Incontri di Analisi Matematica tra Firenze, Pisa e Siena

MathAnalysis(at)UniFIPISI, V

mercoledì 6 dicembre 2023
Università degli Studi di Firenze
Dipartimento di Matematica e Informatica “Ulisse Dini” (DiMaI)

La partecipazione è libera, ti chiediamo però di compilare il modulo di registrazione

PROGRAMMA

14:30 apertura
14:35 Alessandra Lunardi (Università di Parma): Sobolev and BV functions in infinite dimension
15:30 Masayuki Hayashi (Università di Pisa): Modified energies for the generalized derivative NLS
16:00 pausa caffè
16:30 Luigi C. Berselli (Università di Pisa): Energy conservation for incompressible viscous fluids
17:25 Giorgio Saracco (Università di Firenze): Bijections between isoperimetric sets, prescribed curvature sets, and p-Cheeger sets
17:55 chiusura

ABSTRACTS

Luigi C. Berselli: In this talk I will discuss classical and recent results about the energy conservation for Leray-Hopf solutions to the Navier-Stokes equations, satisfying additional assumptions.
In particular, I will focus on Hölder continuous solutions (wrt space variables) and on the technical steps necessary to pass from the periodic case to the Dirichlet problem in various domains with solid boundaries.

Joint work with A. Kaltenbach and M. Růžička

Alessandra Lunardi: In Hilbert or even Banach spaces $X$ endowed with good probability measures there are a few “natural” definitions of Sobolev spaces and of functions with bounded variation. The available theory deals mainly with Gaussian measures and Sobolev and BV functions defined in the whole $X$, while the study and Sobolev and BV spaces in domains, and/or with respect to non Gaussian measures, is largely to be developed.

As in finite dimension, Sobolev and BV functions are tools for the study of different problems, in particular of PDEs with infinitely many variables, arising in mathematical physics in the modeling of systems with an infinite number of degrees of freedom, and of stochastic PDEs through Kolmogorov equations.

In this talk I will describe some of the main features and open problems concerning such function spaces.

Masayuki Hayashi: We prove global existence of solutions to the generalized derivative nonlinear Schrödinger equation in $H^2(\mathbb{T})$. This answers the open problem posed by Ambrose and Simpson (2015). The key in the proof is to extract the terms that cause problems in energy estimates and construct the modified energy so as to cancel the bad terms out by cleverly using integration by parts and the equation.

This talk is based on a joint work with T. Ozawa and N. Visciglia.

Giorgio Saracco: Bijections between isoperimetric sets, prescribed curvature sets, and p-Cheeger sets

Given a planar, open set $\mathrm{\Omega }$, satisfying some weak regularity assumptions, we will consider the following three different problems among subsets of $\mathrm{\Omega }$: the isoperimetric problem for volume $V$; the prescribed curvature problem for curvature $\kappa$; the $p$-Cheeger problem.

We shall see that there exist bijections $\mathfrak{K}$ and $\mathfrak{V}$ between these problems in the following sense: a set is isoperimetric of volume $V$ if and only if it attains the minimum of the prescribed curvature functional for curvature $\mathfrak{K}(V)$; analogously a set is $p$-Cheeger if and only if it is isoperimetric of volume $\mathfrak{V}(p)$.

As a byproduct we infer some convexity properties on the isoperimetric profile, and some fine regularity properties on the contact surface of minimizers.

Based on joint works with Caroccia, Leonardi, Neumayer, and Pratelli.