# Incontri di Analisi Matematica tra Firenze, Pisa e Siena

## MathAnalysis(at)UniFIPISI, I

### Venerdì 17 Maggio 2019

Università degli Studi di Firenze

Dipartimento di Matematica e Informatica (DiMaI)

sede di Viale Morgagni 67/A (Aula Tricerri)

#### PROGRAMMA

11:00 *Apertura*

11:10 **Matteo Novaga** (Università di Pisa): *Anisotropic mean curvature flow
12:00 Alaa Elshorbagy (SISSA e ICTP, Trieste): On the square distance function from a manifold with boundary
*

*pausa pranzo*

*14:30 Stefano Galatolo (Università di Pisa): The existence of Noise Induced Order, a computer aided proof
15:20 M. Patrizia Pera (Università degli Studi di Firenze)
Existence and global bifurcation of periodic solutions for retarded functional differential equations on manifolds
*

*pausa caffé
*

*16:30 Massimo Gobbino (Università di Pisa)
Non-local characterizations of Sobolev spaces and bounded variation functions
17:20 Chiusura*

*ABSTRACTS*

**Alaa Elshorbagy** *(SISSA e ICTP, Trieste)*

On the square distance function from a manifold with boundary

In this talk we aim to characterize arbitrary codimensional smooth manifolds M with boundary embedded in Rn using the square distance function and the signed distance function from M and from its boundary. The results are localized in an open set.

**Stefano Galatolo** *(Università di Pisa)*

*The existence of Noise Induced Order, a computer aided proof*

Dynamical systems perturbed by noise appear naturally as models of physical systems. In several interesting cases it can be approached rigorously by computational methods.

As a nontrivial example of this, we show a computer aided proof to rigorously show the existence of noise induced order in the model of chaotic chemical reactions where it was first discovered numerically by Matsumoto and Tsuda in 1983. We show that in this random dynamical system the increase of noise causes the Lyapunov exponent to decrease from positive to negative, stabi-

lizing the system. The method is based on a certified approximation of the stationary measure in the L1 norm. This is done by an efficient algorithm which is general enough to be adapted to any dynamical system with additive noise on the interval. Time permitting we will also talk about linear response of such systems when the deterministic part of the system is perturbed deterministically.

**Massimo Gobbino** (Università di Pisa)

*Non-local characterizations of Sobolev spaces and bounded variation functions*

We present two approximations of the p-norm of the gradient of a function u(x) through double integrals that do not involve derivatives. In the first one, the so called “horizontal approximation”, the gradient is replaced by finite differences. In the second one, the so called “vertical approximation”, the double integral measures some sort of interaction between the sublevels of u(x). In both cases, the integrand penalizes the pairs of points (x, y) such that

x and y are close to each other, while the difference between u(x) and u(y) is large. We describe a common approach that in both cases leads to compute the pointwise limit and the Γ-limit of these functionals. The main steps are first reducing to dimension one, then localizing the result to an interval, and finally reducing the analysis to the asymptotic behavior of suitable multi-variable inequalities.

(Based on some joint works with M.G. Mora (horizontal approach) and with C. Antonucci, M. Migliorini, and N. Picenni (vertical approach))

**Matteo Novaga** (Università di Pisa)

*Anisotropic mean curvature flow*

I will present existence and uniqueness results for the anisotropic mean curvature flow with arbitrary mobility. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a byproduct of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique “flat flow” in the case of general, possibly crystalline, anisotropies is settled.

**M. Patrizia Pera** (Università degli Studi di Firenze)

*Existence and global bifurcation of periodic solutions for retarded functional differential equa**tions on manifolds*

In this talk, I will present some results on the existence and global bifurcation

of periodic solutions to first and second order retarded functional differential equations on boundaryless smooth manifolds. I will consider both cases of a topologically nontrivial compact manifold (e.g., an even dimensional sphere) and of a possibly noncompact constraint, assuming in the latter case that the topological degree of a suitable tangent vector field is nonzero. The approach is topological and based on the fixed point index theory for locally compact maps on metric Absolute Neighborhood Retracts (ANRs). Finally, I will show how to deduce from our results a Rabinowitz-type global bifurcation result as well as a Mawhin-type continuation principle.