## Seminario di Analisi Matematica

**SEMINARI A.A. 2021/2022**

22/11/2021 |
The aim of my talk is to construct (explicit) heat kernels for some evolution equations which arise in physics, conformal geometry and subelliptic PDEs. A common feature is that the relevant partial differential operator appears in the form L_1 + L_2 - D_t, but the variables of L_1 and L_2 cannot be decoupled. This obstruction results in the fact that the heat kernel of L_1 + L_2 - D_t cannot be written as the product of the heat kernels of the operators L_1 - D_t and L_2 - D_t. One of the highlights will be the construction of the heat kernel for the time-dependent version of an ``extension problem" introduced by Frank, Del Mar Gonzalez, Monticelli and Tan. Such problem plays a pervasive role in conformal CR geometry and, among other things, it serves as a way to define the fractional powers of the conformal sublaplacian on the Heisenberg group. The leitmotif of my talk is emphasising the so far unexplored connection of the relevant hybrid equations with the heat kernel of the generalised operator of Ornstein-Uhlenbeck type in the opening of H\"ormander's groundbreaking 1967 work on hypoellipticity. This is joint wok with Giulio Tralli. |
Università degli Studi di Padova |

29/11/2021 |
In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres |
Universidade de São Paolo |

06/12/2021 |
The analysis of the ergodic behavior of solutions to Hamilton- Jacobi-Bellmann equations has a long history going back to the seminal paper by Lions-Papanicolaou-Varadhan. Since this work, the subject has grown very fast and when the Hamiltonian is of Tonelli type a large number of results have been proved. However, few results are available if the Hamiltonian fails to be Tonelli, i.e., the Hamiltonian is neither strictly convex |
Università degli Studi di Padova |

13/12/2021 |
We consider flows of non-Newtonian heat conducting incompressible fluids in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field |
Università degli Studi di Roma Sapienza |

17/01/2021 |
We show that the rotational Smagorinsky model for turbulent flows can be put in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions identifying a proper weighted spaces and checking some easily verifiable assumptions, at fixed time. We also will briefly discuss the critical role of the exponents present in the model (power of the distance function and power of the curl). |
Università di Pisa |

24/01/2022 |
In this talk we will review some recent results we obtained on the one-dimensionality of the minimizers |
Deutsche Bank |

31/01/2022 |
We present some sharp results for nonnegative solutions of nonlinear PDEs of $p$-Laplace type (possibly anisotropic). These PDEs are critical in the sense that they are associated with the study of critical points of functional inequalities. In this talk we consider critical equations arising from Sobolev and Caffarelli-Kohn-Nirenberg (CKN) inequalities. |
Università degli Studi di Milano |

07/02/2022 |
In this talk we present some recent results concerning the existence and the regularity of weak solutions of the prescribed mean curvature equation in the Lorentz-Minkowski space (for spacelike hypersurfaces), when the mean curvature is in L^p. |
Università degli Studi di Torino |

14/02/2022 |
The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points. |
ICREA & Univerisitat de Barcelona |

21/02/2022 |
I will discuss some recent crystallization results for pairwise interaction energies of systems of particles in the plane. I will focus on the so-called Heitmann-Radin (HR) sticky disc potential that - in its classic form - is defined by V(r)=+infty for r<1, V(r)=-1 for r=1, V(r)=0 elsewhere. For the classic HR functional it has been proven that minimizing configurations are subsets of the regular triangular lattice. First, I will show how this result extends in a suitable sense to the class of quasi-minimizers. Furthermore, I will enrich the classic HR model in order to deal with vectorial crystallization prob |
Istituto per le Applicazioni del Calcolo - CNR |

28/02/2022 | nessun seminario | |

07/03/2022 | nessun seminario | |

14/03/2022 |
We discuss CMC immersions of close surfaces (orientable and of genus larger than 1) in hyperbolic 3-spaces and their moduli spaces in the framework of Teichmuller theory. To this purpose we consider a Donaldson type functional introduced by Gonsalves-Uhlenbeck (2007) and establish uniqueness for the corresponding critical points. By this result we will be able to parametrise the moduli space of the immersions in terms of elements of the tangent bundle of the Teichmuller space. |
Università degli Studi di Roma Tor Vergata |

21/03/2022 |
We discuss the setting of the nonparametric plateau problem and introduce the relaxation of the area functional. We present some recent developments on the analysis of it with the aid of cartesian currents. |
Università di Siena |

28/03/2022 |
In this talk, we present a comparison principle for the Hamilton Jacobi (HJ) equation corresponding to linearly controlled gradient flows of an energy functional defined on a metric space. The main difficulties are given by the fact that the geometrical assumptions we require on the energy functional do not give any control on the growth of its gradient flow nor on its regularity. Therefore this framework is not covered by previous results on HJ equations on infinite dimensional spaces (whose study has been initiated in a series of papers by Crandall and Lions). Our proof of the comparison principle combines some rather classical ingredients, such as Ekeland’s perturbed optimization principle, with the use of the Tataru distance and of the regularizing properties of gradient flows in evolutional variational inequality formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian. Our abstract results apply to a large class of examples, including gradient flows on Hilbert spaces and Wasserstein spaces equipped with a displacement convex energy functional satisfying McCann’s condition. |
Università degli Studi di Padova |

04/04/2022 |
Quantitative rigidity results, besides from their inherent geometric interest, have played a prominent role in the mathematical study of models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for functionals within the framework of nonlinear elasticity. |
## Universität Münster |

11/04/2022 |
Initial-boundary value problems for nonlinear parabolic equations ut = ∆ɸ having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. More recently, solutions have been defined and investigated, which for positive times take values in the space of finite Radon measures. Such solutions are often called Radon measure-valued, to distinguish them from function-valued solutions. In general, whether the solution of the problem is a function for some time t>0 can be regarded as a M-L1 regularizing effect. The aim of this talk is to address well-posedness and regularity results, depending on whether or not the initial data charge sets of suitable capacity (determined by the growth order of ɸ), and on suitable compatibility conditions, describing the behaviour of the singular part of solutions. The diffusion function ɸ is only assumed to be continuous, nondecreasing and at most powerlike: no assumptions about existence or estimates from below of the diffusivity ɸ’ are made (except for some regularization results). This lack of regularity and strong coercivity, along with the possible occurrence of infinitely many nondegenerate intervals where ɸ is constant, requires using more refined compactness arguments and Young measure techniques in the proof of existence, which is in turn obtained by a suitable approximation procedure of the initial measure. The possible occurrence or lack of instantaneous M-L1 regularizing effects, as well as partial uniqueness results, will also be discussed. |
Università Campus Bio-Medico Roma |

02/05/2022 |
This talk is concerned with traveling-wave solutions to nonlinear parabolic equations of forward-backward type. First, a short review of some results where the diffusion coefficient depends on the unknown function u (but not on its derivatives) is given. Then we focus on the case where the diffusion coefficient depends on the spatial derivative of u. A famous example is Perona-Malik’s equation in image processing; however, both a convection and a reaction term are also involved in the discussion. Under quite general assumptions, it is shown the presence of wavefront solutions and their main properties are studied. In particular, such wavefronts exist for every speed in a closed half-line and estimates of the threshold speed are given. The wavefront profiles are also strictly monotone and their slopes are uniformly bounded by the critical values of the diffusion. Joint work with Luisa Malaguti and Elisa Sovrano. |
Università degli Studi di Ferrara |

09/05/2022 |
A classical problem in the regularity theory for vector-valued minimizers of multiple integrals consists in proving their smoothness outside a negligible set, cf. Evans (ARMA ’86), Acerbi & Fusco (ARMA ’87), Duzaar & Mingione (Ann. IHP-AN ’04), Schmidt (ARMA ’09). In this talk, I will show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from nonlinear potential theory. In particular, in the setting of functionals with (p,q)-growth - according to the terminology introduced by Marcellini (Ann. IHP-AN ’86; ARMA ‘89) - I will derive optimal local regularity criteria under minimal assumptions on the data. This talk is partly based on joint work with Bianca Stroffolini (University of Naples Federico II) |
Università di Parma |

16/05/2022 |
We consider the physically relevant fully compressible setting of the Rayleigh-Benard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to the gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure - a stationary statistical solution sitting on the global attractor. In addition, the Birkhoff--Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to the attractor a.s. with respect to the invariant measure. |
Charles University Prague |

23/05/2022 |
The study of free boundary minimal surfaces (namely: of critical points for the area functional in the category of relative cycles), which goes back at least to Courant, has played a distinguished role within the class of geometric variational problems for almost a century. |
ETH, Zürich |

06/06/2022 |
In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is necessarily a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili. |
Université |

27/06/2022 |
We consider complete self-shrinking solitons for the mean curvature flow in R^(n+1), meaning minimal hypersurfaces with respect to a Gaussian conformal background metric. Using comparison geometry, we prove that there is a universal constant bounding the entropies of all such embedded shrinkers with a rotational symmetry. As an application, we prove smooth compactness within this class. Finally, we also show that if we impose an additional reflection symmetry, then in each dimension there are only finitely many such shrinkers. This is joint work with Ali Muhammad and John Ma. |
University of Copenhagen |

04/07/2022 |
Ginzburg-Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau-de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. |
UCLouvain |

**SEMINARI A.A. 2020/2021**

15/03/2021 |
The symmetry of solutions of elliptic equations is a classical and challenging problem in PDE, connected with stability. for solutions which are initially non-symmetric. |
Università di Roma Sapienza |

29/03/2021 |
Sub-Riemannian systems are an important class of nonlinear control systems with linear dependence on controls. Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields. We will discuss the long-time average behaviour of the value function of optimal control problems for sub-Riemannian systems, which cannot be addressed by classical weak KAM theory as the Hamiltonian fails to be coercive in the momentum variable. |
Università di Roma Tor Vergata |

26/04/2021 |
Besides their self-evident geometric significance, which can be traced back at least to Courant, free boundary minimal surfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years thanks to work of Fraser and Schoen, in connection to extremal metrics for Steklov eigenvalues for manifolds with boundary (i. e. for eigenvalues of the corresponding Dirichlet-to-Neumann map). |
ETH, Zürich |

10/05/2021 |
In this talk we introduce a general class of singularly-perturbed elliptic functionals Fε and we study their asymptotic behaviour as the perturbation parameter ε > 0 vanishes. Under suitable assumptions, which in particular allow us to bound Fε by the Ambrosio-Tortorelli functionals, we show that the functionals Fε Γ-converge (up to subsequences) to a free-discontinuity functional of brittle type. Moreover, we provide asymptotic formulas for the limiting volume and surface integrands, which show that the volume and surface contributions of Fε decouple in the limit. If time permits, we will discuss the application of the general convergence result to the setting of stochastic homogenisation. This is joint work with R. Marziani and C. I. Zeppieri (Münster). |
Università di Roma Sapienza |

07/06/2021 |
We consider the problem of finding domains that minimize the first eigenvalue of the Dirichlet Laplacian in a Riemannian manifold under volume constraint (Faber-Krahn minimizers). In the Euclidean setting such domains are balls, and existence and regularity of such domains is trivial. In a non-Euclidean setting very few examples are known. In this talk we will show a general result of existence and regularity of Faber-Krahn minimizers, inspired by the analogous result of existence and regularity of the solutions of the isoperimetric problems in a Riemannian manifold. In particular we will show that Faber-Krahn minimizers are regular in low dimension, and that there |
Universidad de Granada |

21/06/2021 |
Henri Berestycki I report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single prey. We focus on the analysis of stationary states, stability issues, and the asymptotic behavior when the competition parameter becomes unbounded. Existence of solutions is obtained by a bifurcation theory type approach and the segregation analysis rests on a priori estimates and a free boundary problem. We discuss the classification of solutions by using spectral properties of the limiting system. Our results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to have such hostile packs, and on comparing the various territory configurations that arise in this context. These questions lead us to nonstandard optimization problems. |
EHESS, Paris |