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During this year the seminars were organized by Daniele Barbera, Konstantinos Bessas, Luca Briani, and Luciano Sciaraffia.
From PDE's to Geometry: a new path towards the Minkowski Inequality
From PDE's to Geometry: a new path towards the Minkowski Inequality
Luca Benatti (Università di Trento)
Luca Benatti (Università di Trento)
14 October 2021
Abstract
Abstract
The classical Minkowski inequality in Euclidean Spaces establishes a sharp lower bound for the total mean curvature of the boundary in terms of its area. This talk aims to present the extension of the problem to a relevant class of Riemannian manifolds and how an approach through PDE can solve it. We also explore its connection with other related geometric inequalities, such as the Isoperimetric Inequality or the Willmore Inequality.
The Beris-Edward model for liquid crystals with absence of flow
The Beris-Edward model for liquid crystals with absence of flow
Daniele Barbera (Università di Pisa)
Daniele Barbera (Università di Pisa)
orario - luogo
Abstract
Abstract
In this talk I’ll present my master’s thesis project, focused on the Beris-Edward model. This system studies the behaviour of a molecule in a flux of nematic liquid crystals, a state of matter intermediate between the solid state and the liquid one. We consider the case with absence of flow and with energy functional unbounded from below. At the beginning, we study the stationary problem. We prove the existence of a least energy solution (when the domain is all the space) and of ground states with several open sets and boundary conditions. The last part is devoted to the evolution problem in all the space, where we establish local and global existence with small initial data. The results come from several applications of Strichartz-type estimates and by contraction arguments.
On the convex components of a set
On the convex components of a set
Giorgio Stefani (University of Basel)
Giorgio Stefani (University of Basel)
18 November 2021
Abstract
Abstract
The convex components of a set (with non-empty interior) are not uniquely determined in general. In this talk, after having recalled some elementary (quantitative) monotonicity properties of the Euclidean perimeter, we study some lower bounds on the minimal number of convex components of an arbitrary set, showing the sharpness of our results with some explicit examples. This is a joint work in collaboration with F. Giannetti.
On modified quasilinear Schrödinger equations: a variational approach
On modified quasilinear Schrödinger equations: a variational approach
Caterina Sportelli (Università di Bari)
Caterina Sportelli (Università di Bari)
25 November 2021
Abstract
Abstract
"In the last years, quasilinear Schrödinger-type equations became a topic of wide interest and they have been derived as a model of several physical phenomena. In this talk, after having recall some classical variational tools, we study the existence of solutions of a generalized version of the quasilinear modified Schrödinger equation in which the coefficient of the principal part depends also on the solution itself.
In particular, the analysis of the interaction of two different norms in a suitable Banach space and a modified version of the Mountain Pass Theorem, allow us to prove the existence of a weak bounded solution.
This is a joint work with Anna Maria Candela and Addolorata Salvatore."
Curvature varifolds and biological membranes
Curvature varifolds and biological membranes
Katharina Brazda (University of Vienna)
Katharina Brazda (University of Vienna)
2 December 2021
Abstract
Abstract
Varifolds are measures that describe generalized surfaces in geometric measure theory. Due to their compactness properties, varifolds provide a favorable framework to study geometric variational problems with the Direct Method. An example of a geometric energy functional that also involves curvature is the Canham-Helfrich energy. Its minimizers model the equilibrium configurations of biological membranes, like the famous biconcave shape of human red blood cells. After a gentle introduction to the theory of curvature varifolds, I present an existence result for multiphase membranes, that I obtained in collaboration with Luca Lussardi (Torino) and Ulisse Stefanelli (Vienna).
The first eigenvalue of the (fractional) Dirichlet-Laplacian vs. the inradius.
The first eigenvalue of the (fractional) Dirichlet-Laplacian vs. the inradius.
Francesca Bianchi (Università di Ferrara)
Francesca Bianchi (Università di Ferrara)
orario - luogo
Abstract
Abstract
We discuss for geometrical estimates for the first eigenvalue of the Dirichlet-Laplacian of planar sets. In particular, we focus on a lower bound in terms of the inradius, for convex sets and simply connected ones. We also discuss the existence of fractional counterparts for such an estimate.
A comparison principle for the Lane-Emden equation and applications to geometric estimates
A comparison principle for the Lane-Emden equation and applications to geometric estimates
Anna Chiara Zagati (Università di Ferrara)
Anna Chiara Zagati (Università di Ferrara)
orario - luogo
Abstract
Abstract
We introduce a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the p−Laplacian. Then we apply such a comparison principle to obtain a variety of results, as “hierarchy” of solutions, sharp pointwise double-side estimates for positive solutions in convex sets, and a sharp geometric estimate on the generalized principal frequencies of convex sets, which provides a generalization of the so-called Hersch-Protter inequality.
Nonlinear Schrödinger ground states on metric graphs
Nonlinear Schrödinger ground states on metric graphs
Filippo Boni (Politecnico di Torino & Università di Torino)
Filippo Boni (Politecnico di Torino & Università di Torino)
orario - luogo
Abstract
Abstract
Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose–Einstein condensation. In the first part of the talk, we introduce the problem of the existence of ground states at fixed mass for a focusing nonlinear Schrödinger equation with a standard nonlinearity of power type, highlighting how the topology of the graph can affect the existence results. In the second part, we study the same minimization problem for a nonlinear Schrödinger equation with two focusing power–type nonlinear terms: a pointwise one, located at the vertices of the graph, and a standard one. We show that existence and non–existence results strongly depend on the interplay between the two nonlinearities and on the topological and metric properties of the graph.
Distributional fractional Sobolev and BV spaces
Distributional fractional Sobolev and BV spaces
Giovanni Comi (Università di Pisa)
Giovanni Comi (Università di Pisa)
orario - luogo
Abstract
Abstract
Differently from their integer versions, the fractional Sobolev spaces $W^{\alpha,p}(R^n)$ do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and divergence already existing in the literature, we introduce via a distributional approach the new spaces $BV^{\alpha,p}(R^n)$ of $L^p$ functions with bounded $\alpha$-fractional variation in $R^n$, for $\alpha \in (0,1)$ and $p \in [1, \infty]$. In addition, we define in a similar way the distributional fractional Sobolev spaces $S^{\alpha,p}(R^n)$, which extend naturally $W^{\alpha,p}(R^n)$. The properties of these spaces have been analyzed in a series of papers in collaboration with Giorgio Stefani, Elia Bruè, Mattia Calzi and Daniel Spector. In this talk, we shall focus ourselves on the absolute continuity property of the fractional variation with respect to a suitable Hausdorff measure, and subsequently on the fractional Leibniz rules involving $BV^{\alpha,p}$ and $S^{\alpha,p}$ functions. As an application of these results we will show the well-posedness of the boundary-value problem for a general $2\alpha$-order fractional elliptic operator in divergence form.
Free boundary regularity in the one-phase Stefan problem: a recent perturbative approach
Free boundary regularity in the one-phase Stefan problem: a recent perturbative approach
Nicolò Forcillo (Università di Bologna)
Nicolò Forcillo (Università di Bologna)
orario - luogo
Abstract
Abstract
In this talk, we focus on the free boundary regularity in the one-phase Stefan problem. Specifically, we describe a recent perturbative approach to the study of this. In the first part, we introduce some general notions and results connected with free boundary problems. In the second part, we deal with the specific topic of the talk, providing the key steps of the method. This talk is based on a joint work with D. De Silva and O. Savin.
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