Schedule and abstracts

Schedule

Thursday Morning

• 9:30 - 10:15 Malizia

• 10:15 - 11:00 De Ponti

• 11:00 - 11:30 Coffee break

• 11:30 - 12:15 Borghini

• 12:15 - 13:00 Luzzi

Friday Morning

• 9:30 - 10:15 De Filippis

• 10:15 - 11:00 Giovannardi

• 11:00 - 11:30 Coffee break

• 11:30 - 12:15 Ognibene

• 12:15 - 13:00 Malagutti

Martina Bellettini

Decomposability bundle of Radon measures on Rn

The decomposability bundle of a Radon measure μ on Rn, introduced by Alberti and Marchese, can be viewed as a kind of “tangent bundle” for measures. It describes the tangential directions associated with possible decompositions of μ into curves. In [G. Alberti and A. Marchese. “On the differentiability of Lipschitz functions with respect to measures in the Euclidean space”] and later works, the authors show that this bundle precisely characterizes the vector subspaces v for which vμ is a 1-dimensional flat chain. In this work, we extend the discussion from curves to surfaces of arbitrary dimension k. We explore whether the previous characterization can be generalized to k-dimensional flat chains. Notably, the characterization does hold when k = n − 1, but it fails for all intermediate dimensions.


Konstantinos Bessas

Regularity aspects of a class of nonlocal total variation-based image denoising models

Image denoising is a core problem in image processing, consisting in finding a regular approximation of a given degraded image.  In this seminar we will first survey the deeply investigated class of total-variation based denoising models highlighting their main features.  We will then focus our attention on denoising models whose regularizing term is a nonlocal total variation induced by a singular kernel. Specifically, we will study the fidelity of the solutions of these models to the initial data as well as their smoothness. Finally, we will discuss the relationship between the jump sets of the regularized images and the ones of the initial data.  Part of the results that we will present was obtained in collaboration with Giorgio Stefani (University of Padova) and part is based on an ongoing project with Antonin Chambolle (CEREMADE, CNRS & Université Paris-Dauphine).

Gaia Bombardieri

Mean Curvature Flow in the Heisenberg group (Ricci unbounded from below)

In this talk, we investigate the Mean Curvature Flow in the Heisenberg group, for a suitable class of mean convex hypersurfaces. The Heisenberg group is fundamental model of sub-Riemannian metric space without Ricci curvature bounded from below in a Riemannian sense. We establish existence results for this flow using a level set approach under axisymmetric or non-characteristic assumptions. Furthermore, we prove that every smooth, compact characteristic hypersurface with a bounded mean curvature is not self-similar under the horizontal mean curvature flow, implying that the Pansu sphere does not evolve self-similarly either. This highlights fundamental differences from the Euclidean setting. This talk is based on joint works with Luca Capogna, Mattia Fogagnolo and Valentina Franceschi.


Stefano Borghini

Serrin problem for ring-shaped domains.

Let Ω ⊂ Rn be a bounded domain and u : Ω → R be a function with constant nonzero Laplacian and such that u = 0 on the boundary ∂Ω. A celebrated result due to Serrin states that, if one assumes the additional hypothesis that the normal derivative of u is constant on ∂Ω, then Ω must be a ball and u is rotationally symmetric. We discuss the characterization of rotationally symmetric solutions of Serrin’s problem on a ring-shaped domain. In contrast with Serrin’s result, we show that having locally constant Neumann boundary data is not sufficient for this purpose. Nevertheless, we prove that rotational symmetry can be forced by means of an additional assumption on the structure of the set of maximum points. This talk is based on joint works with Virginia Agostiniani, Chiara Bernardini and Lorenzo Mazzieri.


Filomena De Filippis

μ-ellipticity and nonautonomous integrals

µ-ellipticity describes certain degenerate forms of ellipticity typical of convex integrals at linear or nearly linear growth, such as the area integral or the iterated logarithmic model. The regularity of solutions to autonomous or totally differentiable problems is classical after Bombieri, De Giorgi and Miranda, Ladyzhenskaya and Ural’tseva and Frehse and Seregin. The anisotropic case is a later achievement of Bildhauer, Fuchs and Mingione, Beck and Schmidt and Gmeineder and Kristensen. However, all the approaches developed so far break down in presence of nondifferentiable ingredients. In particular, Schauder theory for certain significant anisotropic, nonautonomous functionals with Hölder continuous coefficients was only recently obtained by C. De Filippis and Mingione. We will see the validity of Schauder theory for anisotropic problems whose growth is arbitrarily close to linear within the maximal nonuniformity range, and discuss sharp results and insights on more general nonautonomous area type integrals. From recent, joint work with Cristiana De Filippis (Parma) and Mirco Piccinini (Pisa).


Nicolò De Ponti

Unique continuation at infinity: Carleman estimates on general warped cylinders

We discuss a vanishing result for solutions of $|\Delta u| \leq q_1 |u| + q_2 |\nabla u|$ that decay to zero at infinity on a Riemannian manifold. The main new ingredient that we present is a Carleman estimate suitable for dealing with general warped cylindrical ends and potential functions $q_1, q_2$. We also discuss some geometric applications of our result. Joint work with Stefano Pigola and Giona Veronelli.


Guido Drei

Global hypoellipticity on products of compact Lie groups

The purpose of the talk is to introduce Fourier analysis on compact Lie groups. We recall a necessary and sufficient condition for global hypoellipticity of left-invariant pseudo-differential operators. In particular, we present some simple but meaningful examples of first-order operators on the torus and the 3-sphere and we study conditions for them to be globally hypoelliptic. As conjectured by S.Greenfield and N.Wallach, every smooth closed manifold that admits a globally hypoelliptic vector field is diffeomorphic to a torus, and the vector field is conjugated to a constant vector field satisfying a Diophantine condition. However, it is possible to construct global hypoelliptic first-order operators on a product T ^(r+1) × (S^3)^s, by considering suitable zero-order perturbations of complex vector fields. Some examples of global hypoelliptic operators arise very naturally, for instance by considering the heat operator. A very important problem in this research field is the improvement of techniques for the construction of parametrices, which is not a simple issue when dealing with products of groups, and not yet completely known.


Marco Flaim

On a parabolic curvature lower bound generalizing Ricci flows

Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. A number of results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals, which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport methods and understand it heuristically as Ricci non-negativity of the space-time. We show some inequalities that are new even for Ricci flows and are related to the Hamiltonian perspective on the $L$ distance.


Mattia Fogagnolo

Mean curvature flow and Ricci curvature bounded from below

The mean curvature flow evolves a hypersurfaces with a velocity proportional to its mean curvature vector. If the initial hypersurface is mean convex (positive mean curvature) such flow can be described by the level sets of a spatial function, called arrival time function, that solves a degenerate elliptic equation.  In this talk I will illustrate a new sharp gradient estimate for such a function, implying a sharp extinction time estimate for the mean curvature flow in terms of the initial mean curvature and of the ambient Ricci curvature lower bound. The inequalities are saturated along the mean curvature flow of a geodesic sphere in space forms. The talk is based on a work in progress with Gaia Bombardieri and Valentina Franceschi.


Gianmarco Giovannardi 

The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group

We will deal with the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body, for t-graphs on a bounded domain \Omega  in the Heisenberg group H^n . When the prescribed datum  is constant and strictly smaller than the Finsler mean curvature of the boundary of \Omega, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme. This is joint work with A. Pinamonti, J. Pozuelo, S. Verzellesi.


Luciano Luzzi

Isosystolic Inequalities in $\mathbb{C}P^n$

In this talk, we introduce the \emph{holomorphic $k$-systole} on complex projective space, defined as the infimum of areas of homologically non-trivial holomorphic $k$-chains. Via a variational analysis of this functional, we contrast its behavior with the classical homological $2k$-systole to derive geometric and analytic properties of Hermitian metrics in a neighborhood of the Fubini-Study metric. As an application, we present restrictions on generalizing classical results in Kähler geometry to general Hermitian metrics.  


Alberto Maione

H-compactness for nonlocal linear operators in fractional divergence form

In this talk, we present the mathematical theory of the homogenisation of composite materials, from its origins in the 1970s to some recent applications. In particular, we discuss a new H-compactness result for possibly non-symmetric and nonlocal linear operators in fractional divergence form. In the second part, we focus on symmetric operators and show that the H-convergence is equivalent to the Γ-convergence of the associated nonlocal energies. This research is carried out in collaboration with Maicol Caponi (University of L'Aquila) and Alessandro Carbotti (University of Salento).


Marcello Malagutti

Scattering for the Steklov problem on an infinite wedge

The Steklov problem is the spectral problem for the Dirichlet-to-Neumann map (DtN) for the Laplacian–an operator which appears, for example, in the Calderon problem and the Sloshing problem. Motivated by the Steklov problem on higher dimensional, piecewise smooth Lipschitz domains, we consider the massive DtN map (i.e. the DtN map for -\Delta +M^2) on infinite, asymptotically conic, domains. We develop a scattering theory for the DtN map by proving a limiting absorption principle for this operator and construct the associated scattering matrix. Using this as a tool, we describe the spectrum of the massive DtN on a piecewise smooth, bounded, Lipschitz domain in two dimensions. Based on joint work with Jeffrey Galkowski and Ruoyu P.T. Wang.


Francesco Malizia

Min-max theory and Yamabe metrics on conical manifolds

While the Yamabe problem (that is, finding metrics with constant scalar curvature on a given conformal class) has been completely solved on closed smooth manifolds, its extension to singular spaces presents significant new challenges. In this talk, we consider the Yamabe problem on manifolds with isolated conical singularities. In particular, we prove the existence of Yamabe metrics on four-manifolds possessing finitely-many conical points with Z/2Z-group, using for the first time a min-max scheme in the singular setting. In our proof, we exploit recent positive mass theorems in the conical setting and study how the mass of the conformal blow-up diverges as the blow-up point approaches the singular set. This is based on a joint work with Mattia Freguglia and Andrea Malchiodi.


Roberto Ognibene

Harmonic maps and optimal partition problems

In their 1992 foundational paper, M. Gromov and R. Schoen developed a framework for the regularity theory of harmonic maps with values into possibly singular target spaces. A suitable choice of the singular target allows to recover solutions to the optimal partition problem, namely energy-minimizing vector-valued functions for which at most one component is nonzero at each point. I will explain this correspondence and introduce the notion of free boundary, which naturally emerges as the interface separating the positivity sets of different components. I will then review what is known about its regularity and geometrical structure (mainly focusing on the works of Gromov–Schoen, Conti–Terracini–Verzini, and Caffarelli–Lin) and, finally, I will present recent results obtained in collaboration with B. Velichkov (Pisa).


Julián Pozuelo

Existence of isoperimetric regions in sub-Finsler nilpotent groups

In this talk, we will discuss the method developed by Morgan to prove the existence of isoperimetric regions in sub-Finsler nilpotent Lie groups. The main novelty is a new and elementary proof of a deformation lemma based on the existence of Cheeger sets in metric measure spaces and the use of the translations of the space. This lemma will play a key role in proving properties of isoperimetric regions, such as the coincidence between the topological and essential boundaries, the boundedness, and the existence, and provides a direct proof of the locally Lipschitzianity of the isoperimetric profile.


Gianluca Somma

Diameter of small sub-Riemannian balls

A natural question in metric geometry is whether, and how, the diameter of a ball is related to its radius. In the setting of C^{1,1} sub-Riemannian manifolds, we prove that, for sufficiently small balls (locally uniformly), the diameter equals twice the radius. Moreover, when the regularity of the structure is lowered to C^0, the diameter can be made arbitrarily close to twice the radius. This talk is based on a joint work with Marco Di Marco and Davide Vittone.


Giorgio Stefani

A geometrical approach to the sharp fractional Hardy inequality on convex sets

We investigate the sharp constant in the fractional Hardy inequality on convex sets for p=1. Our approach reformulates the sharp constant equivalently as the Cheeger constant for the fractional perimeter and the Lebesgue measure with an appropriate weight. As a by-product, we derive new lower bounds for the sharp constant in the one-dimensional case, even for non-convex sets, some of which optimal in the case p=1. This is a joint work with F. Bianchi and A. C. Zagati.