n>2$, $k(|x|)>0$, and its generalization to the $p$-Laplace case. We focus in particular on the critical case $q=2^*=\frac{2n}{n-2}$. Our goal is to find conditions on $k$ ensuring existence and multiplicity of ground states with fast decay, i.e. solutions $u(x)$ defined and positive in the whole of $\mathbb{R}^n$ and decaying as $|x|^{-(n-2)}$ for $|x|$ large. Using Fowler transformation we pass from (1) to a two dimensional dynamical system so that we can apply phase plane techniques such as invariant manifold theory, shooting, Melnikov theory. In particular the search of ground states with fast decay is translated on the search of homoclinic trajectories.

I will illustrate a recent result obtained in collaboration with B. Velichkov and L. Spolaor concerning the regularity of the free boundaries in the two phase Bernoulli problems. The new point is the analysis of the free boundary close to branch points, where we show that it is given by the union of two C^1 graphs. This complete the analysis started by Alt Caffarelli Friedman in the 80’s.

Il filo di questo seminario, del tutto espositivo, segue, in una particolare direzione, la relazione che intercorre tra disuguaglianze differenziali e integrali: a partire dalla nota disuguaglianza di Jensen (e, tempo permettendo, gli spazi di Orlicz), attraverso classici argomenti di subarmonicità nello studio di alcuni integrali singolari, quindi il metodo di Burkholder per stime di operatori su spazi di martingale, e arrivare infine al metodo delle funzioni di Bellman di Nazarov, Treil e Volberg, che ha le sue radici nella teoria del controllo stocastico ottimale.

In this talk we consider the functional whose critical points are solutions of the fractional CR Yamabe-type equation on the CR sphere. Due to the lack of compactness for the associated critical Sobolev embedding, the functional does not satisfy the Palais-Smale condition. By adapting a classical arguments by Struwe and by making use of some recent commutator estimates, which allow us to deal with our non-local setting, we obtain a characterization of the Palais-Smale sequences. Then, as an application, we prove a multiplicity result for the related equation. This is joint work with A.Maalaoui and V.Martino.

In this talk we study the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.

We present some new regularity results for the orthotropic harmonic functions, which are the minimizers of a egenerate and anisotropic variant of the Dirichlet functional. These results have been obtained in collaboration with L. Brasco (Ferrara), V. Julin (Jyvaskyla), C. Leone (Naples) and A. Verde (Naples).

In this talk we discuss some extensions of the classical Krylov-Safonov Harnack inequality. After reviewing the standard regularity theory, we will introduce a weaker notion of viscosity solutions. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. Roughly, our viscosity solutions satisfy comparison in a neighborhood of a touching point whose size depends on the properties of the test functions. As an application, we recover the C^{1,\alpha} estimates of Almgren and Tamanini for quasi-minimizers of the perimeter functional. We also establish the regularity of the free boundary for almost minimizers of one-phase type problems.

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

The talk will be concerned with s-minimal surfaces, that is, hypersurfaces of R^n with zero nonlocal mean curvature. These are the equations associated to critical points of the fractional s-perimeter. We will present a recent result in collaboration with M. Cozzi in which we establish, in any dimension, a gradient estimate for nonlocal minimal graphs. It leads to their smoothness, a result that was only known for n=2 and 3 (but without a quantitative bound); in higher dimensions only their continuity had been established. We will also present a work with E. Cinti and J. Serra in which we prove that half spaces are the only stable s-minimal cones in R^3 for s sufficiently close to 1.

In this talk we present some recent results obtained in collaboration with B. Franchi and P. Pansu about Poincaré and Sobolev inequalities in Heisenberg groups (some results are new also for Euclidean spaces). For $L^p$, $p>1$, the estimates are consequence of singular integral estimates. I would like to concentrate the seminar, in particular, to the limiting case $L^1$, where the exterior Rumin-differential of a differential form is measured in $L^1$ norm. Unlike for $L^p$, $p>1$, the estimates are doomed to fail in top degree. In the limiting case, the singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis and Lanzani-Stein in Euclidean spaces, and to Chanillo-Van Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups.

We present several Mean Value formulas for solutions to linear second order PDEs endowed with smooth ''local fundamental solutions''. We then show how these formulas can be used to obtain Liouville Theorems for entire solutions. Our formulas are, in general, weighted average formulas. The relevant weights are ''densities with the mean value property'' a notion playing a central role in rigidity and stability problems. The results we present, related to the Mean value formulas, are obtained in collaboration with Giovanni Cupini. The ones related to the Liouville Theorems are joint works with Alessia Kogoj.

The notion of weakly monotone functions was introduced, in the setting of Sobolev spaces, by J.Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. We propose a criterion for the continuity of weakly monotone functions in terms of the decreasing rearrangement of their gradient. We also prove the continuity of weakly monotone functions whose gradient is in suitable rearrangement-invariant spaces. In particular, weakly monotone functions with gradient belonging to an Orlicz space or to a Lorentz space are discussed. These results are contained in joint works with Andrea Cianchi.

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In this seminar we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L.

We will discuss the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including L_a = div(|y|^a \nabla), with -1<a<1 and their perturbations. As they belong to the Muckenhoupt class A_2, these operators appear in the seminal works of E. Fabes, C. Kenig, D. Jerison and R. Serapioni and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension. Our goal is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of R. Hardt, L. Simon, Q. Han and F.-H. Lin, giving several applications in the context on solutions to non-local elliptic equations with fractional Diffusions. This is a joint work with Y. Sire and S. Terracini.

The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the Moser-Trudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and P-D. Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy, we use these estimates to compute the Leray-Schauder degree of the Moser-Trudinger equation (via a suitable use of the Poincaré-Hopf theorem), hence proving that for any bounded non-simply connected domain the Moser-Trudinger inequality admits critical points of arbitrarily high energy. In a work in progress with F. De Marchis, O. Druet, A. Malchiodi and P-D. Thizy, we expect to use a variational argument to treat the case of a closed surface.

We deal with non negative s-harmonic functions in a cone C of R^n (with vertex at the origin), which satisfy 0-Dirichlet boundary conditions in the complement of the cone. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half sphere. These conic functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions.

I will talk about a conjecture of Pólya and Szegö on minimal electrostatic capacity sets in convex shape optimization. The functional, associated to the conjecture, involves capacity and perimeter. We will focus on the generalized solutions of the corresponding Euler-Lagrange equation and talk about recent joint work with Nicola Fusco.

In this seminar we illustrate some results of maximal regularity for the Cauchy-Dirichlet mixed problem, with a fractional time derivative of Caputo type in spaces of continuous and Hölder continuous functions. In questo seminario presentiamo alcuni risultati di regolarità massimale per il problema misto di Cauchy-Dirichlet, con una derivata temporale frazionaria di Caputo, in spazi di funzioni continue e hölderiane.

I discuss an approach, based on generalized Mather measures, to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such PDEs, which are natural extensions of Mather measures, originally due to J. Mather. Using the viscosity Mather measures, we can show that the whole family of solutions $v^\lambda$ of the discounted problem, with the discount factor $\lambda$, converges to a solution of the ergodic problem as $\lambda$ goes to 0. This is based on joint work with Hiroyoshi Mitake (Hiroshima University) and Hung V. Tran (University of Wisconsin, Madison).

Nel 2009, Caffarelli, Roquejoffre e Savin hanno introdotto una nozione non locale di perimetro di insiemi, detto perimetro frazionario. Dalla variazione prima del perimetro si ottiene la curvatura media frazionaria di un insieme, che è definita da un operatore integrale con nucleo singolare. Da allora, vari autori hanno studiato queste nozioni, ottenendo ad esempio proprietà di regolarità per superfici minime non locali, esistenza di superfici di tipo Delaunay a curvatura frazionaria costante, e disuguaglianze isoperimetriche. Più recentemente, è stato considerato il moto di superfici secondo la curvatura media frazionaria, che è il flusso gradiente del perimetro non locale, ottenendo risultati di esistenza e unicità per soluzioni deboli e proprietà di invarianza. Dopo aver richiamato queste proprietà, ci soffermeremo su un risultato in collaborazione con E. Cinti ed E. Valdinoci, che dimostra l'esistenza di superfici che sviluppano singolarità di tipo "collo di bottiglia" (neckpinch). E' interessante notare che, come conseguenza della natura non locale della curvatura frazionaria, tali singolarità si sviluppano in qualunque dimensione, inclusa quella orrispondente al caso di curve nel piano. In questo aspetto l'evoluzione si differenzia da quella classica, dove le curve si contraggono a un punto senza sviluppare singolarità in base al teorema di Grayson.

In questo seminario, dopo aver introdotto la nozione di level convessita' ed il ruolo che essa riveste nei problemi di Calcolo delle Variazioni in L^\infty, si studiera' l'inviluppo semicontinuo di un funzionale della forma $$F(u)=\supess_{\Omega} f(x,Du(x))$$ su $W^{1,\infty}(\Omega)$ rispetto la topologia debole* e si dimostrera' che esso soddisfa la proprieta' di level convessita'. A tal fine si rappresenteranno i sottolivelli del funzionale rilassato per mezzo di opportune pseudo-distanze associate al funzionale $F$.

Discuteremo alcune versioni quantitative del Teorema di Alexandrov della bolla di sapone, che afferma che le sfere sono le sole ipersuperfici chiuse embedded a curvatura media costante. In particolare, considereremo ipersuperfici con curvatura media vicina ad una costante e descriveremo in maniera quantitativa la vicinanza ad una singola sfera o ad una collezione di sfere tangenti di raggio uguale in termini dell'oscillazione della curvatura media. Inoltre considereremo il problema analogo in ambito nonlocale, mostrando come l'effetto nonlocale implichi una maggiore rigidità del problema e prevenga la formazione di più bolle.

In this talk I present the main results of a recent paper in collaboration with E. Spadaro (U. Roma La Sapienza) on the regularity of the free boundary for a class of lower dimensional obstacle problems, including the classical scalar Signorini problem. We prove the first results concerning the global structure of the free boundary, in particular showing its local finiteness and its rectifiability.

Among N-dimensional open sets with given measure, balls (uniquely) minimize the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary conditions. We review this classical result and discuss some of its applications. Then we show how this can be enhanced by means of a quantitative stability estimate. The resulting inequality, first conjectured by Nadirashvili and Bhattacharya & Weitsman, is sharp. The results presented are contained in a paper in collaboration with Guido De Philippis and Bozhidar Velichkov.

The convexity of the integrand of a functional of the calculus of variations is equivalent to the lower semicontinuity of the functional in the scalar case, but it is only a sufficient condition in the vectorial case. So, it is not satisfied by many interesting examples to which the existence theorems apply. Moreover, the convexity of the integrand turns out to be a too strong and unrealistic assumption in applications, as for instance in mathematical models in nonlinear elasticity (Ball 1977). In the vectorial framework more appropriate and weaker conditions than the convexity are the polyconvexity and the quasiconvexity. Under these assumptions, many results were proved concerning the partial regularity of minimizers (regularity on open sets of full measure), but the results concerning the everywhere regularity are very few and mainly in low dimensions (n=N=2). We will discuss recent everywhere regularity results of vectorial minimizers for some classes of polyconvex and quasiconvex functionals (n,N >2) obtained in collaboration with F. Leonetti and E. Mascolo (local boundedness) and with them and M. Focardi (Holder continuity). The proofs rely on the power and elegant (typically scalar) method by De Giorgi (1957).

The Dirichlet space on the polydisc consists of analytic functions defined on the cartesian product of n-copies of a disc, having finite Sobolev norm. In the one-dimensional case (d = 1) the Carleson measures were first described by Stegenga (’80) in terms of capacity, further development was achieved in papers by Arcozzi, Rochberg, Sawyer, Wick and others. Following Arcozzi et al. we consider the equivalent problem in the discrete setting - characterization of trace measures for the Hardy operator on the polytree. For d = 2 we present a description of such measures in terms of bilogarithmic capacity (which, in turn, gives the description of Carleson measures for the Dirichlet space on the bidisc in the sense of Stegenga). We also discuss some arising combinatorial problems. This talk is based on joint work with N. Arcozzi, K.-M. Perfekt, G. Sarfatti.

In this talk I will present different semilinear wave-type problems with time-variable coefficients. Main discussion will concern the influence of such coefficients on the critical exponents which characterize the equation. The analysis of global existence and blow-up below or above this critical exponent will follows.

Illustreremo una serie di risultati in collaborazione con vari coautori sul problema della regolarità delle curve minime per la lunghezza negli spazi di Carnot-Caratheodory. Discuteremo l'esistenza di tangenti in ogni punti a valcuni risultati algebrici sulle cosiddette curve abnormali.

I will present a class of boundary conditions for Kramers-Fokker-Planck operators which guarantees subelliptic estimates similar to the whole space problem.

In questo seminario introduciamo le derivate frazionarie di Riemann-Liouville e di Caputo, con alcune delle loro principali proprietà. Concludiamo illustrando alcuni risultati di regolarità massimale per problemi misti al contorno, in cui compaiono tali derivate.

We establish new fractional Poincaré inequalities encoding geometry of conformally flat manifolds with finite total Q-curvature. The method of proof is based on some improvement of the standard Poincare inequality and harmonic analysis techniques. We will give a description of the underlying geometry and in particular the role of the Q-curvature.

abstract: We consider the subelliptic eikonal equation, i.e. the eikonal equation associated with a family of (real) smooth vector fields satisfying the Hoermander bracket generating condition on a neighborhood of an open bounded set with smooth boundary. We study the regularity and the singularities of the viscosity solution of the homogeneous Dirichlet problem for such an equation.

Il Soap Bubble Theorem (SBT) stabilisce che una superficie compatta con curvatura media costante è una sfera. Per dimostrare questo risultato, A. D. Alexandrov ha inventato il suo principio di riflessione, che è stato in seguito perfezionato da J. Serrin nel metodo dei piani mobili, per ottenere la simmetria radiale per una classe di problemi sovra-determinati. H. F. Weinberger ha fornito una dimostrazione del risultato di Serrin basata su alcune identità e disuguaglianze integrali. R. C. Reilly ha infine fatto vedere come il metodo di Weinberger può essere usato per ottenere un'altra dimostrazione del SBT. Nel mio seminario, seguendo le orme di Weinberger e Reilly, farò vedere come i due risultati di simmetria discendano da due identità integrali per la rigidità torsionale di una sbarra. Le due identità saranno poi usate per ottenere risultati di stabilità della configurazione sferica nei due problemi ed in altri problemi analoghi.

In this seminar some recent results concerning Harnack inequalities will be presented for several classes of sub-elliptic operators. We will start by considering a class of sub-elliptic operators, in divergence form, with low-regular coefficients under global doubling and Poincaré assumptions; for these operators a non-homogeneous invariant Harnack inequality will be shown. As a consequence, we will prove the solvability of the Dirichlet problem (in a suitable weak sense). In the second part, we will consider a class of hypoelliptic non-Hormander operators for which we have been able to construct a Green function; with a completely different approach with respect to the case of doubling metric spaces, we will conclude by showing (by means of techniques of Potential Theory) how the solvability of the Dirichlet problem has been a fundamental tool in order to prove a homogeneous Harnack inequality in the framework of harmonic spaces.

In this presentation, we will analyze a p-Laplacian problem set in a ball of R^N, with homogeneous Neumann boundary conditions. The equation involves a nonlinearity g which is (p-1)-superlinear at infinity, possibly supercritical in the sense of Sobolev embeddings. The nonlinearity allows the problem to have a constant non-zero solution. In this setting, we prove via shooting method the existence, multiplicity, and oscillatory behavior (around the constant solution) of non-constant, positive, radial solutions. We show that the situation changes drastically depending on p>1. For example, in the prototype case g(s)=s^{q-1}, if p>2, the problem has infinitely many solutions for q>p. While, if p=2, the problem admits at least k non-constant solutions provided that q-2 is bigger than the (k+1)-th radial eigenvalue of the Laplacian with Neumann boundary conditions. Finally, for 1<p<2 a surprising result is found, as non-constant solutions with the same oscillatory behavior appear in couples when the radius of the domain is big enough. We will try to give a unified description and motivation for these three different situations. This is a joint work with Alberto Boscaggin (Università di Torino) and Benedetta Noris (Universitè de Picardie Jules Verne). [A. Boscaggin, F. Colasuonno, B. Noris, Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions, preprint] [F. Colasuonno, B. Noris, A p-Laplacian supercritical Neumann problem, Discrete Contin. Dyn. Syst., Vol. 37 n. 6 (2017) 3025-3057]

We present some recent results in the study of the fractional Allen-Cahn equation. In particular, we are interested in the analogue, for the fractional case, of a well known De Giorgi conjecture about one-dimensional symmetry of bounded monotone solutions. In dimension n=2 and for any fractional power 0<s<1 of the Laplacian, the conjecture is known to be true. In this seminar, we will address the 3-dimensional case. Depending wheter s is below or above 1/2, we need to exploit different techniques and ingredients in the proof of the one-dimensional symmetry. In particular, when s<1/2, some properties of the so-called nonlocal minimal surfaces, will play a crucial role. This talk is based on several papers in collaboration with X. Cabré, J. Serra, and E. Valdinoci.

The aim of this seminar is to present some results about the isoperimetric problem in Carnot-Carathéodory spaces connected with the Heisenberg geometry. The Heisenberg group is the framework of an open problem about the shape of isoperimetric sets, known as Pansu’s conjecture. We start by studying the isoperimetric problem in Grushin spaces and Heisenberg type groups, under a symmetry assumption that depends on the dimension. We emphasize a relation between the perimeter in these two types of structure. We conclude by presenting some recent results about constant mean curvature surfaces (hence about isoperimetric sets) in the Riemannian Heisenberg group, focusing our attention on the subriemannian limit.

In this talk we analyze the local solvability property of a class of degenerate second order partial differential operators with smooth and non-smooth coefficients. The class under consideration exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. In particular we shall show the local solvability property of the class in the neighborhood of a set where the principal symbol of the operator can possibly change sign (which is a property that can negatively affect the local solvability of the operator).

It is well known that real regular bounded KP (n-k,k)-line solitons are associated to soliton data in the totally non-negative part of the Grassmannian Gr(k,n) and that, in principle, they may be obtained in a certain limit from regular real quasi--periodic KP solutions. The latter class of KP solutions correspond to algebraic geometric data a la Krichever on regular M-curves according to a theorem by Dubrovin-Natanzon. In this talk I shall present some new results recently obtained in collaboration with P.G. Grinevich (LITP-RAS and Moscow State University). The purpose of our research is the connection of such two areas of mathematics using the real finite gap theory of the KP equation. I shall explain how we associate to any KP soliton data in the real totally nonnegative part of Gr(k,n) the rational degeneration of an M-curve of genus g=k(n-k) and the effective KP divisor.

Let $\Omega$ be a domain in ${\mathbb{R}^N$. A density with the mean value property for non-negative harmonic functions in $\Omega$ is a positive l.s.c. function $w$ such that, for a suitable $ x_0 \in \Omega $, $$ u(x0) = \frac{1}{w(Ω)} \nt_{\Omega} u(y)w(y)dy $$ for every non-negative harmonic function $u$ in $\Omega$. In this case we say that $(\Omega,w,x_0)$ is a $\Delta$-triple. Existence of $\Delta$-triples on every sufficently smooth domain has been proved in 1994-1995, by Hansen and Netuka, and by Aikawa. Very recently, we have given positive answers to the following inverse problem: “Let $ (\Omega,w,x_0)$ and $(D,w',x_0)$ be $\Delta$-triples such that $\frac{w }{w(\Omega)= \frac {w'}{w'(D)} in $D ∩Ω$. Then is it true that $ \Omega = D$?” Our result contains, as particular cases, several classical potential theoretical characterizations of the Euclidean balls. Densities with the mean value property for solutions to wide classes of Picone’s elliptic-parabolic PDEs have appeared in literature since the 1954 pioneering work by B.Pini on the mean value property for caloric functions. In this talk we present an abstract inverse problem Theorem allowing to extend the previously recalled result on the $ \Delta$-triples to elliptic, parabolic and sub-elliptic PDEs. The results have been obtained in collaboration with Giovanni Cupini (Universita' di Bologna).

We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on (−∞;∞). Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too. Keywords: Inverse problem; First-Order problem, Second-Order problem, c0−semigroup, Periodic Solution. Joint work with: Mohammed AL Horani; Mauro Fabrizio; Hiroki Tanabe

In this talk we discuss a rigidity result for a class of real hypersurfaces in C^2 with constant Levi curvature. Following old techniques due to Jellett, we consider the boundaries of starshaped domains which satisfy a suitable condition. We provide as application an Aleksandrov-type result for domains with circular symmetries. This is a joint work with V. Martino.

Nel seminario presenteremo un risultato di minimalità locale per un’energia ottenuta come limite del modello di Ohta-Kawasaki. Utilizzando tale risultato mostreremo che le configurazioni tridimensionali periodiche, strettamente stabili per il funzionale dell’area, sono esponenzialmente stabili sia per il flusso non locale di Mullins-Sekerka che per quello di Hele-Shaw.

Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means. Joint work with Amalia Culiuc and Yumeng Ou and partly with Jose Manuel Conde-Alonso, Yen Do and Gennady Uraltsev.

Abstract. Let L be the Laplacian on R^n . The investigation of necessary and sufficient conditions for an operator of the form F (L) to be bounded on L^p in terms of smoothness properties of the spectral multiplier F is a classical and very active research area of harmonic analysis, with long-standing open problems (e.g., the Bochner–Riesz conjecture) and connections with the regularity theory of PDEs. In settings other than the Euclidean, particularly in the presence of a sub- Riemannian geometric structure, the natural substitute L for the Laplacian need not be an elliptic operator, and it may be just sub-elliptic. In this context, even the simplest questions related to the L^p -boundedness of operators of the form F (L) are far from being completely understood. I will survey recent results dealing with the case of sub-Laplacians on 2-step Carnot groups, complex and quaternionic spheres, and Grushin operators.

In this talk we will present an overview of some recent results on α-harmonic maps which are horizontal with respect to a given plane distribution.

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the successful study of polynomial dynamics. It states that, for a complex polynomial $f$ with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of the theorem for entire functions with bounded postsingular set. If such $f$ additionally has finite order of growth, then our result states precisely that every periodic hair of $f$ lands at a repelling or parabolic point, and again conversely every repelling or parabolic point is the landing point of at least one periodic hair. (Here a \emph{periodic hair} is a curve consisting of escaping points of $f$ that is invariant under an iterate of $f$.) For general $f$ with bounded postsingular set, but not necessarily of finite order, the role of hairs is taken by more general connected sets of escaping points, which we call \emph{dreadlocks}. This is joint work with Lasse Rempe-Gillen.

We present a review on some regularity results I obtained in the last 10 years for elliptic equations whose prototype is the p(x)-Laplacian; they can be interpreted as the Euler-Lagrange equations of integral functionals appearing in the mathematical modelling of strongly anisotropic materials. Under suitable continuity assumptions on the function p, the results I'm going to present include: - Hoelder continuity results in the scalar case (also for the obstacle problem) - Calderon-Zygmund estimates for a class of obstacle problem with variable growth exponent - global regularity and stability of solutions to elliptic equations with non-standard growth - Lipschitz estimates for systems (thus in the vectorial setting) with ellipticity conditions at infinity

In questa conferenza presenteremo alcune stime L^p-L^2, per valori di p compresi fra 1 e 2, per i proiettori spettrali congiunti associati all’operatore di Laplace--Beltrami e a un sublaplaciano definito su sfere complesse e quaternioniche. Discuteremo, in particolare, il ruolo giocato dalle stime ottimali per le autofunzioni congiunte e illustreremo alcuni problemi connessi all’alta concentrazione delle armoniche sferiche.

I will present two (unconventional) overdetermined problems. Let $n\geq 3$ and $\Omega$ be a bounded domain in $R^n$. First: if the Newtonian potential $u$ of $\Omega$ has two homothetic convex level sets, then $\Omega$ is a ball. Second: if the Newtonian potential $u$ of $\Omega$ is $\frac{1}{2-n}$-concave (i.e. $u^{(1/(2−n)}$ is convex), then Ω is a ball. The result can be extend to the $p$-capacity potential for $p\in(1,n)$.

Presenterò alcuni risultati di maggiore differenziabilità frazionaria per soluzioni di equazioni ellittiche non lineari in forma di divergenza del tipo divA(x;Du) = divG; contenuti in [1]. L’operatore A(x;z) ha crescita quadratica rispetto alla variabile z e la mappa parziale A(.; z) appartiene ad una opportuna classe di Besov. Proviamo che le proprietà di differenziabilità frazionaria di G si trasferiscono al gradiente della soluzione senza perdita nell’ordine di differenziazione. References [1] A. Baison, A. Clop, R. Giova, J.Orobitg, A. Passarelli di Napoli. Fractional differentiability for solutions of non linear elliptic equations - arxiv Preprint 2016.

Mentre è ben nota la caratterizzazione geometrica dell'ipoellitticità C^\infty di una somma Q di quadrati di campi vettoriali, per l'ipoellitticità analitica la questione è largamente aperta. A questo riguardo, F. Treves [1996] ha formulato una congettura secondo cui l'ipoellitticità analitica dipenderebbe dalla "regolarità" simplettica di un'opportuna stratificazione dell'insieme caratteristico di Q. In questo seminario, si discuterà un risultato, ottenuto in collaborazione con P. Albano e A. Bove, che mostra come la suddetta stratificazione non sia in realtà sufficiente a garantire l'ipoellitticità analitica di Q.

Saranno presentati alcuni risultati di esistenza, unicità e regolarità per $t$- grafici e grafici intrinseci nel gruppo di Heisenberg. Inoltre saranno discussi alcuni problemi aperti in questo ambito, con particolare riguardo al problema di Bernstein.

We show how to apply Harmonic Spaces Potential Theory in studying Dirichlet problem for a general class of evolution hypoelliptic PDEs of second order. We construct Perron-Wiener solution and we show a new regularity criterion for the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan. The class of operator to which our results apply contains the Heat operators on stratified Lie groups and the prototypes of the Kolmogorov operators.

In this talk I'll show that the Heisenberg group contains a measure zero set $N$ such that every Lipschitz function $f: H^n\to R$ is Pansu differentiable at a point of $N$. This is a joint work with G. Speight (University of Cincinnati).

We give an survey of some recent results concerning the structure of bases and frames generated by unitary group orbits in Hilbert spaces. Invariant subspaces can be characterized, by means of Fourier intertwining operators, as modules whose rings of coefficients are given by group von Neumann algebras. It can be shown that these modules are naturally endowed with an unbounded operator-valued pairing which defines a noncommutative Hilbert structure. Roughly speaking, each orbit defines a point in such a Hilbert module, and the noncommutative pairing defines the analogous of a scalar product. Frames and bases obtained by countable families of orbits can then be characterized in terms of new notions of noncommutative reproducing systems, for which a full theory of linear expansions has been developed. Motivations for this study come from problems in approximation theory, concerning group generalizations of wavelets and multiresolution analysis, and issues of regular sampling in shift-invariant spaces such as generalized Paley Wiener spaces.

The Newtonian potential of a homogeneous body D is proportional, outside D, to the Newtonian potential of a mass concentrated at a point x_0 in D if and only if D is a Euclidean ball centered at x_0. The if part simply follows from Gauss Mean Value Theorem for harmonic functions. The only if part is a Theorem by Aharonov, Schiffer and Zalcman. Aim of this talk is to present an extension of the previous result to non-homogeneous bodies, obtained in collaboration with Giovanni Cupini.

In this talk, we review semiclassical analysis for systems whose phase space is of arbitrary (possibly infinite) dimension. An emphasis will be put on a general derivation of the so-called Wigner classical measures as the limit of states in a non-commutative algebra of quantum observables. In the remaining time, the related problem of quantization will be introduced; and we will follow the projective approach of Ammari and Nier.

(joint work with Paulo D.~Cordaro) We study the properties of the Green operator for an analytic linear PDO such that both it and its formal adjoint are globally sub-elliptic and globally analytic-hypoelliptic (GAH) in the torus. We introduce the class of M\'etivier operators, $ \mathscr{M}_{\varepsilon}(\mathbb{T}^{N})$, study the properties of its perturbations and of its analytic vectors and show that when the Green operator of $ P(x,D)$ belongs to a well defined class of analytic pseudodifferential operators on the torus then $ P(x,D) \in \mathscr{M}_{\varepsilon}(\mathbb{T}^{N})$. We present some examples of linear PDO in such class.\\ We also study (joint work with N. ~Braun Rodrigues, Paulo D.~Cordaro and M.~R.~Jahnke) the perturbation problem and the Gevrey regularity of the Gevrey vectors for a class of globally analytic hypoelliptic H\"ormander's operators defined on the $N$-dimensional torus introduced by P.~D.~ Cordaro and A.~A.Himonas.

We overview some recent results involving the fractional p-Laplacian operator and discuss the solvability of a related Brezis-Nirenberg problem.

In the present talk we are interested in a singular limit problem for a compressible Navier-Stokes-Korteweg system under the action of strong Coriolis force. This is a model for compressible viscous capillary fluids, when the rotation of the Earth is taken into account. Supposing both the Mach and Rossby numbers to be proportional to a small parameter $\veps$, we are interested in the asymptotic behavior of a family of weak solutions to our model, for $\veps$ going to $0$. We consider this problem in the regimes of both constant and vanishing capillarity: we prove the convergence of the model to $2$-D Quasi-Geostrophic type equations for the limit density function. The case of variations of the rotation axis will be discussed as well.

In this talk I will discuss the validity of the Hopf lemma for certain degenerate-elliptic equations at characteristic boundary points. In the literature there are some positive results under the assumption that the boundary of the domain reflects the underlying geometry of the specific operator. I will mainly focus on conditions on the boundary which are suitable for some families of degenerate operators, also in presence of first order terms. This is a joint work with Vittorio Martino.

We consider 1-D Schrodinger equation with cubic nonlinearity and Hamiltonian with short range potential without zero resonances. We prove existence and scattering of small data solutions.

We consider semilinear parabolic equations involving an operator that is $X$-elliptic with respect to a family of vector fields $X$ with suitable properties. The vector fields determine the natural functional setting associated to the problem and the admissible growth of the non-linearity. We prove the global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity. These results were obtained in collaboration with Alessia E. Kogoj.

In un lavoro in collaborazione con P.G. Grinevich troviamo la connessione fra la teoria delle Grassmanniane totalmente positive e le degenerazioni razionali di M-curve usando la teoria dell'equazione di Kadomtsvev-Petviashvili (KP). Ad ogni punto della grassmanniana reale totalmente positiva $GR^{TNN}_+ (N,M)$ associamo la degenrazione razionale di una M-curva di genere minimo $g=N(M-N)$ e ricostruiamo i dati algebro-geometrici alla Krichever per la corrispondente soluzione multi-solitonica dell'equazione KP. Nel presente seminario spiego l'idea alla base di tale costruzione.

We present some recent results on the local boundedness and Lipschitz continuity of weak solutions to quasilinear systems and/or local minimizers of vector-valued integral functionals. We consider systems and functionals under non standard p-q growth conditions. Moreover, in this context, the existence of weak solutions is also examined.

I would like to describe in some way an asymptitoc behaviour of solutions to Cauchy problems for degenerate parabolic equations with p-Laplacian and gradient absorption term in generalized Baouendi-Grushin type settings. Among qualitative properties under the consideration will be existence, Holder continuity and some sharp estimates of radii of the supports of the solutions and essential maximums of solutions. Also the criterion for the mass decay fenomenon will be discussed.

Starting from a Kolmogorov equations arising in finance we present a method to obtain approximate solutions of the related Cauchy problem. Error estimates for small time strongly depend on the regularity of the final datum. This is our motivation to define for an homogeneous Kolmogorov operator suitable Holder spaces of every order and prove a Taylor type formula. We also compare our definitions and results with the rich related literature.

The Hardy space of slice regular functions on the quaternionic unit ball H^2(B) is a reproducing kernel Hilbert space. In this talk, after an appropriate introduction to the subject, we will see how this property can be exploited to construct a Riemannian metric on B and we will study the geometry arising from this construction. We will also see that, in contrast with the example of the Poincaré metric on the complex unit disc, no Riemannian metric on B is invariant with respect to all slice regular bijective self maps of B. The results presented are obtained in collaboration with Nicola Arcozzi.

In questo seminario tratteremo il problema di caratterizzare lo sviluppo asintotico per tempo piccolo del nucleo del calore p_t(x, y) associato al Laplaciano sub-Riemanniano. In particolare, dopo aver ricordato i risultati noti nel caso Riemanniano e sub-Riemanniano, esamineremo l'asintotica del nucleo del calore quando y e' nel cut locus di x (tipicamente quando la distanza sub-Riemanniana d^2(x,\cdot) non e' differenziabile in y). Mostreremo come l'asintotica di p_t(x,y) riflette la struttura delle geodetiche che collegano x con y. Questi risultati (collaborazione con U. Boscain e R. Neel) sono ottenuti estendendo all'ambito sub-Riemanniano una idea di Molchanov per il caso Riemanniano.

We will show that the CR-Yamabe equation has several families of infinitely many changing sign solutions, each of them having different symmetries. The problem is variational but it is not Palais-Smale: using different complex group actions on the sphere, we will find many closed subspaces on which we can apply the minmax argument.

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem associated with integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.

It is known that the boundary values of the resolvent of selfadjoint Schrödinger operator with a slowly decreasing potential is smooth at the threshold zero. In this talk, I shall show that in a more general setting, one has in fact some Gevrey estimates for the resolvent. As applications, we show that local energies of solutions to the associated heat equation and the Schrödinger equation decay subexponentially.

In mathematical physics and dynamical systems one encounters non self-adjoint operators with spectrum not intersecting a strip, called spectral gap. The estimates of the norm of the resolvent of P in the strip play an important role in the investigation of the local decay of the energy, the analysis of the scattering resonances and the analytic continuation of the dynamical zeta function. We will discuss results and open problems concerning the estimate of the inverse of a holomorphic function without zeros in a strip and the estimates of the cut-off resolvent of the Dirichlet Laplacian for trapping perturbations.

We will show that the injection of a suitable subspace of the space of Legendrian loops into the full loop space is an S^1-equivariant homotopy equivalence. Moreover, since the smaller space is the space of variations of a given action functional, we will compute the relative Contact Homology of a family of tight contact forms on the three-dimensional torus.

Some L^p-Liouville theorems for several classes of evolution equations will be presented. The involved operators are left invariant with respect to Lie group composition laws in R^{n+1}. Results for both solutions and sub-solutions will be given.

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space $\mathbb{H}^n\times \R^+$. This is a joint work with Jinggang Tan.

Bi-Lipchitz maps between metric spaces do not change the Hausdorff dimension of sets, while Hölder continuous maps distort dimension in a controlled way. In this talk we will consider this type of questions for the class of Sobolev mappings in the setting of foliated metric spaces. The results come from recent joint works with R. Monti, J. Tyson, K. Wildrick.

We give a classification of the unitary irreducible representations (unirreps) of the Dynin-Folland Group, a.k.a. the Heisenberg Group of the Heisenberg Group, and discuss the modulation spaces induced by these unirreps on their corresponding co-adjoint orbits via the Weyl-Pedersen calculus. We compare these spaces with both the classical modulation spaces and the co-orbit modulation spaces induced by the Dynin-Folland group.

Si considera il problema di Cauchy per una equazione di tipo Schr\"odinger con una Hamiltoniana dipendente dal tempo ed un termine di convezione. Si provano condizioni necessarie e sufficienti per la buona posizione in spazi di Sobolev e di Gevrey.

The Heisenberg group is equipped with a foliation by horizontal lines that differs substantially from the standard foliation of Euclidean space by lines. We investigate the behavior of Sobolev mappings on this foliation. It is a fundamental property of Sobolev mappings that they are absolutely continuous on almost every line; we estimate the quantity of lines whose image under a Sobolev mapping has dimension at least a fixed number d>1.

Let M be a closed manifold, E, F be complex vector bundles over M, and P be a pseudodifferential operator mapping smooth sections of E to smooth sections of F. I will first discuss implications on the relation between E and F when P is elliptic, then implications on the relations between these vector bundles and M when E and F are line bundles, M a surface and P a first order globally hypoelliptic differential operator of principal type. At the end of the talk I will return to ellipticity and discuss some open problems. The talk is based on joint work with H.Jacobowitz (Indiana Univ. Math J., 2002 and TAMS, 2003) and with A.P.Bergamasco and S.L.Zani (Comm. PDE, 2012).

In questo seminario proveremo l'esistenza di (infinite) soluzioni a segno non costante per l'equazione di Yamabe CR. Il problema e' variazionale, ma il funzionale associato non soddisfa le condizioni di compattezza di Palais-Smale; mediante una opportuna azione di gruppo si costruira' un sottospazio sul quale sara' comunque possibile applicare un argomento di minimax di tipo Ambrosetti-Rabinowitz. Il risultato risolve una questione rimasta aperta dopo la classificazione delle soluzioni positive fatta da Jerison-Lee negli anni '80.

We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be zero or infinity. Therefore, those level sets (called \textsf{vertical curves}) can be of rough nature and not belong to the class of intrinsic regular manifolds.

Presenteremo due stime integrali che determinano in particolare la regolarita` lipschitziana di funzioni superiormente limitate e convesse rispetto a campi di Hörmander. Gli argomenti utilizzati si basano sia sulla geometria indotta dai campi di Hörmander, che da stime integrali per sottosoluzioni di sublaplaciani.

Partirò illustrando una classica disuguaglianza isoperimetrica di Michael e Simon nel caso particolare delle ipersuperfici regolari dello spazio Euclideo n-dimensionale. Questo risultato verrà poi commentato avendo come scopo l'individuazione degli ingredienti-chiave necessari alla sua generalizzazione al contesto non-Euclideo dei gruppi di Carnot. In particolare, illustrerò la cosidetta "disuguaglianza di monotonia". Quindi, dopo aver introdotto le principali notazioni concernenti i gruppi di Carnot, cercherò di illustrare una tecnica che permette di generalizzare a questo setting la disuguaglianza di monotonia, ma in una versione localizzata. Darò poi alcune applicazioni, tra le quali la più importante è una versione generale della disuguaglianza isoperimetrica di Michael e Simon valida per ipersuperfici compatte -con bordo- dei gruppi di Carnot. I risultati di questo seminario si possono trovare, tra gli altri, nel preprint "Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups", reperibile sul sito Arxiv all'indirizzo: http://arxiv.org/pdf/0910.5656