Daniele Barbera (UNIPI) 

L^p-L^q Maximal Regularity estimate for the Beris-Edward model in the half-plane

In this seminar we will consider the Beris-Edward model for the study of nematic liquid crystals in the half-plane R^N_+ with N\ge 2. Using the L^p-L^q Maximal Regularity Theory, we will prove the existence and uniqueness of a solution for the resolvent system and the corresponding existence and uniqueness of the solution for the linear evolution problem for p,q\in(1,\infty). As an application, we will see the existence of a local solution for the general model for small initial data for p>2 and q>N.

Camillo Brena (SNS)

Introduzione agli spazi RCD

Lo scopo di questo seminario è di introdurre brevemente gli spazi RCD. Inizierò con un esempio che motiva l’interesse verso l’analisi in ambiente non-liscio per poi definire gli spazi RCD ed esporne le proprietà più rilevanti

Michele Caselli (SNS)

Non-local approximation of minimal surfaces and existence of infinitely many minimal surfaces in 3-manifolds

In this talk, I will describe a possible path towards a proof of Yau's conjecture (on the existence of infinitely many minimal surfaces in every Riemannian 3-manifold) based on nonlocal approximation of minimal surfaces. On one hand, in a work in preparation with J. Serra and E. F. Simon, we show that the analog of Yau's conjecture for nonlocal minimal surfaces holds. This is because Morse theory for nonlocal minimal surfaces seems to be flawless as finite-dimensional Morse theory, at least from the compactness perspective. On the other hand, in a recent work by H. Chang, S. Dipierro, J. Serra, and E. Valdinoci the authors prove that one can successfully pass nonlocal minimal surfaces to the limit to obtain classical minimal surfaces, keeping strong curvature and separation estimates. These two works seem to point to the fact that one could recover Yau's conjecture from its nonlocal analog.

Lorenzo Ferreri (SNS)

Asymptotic properties of an optimal principal eigenvalue with dirichlet boundary conditions

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth
bounded domain Ω ⊂ R^N , where the bang-bang weight equals a positive constant m on
the favourable region E ⊂ Ω and a negative constant −m on Ω \ E.
The corresponding positive principal eigenvalue provides a threshold to detect persis-
tence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP
equation in population dynamics. In particular, we study the minimization of such eigen-
value with respect to the shape and position of E in Ω.
We provide a sharp asymptotic expansion of the optimal principal positive eigenvalue
and a sharp spherical asymmetry estimate of the optimal favourable regions in the singularly
perturbed regime in which the volume of E vanishes. In particular, we deduce that, up to
subsequences, the optimal regions E concentrate at a point maximizing the distance from ∂Ω.
This is a joint work with Gianmaria Verzini

Mattia Freguglia (SNS)

Alcuni problemi riguardanti l’approssimazione diffusa del funzionale elastica

In questo seminario parlerò di una congettura proposta da Bellettini e Paolini come variante di una congettura precedente di De Giorgi sulla possibilità di approssimare, nel senso della Gamma-convergenza, in modo diffuso la somma del funzionale Perimetro e del funzionale di Willmore su insiemi con bordo di classe C^2. Questo problema è stato risolto da Röger e Schätzle in dimensione n=2,3 e indipendentemente da Nagase e Tonegawa in dimensione n=2, ma rimane aperto in dimensione più alta. Dopo aver introdotto il problema mi concentrerò sul caso n=2 e discuterò le difficoltà che emergono provando a estendere il risultato a insiemi meno regolari. Alla fine proporrò un possibile candidato a essere il Gamma-limite dei funzionali diffusi su tutto L^1 spiegando quali sarebbero gli step mancanti per ottenere una dimostrazione. 

Nicola Picenni (SNS)

How to recognize constant functions through nonlocal functionals

We describe a problem proposed by H. Brezis concerning the characterization of constant functions through double integrals that involve their difference quotients. In particular, we focus on the problem of finding necessary and sufficient conditions under which such characterization holds true.
We present and motivate a natural necessary condition proposed by R. Ignat and we show with a counterexample that it is not sufficient.
On the other hand, we describe some additional conditions under which the answer to the question turns out to be postive.
Finally, we discuss many intermediate cases that remain open, and how they motivate further questions in measure theory.
Some parts of the talk are based on a joint work with M. Gobbino.