Venerdì 24 Gennaio 2020, ore 11, Aula 430

Gradient of the single layer potential and rectifiability

Carmelo Puliatti (Universitat Autònoma de Barcelona, puliatti@mat.uab.cat)

Abstract: An important theorem by Nazarov, Tolsa and Volberg asserts that a measure $\mu$ on $\mathbb{R}^{n+1}$ which has $n$-polynomial growth both from above and from below (also known as $n$-Ahlfors-David regular measure) and whose associated $n$-Riesz transform is bounded on $L^2(\mu)$ is $n$-uniformly rectifiable.

In this talk I will discuss an analogous result for the operator

where $A$ is an $(n+1)\times(n+1)$-uniformly elliptic matrix with Hölder continuous coefficients and $\mathcal{E}_A(\cdot,\cdot)$ is the fundamental solution to the equation

This is a joint work with Laura Prat and Xavier Tolsa, and it is also motivated by an application to the study of elliptic measure.

I will also present a recent closely related article which contains a quantitative rectifiability criterion for more general Radon measures in terms of $T_\mu$. This applies to a two-phase problem for the elliptic measure.

## Past Talks

Lunedì 28 Ottobre 2019, ore 11, Aula 2AB45

On the geometry of soap films and soap bubbles

Andrea Marchese (Università di Pavia)

Abstract: I will discuss a recent result establishing that area minimizing m-dimensional currents modulo p in any codimension are regular submanifolds outside a singular set of Hausdorff dimension at most m-1, moreover the singular set is rectifiable and of locally finite Hausdorff measure for odd values of p. Joint work with Camillo De Lellis, Jonas Hirsh, and Salvatore Stuvard.

Lunedì 11 Novembre 2019, ore 12 (non 11!), Aula 2AB45

Some results on the flow of vector fields of BV class

Stefano Bianchini (Sissa, Trieste)

Abstract: This seminar will try to present 3 results.

1) A general theory for uniqueness of Lagrangian representations
Given a vector field $\rho (1,{\mathbf b}) \in L^1_{\mathrm{loc}}({\mathbb R}^+\times {\mathbb R}^{d},{\mathbb R}^{d+1})$ such that ${\mathrm{div}}_{t,x} (\rho (1,{\mathbf b}))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,{\mathbf b}) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to {\mathbb R}^d$, $\dot \gamma (t)= \b(t,\gamma(t))$. We give conditions in terms of a local structure of the representation $\eta$ on suitable sets in order to prove that there is a partition of ${\mathbb R}^{d+1}$ into disjoint trajectories $\wp_{\mathfrak a}$, ${\mathfrak a} \in {\mathfrak A}$, such that the PDE
$$ {\mathrm{div}}_{t,x} \big( u \rho (1,{\mathbf b}) \big) \in \mathcal M({\mathbb R}^{d+1}), \qquad u \in L^\infty({\mathbb R}^+\times {\mathbb R}^{d}), $$
can be disintegrated into a family of ODEs along $\wp_{\mathfrak a}$ with measure r.h.s.. The decomposition $\wp_{\mathfrak a}$ is essentially unique.

2) The application to BV vector fields We show that ${\mathbf b} \in L^1_t({\mathrm{BV}}_x)_{\mathrm{loc}}$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible ${\mathrm{BV}}$ vector fields.

3) Differentiability of the flow

The unique flow generated in point 2 enjoys a weak differentiability property, and its weak derivative satisfies the equation for the Jacobian matrix in this weak setting.

Lunedì 25 Novembre 2019, ore 11, Aula 2AB45

Geometry of 1-codimensional measures in the Heisenberg groups

Andrea Merlo (Pisa)

Abstract: Characterisation of rectifiable measures in Euclidean spaces through the existence of the density has been a longstanding problem for Geometric Measure Theory until the complete answer by D. Preiss in 1987. The question of how in more general metric spaces existence of density can affect any kind of gain in terms of regularity of the measure is a completely open problem. In this talk I will discuss how the mere existence of the 1-codimensional density for a measure in the Heisenberg groups endowed with the Koranyi metric implies that almost everywhere the tangents to the measure are flat.

Lunedì 9 Dicembre 2019, ore 11, Aula 2AB45

Minimal bubble clusters in the plane with double density

Valentina Franceschi (Sorbonne Université, Parigi)

Abstract: We present some results about minimal bubble clusters in the plane with double density. This amounts to find the best configuration of $m\in \mathbb N$ regions in the plane enclosing given volumes, in order to minimize their total perimeter, where perimeter and volume are defined by suitable densities. We focus on a particular structure of such densities, which is inspired by a sub-Riemannian model, called the Grushin plane.
After an overview concerning existence of minimizers, we focus on their Steiner regularity, i.e., the fact that their boundaries are made of regular curves meeting at 120 degrees. We will show that this holds in a wide generality.
Although our initial motivation came from the study of the particular sub-Riemannian framework of the Grushin plane, our approach works in wide generality and is new even in the classical Euclidean case.

Mercoledì 18 Dicembre 2019, ore 11, Aula 430
(Attenzione: non di lunedì e in un’aula diversa dal solito)

A rectifiability result for sets of finite perimeter in Carnot groups

Sebastiano Don (University of Jyväskylä)

Abstract: After an introduction to the regularity problem for sets of finite perimeter in Carnot groups, we prove that the reduced boundary of a set of finite perimeter in a Carnot group can be covered by a countable union of sets satisfying a “cone property”. We show that this weak notion of rectifiability implies the intrinsic Lipschitz rectifiability in a class of Carnot groups including all filiform groups. This is a joint work with Enrico Le Donne, Terhi Moisala and Davide Vittone.