Data: 05/06/2019

Ora: 14:30:00

Descrizione:

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $\BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $\BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini (SISSA, Trieste).

]]>Data: 22/05/2019

Ora: 14:30:00

Descrizione:

In this talk, we survey some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation

]]>$$

{\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},

$$

in a bounded Lipschitz domain $\Omega \subset \RR^N$, with $a,b>0$ parameters.

This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids.

In this talk, we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.

This seminar is supported by the INdAM-GNAMPA Project 2019 “Il modello di Born-Infeld per l’elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni”.

Data: 20/05/2019

Ora: 14:30:00

Descrizione:

In this talk I will introduce the concept of Multi-Marginal Optimal Mass transportation (MOT) with the emphasis on repulsive cost functions. Then I will outline the Monge problem, discuss it's difficulty in the MOT setting, and present some nonexistence results that are joint work with Augusto Gerolin and Tapio Rajala.

]]>Data: 30/04/2019

Ora: 14:30:00

Descrizione:

Solutions of nonlinear nonlocal Fokker–Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for L^2 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.

]]>Data: 08/04/2019

Ora: 14:30:00

Descrizione:

I will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Operator, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore, I will discuss necessary and sufficient conditions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits, I will also discuss regularity estimates for solutions to corresponding Poisson problems.

]]>