The purpose of this workshop is to bring together young researchers and Italian experts on the subject of vector fields and related geometric structures in singular settings, such as Carnot-Carathéodory or Gaussian spaces, where usual Euclidean analysis techniques meet several difficulties.

Location

Meetings will take place at Aula 6 of Dipartimento di Matematica e Informatica, via Machiavelli 30, Ferrara.

Confirmed speakers

• Davide Addona (Università degli studi di Ferrara)
• Elia Bruè (Scuola Normale Superiore)
• Vito Buffa (Università degli studi di Ferrara)
• Giovanni Eugenio Comi (Scuola Normale Superiore)
• Alessandra Lunardi (Università degli studi di Parma)
• Giorgio Menegatti (Università degli studi di Ferrara)
• Giorgio Stefani (Scuola Normale Superiore)
• Davide Vittone (Università degli studi di Padova)

Organizers

• Valentino Magnani (Università degli studi di Pisa)
• Michele Miranda Jr (Università degli studi di Ferrara)
• Dario Trevisan (Università degli studi di Pisa)

Acknowledgments

• INdAM (progetto di ricerca GNAMPA 2017 ''campi vettoriali, superfici e perimetri in geometrie singolari'')
• Università degli studi di Ferrara
• Università degli studi di Pisa

Tuesday, 27th

11:00 -- 11:40 Addona

11:50 -- 12:30 Vittone: On the rank-one theorem for BV functions

Lunch break

14:30 -- 15:10 Comi: The Gauss-Green theorem in stratified groups

15:20 -- 16:00 Menegatti: Sobolev classes and bounded variation functions on domains of Wiener spaces

Coffee break

16:40 -- 17:20 Stefani: Heat and entropy flows in Carnot groups

Wednesday, 28th

9:00 -- 9:40 Bruè: Approximation in Lusin’s sense of Sobolev functions by Lipschitz functions and applications.

9:50 -- 10:30 Buffa: BV Functions in Metric Measure Spaces: new insights into integration by parts formulæ, and traces

Coffee break

11:10 -- 11:50 Lunardi: Funzioni BV in spazi di Hilbert

Alessandra Lunardi (Università degli studi di Parma)

Funzioni BV in spazi di HIlbert

Si introducono e si studiano funzioni a variazione limitata definite su uno spazio di Hilbert dotato di una misura di probabilità "buona", ovvero che permetta di fare integrazioni per parti lungo direzioni opportune. Particolare attenzione è dedicata alle funzioni caratteristiche di insiemi misurabili, e quindi agli insiemi di perimetro finito. Si stabiliscono proprietà e caratterizzazioni di base, e si danno esempi in alcune situazioni significative.

Davide Vittone (Università degli studi di Padova)

On the rank-one theorem for BV functions

In 1993 G. Alberti proved a celebrated result, conjectured by L. Ambrosio and E. De Giorgi, concerning a rank-one property for the singular part of the derivative of a vector-valued map with bounded variation. We will discuss a recent elementary proof of this result together with some applications to BV functions in sub-Riemannian Carnot groups. These are joint works with S. Don and A. Massaccesi.

Giovanni Eugenio Comi (Scuola Normale Superiore)

The Gauss-Green theorem in stratified groups

The Gauss-Green formula is of significant relevance in many areas of mathematical analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields. In the Euclidean setting it has been shown by Silhavy (2005) and Chen, Torres and Ziemer (2009) that Gauss-Green formulas hold for sets of finite perimeter and L^{\infty}-divergence measure fields, i. e. essentially bounded vector fields whose distributional divergence is a Radon measure. We extend these results to the context of stratified groups. In particular, we prove the existence of generalized normal traces on the reduced boundary of sets of locally finite h-perimeter without requiring De Giorgi's rectifiability theorem to hold. This is a joint work with V. Magnani.

Giorgio Menegatti (Università degli studi di Ferrara)

Sobolev classes and bounded variation functions on domains of Wiener spaces

We consider problems connected to W^{1,p}(O) and BV(O) for O convex set in a Wiener space (Banach separable space with Gaussian measure); we focus our analysis on the approximation of functions with regularizing sequences, in particular by considering an extension of a result obtained by Barbu and Röckner in the Euclidean case.

Giorgio Stefani (Scuola Normale Superiore)

Heat and entropy flows in Carnot groups

After the work of Ambrosio, Gigli and Savaré, it is well-known that in any CD(K,+\infty) space, i.e. a space with Ricci curvature bounded from below in the sense of Sturm-Lott-Villani, the gradient flow of the Boltzmann entropy and the heat flow coincide. In 2014 Juillet proved that this correspondence holds also in the Heisenberg groups of any dimension, although these groups are not CD(K,+\infty) spaces. It was an open problem to establish the same correspondence in any Carnot group. In this talk, we give a positive answer to this question. This is a joint work with L. Ambrosio.

Elia Bruè (Scuola Normale Superiore)

Approximation in Lusin’s sense of Sobolev functions by Lipschitz functions and applications.

We say that a real valued function f, defined in a metric measure space, is approximable in a Lusin sense by Lipschitz functions if, for every \epsilon>0, there exists a Lipschitz function that coincides with f outside a set of measure less than \epsilon. In Euclidean spaces, more generally in metric measure spaces satisfying the doubling and Poincarè inequality, Sobolev functions fulfill this approximation property in a quantitative form.

In a joint work with L. Ambrosio and D. Trevisan we extend these results to a class of non-doubling metric measure structures. Our strategy relies upon pointwise estimates for heat semigroups and applies to Gaussian and RCD(K,\infty) spaces. As a consequence, we prove a first quantitative stability estimate for regular Lagrangian flows associated to Sobolev vector fields in an infinite dimensional setting.

Davide Addona (Università degli studi di Ferrara)

TBA

TBA

Vito Buffa (Università degli studi di Ferrara)

BV Functions in Metric Measure Spaces: new insights into integration by parts formulæ, and traces

We adapt the tools from the differential structure developed by N. Gigli in order to give a definition of BV functions on RCD(K,\infty) spaces via suitable vector fields and then establish an extended Gauss-Green formula on a class of "regular" domains, which features the "normal trace" of vector fields with finite divergence measure. Then, we pass to the more classical context of a doubling metric measure space supporting a Poincaré inequality, where we reformulate the theory of "rough traces" of BV functions (after V. Maz'ya)in comparison with the Lebesgue-points characterization, and discuss the conditions under which the respective notions of trace coincide. Based on a joint work with M. Miranda Jr.