06 Jul 2020

• Title: Monotonicity formula and its applications to the unique continuation property for elliptic problems

• Abstract: I will first introduce the monotonicity formula in the cases of harmonic functions and very general perturbed elliptic problems by introducing the so-called Almgren’s frequency function and deriving the Pohozaev identity. Then I will show some of its applications, as the unique continuation property for second order elliptic equations. Finally I will present an approximation argument which I developed in order to prove the unique continuation property when the domain is highly non-smooth due to the presence of a crack.

29 Jun 2020

• Title: Introduction to calculus on metric measure spaces and the problem of the parallel transport

• Abstract: In this seminar, we give an overview of the functional analytic objects involved in the problem of defining the parallel transport on metric measure spaces satisfying Ricci lower bounds.In Riemannian geometry, given a smooth curve and a tangent vector at the initial point, one can construct a smooth vector field along the curve having zero covariant derivative along the curve and the prescribed initial condition.In the first part of the talk, we introduce the objects that play the role of the non smooth counterpart of curves and tangent vectors, namely test plans and vector fields (L^2 integrable with respect to the reference measure).In the second part, we discuss what are the non smooth counterpart of vector fields along a curve in this context, that are Sobolev vector fields along a test plan. We discuss two possible definitions of such objects, either in a distributional way or as the closure of  “test” vector fields along a test plan in the Sobolev norm.

22 Jun 2020

• Title: Some classification results for translating solitons of the mean curvature flow

• Abstract: Mean curvature flow is arguably the most natural evolution equation for hypersurfaces. It is a parabolic equation and shares several nice properties with the heat equation. However, it is not linear and its solutions typically develop singularities in finite time. Solitons, special solutions that evolve self-similarly in time, are particularly important because of their role in the singularity analysis and because of their relationship with the theory of minimal hypersurfaces.I will discuss some recent result and open problems, focusing in particular on traslating solitons.

25 May 2020

• Title: Supremal variational problems and absolute minimizers: some properties

• Abstract: The aim of this seminar is to introduce a special class of variational problems: the one of the so called supremal variational problems. I will start presenting the most famous supremal functional, the one associated to the infinity-Laplacian equation. I will define the class of the absolute minimizers associated to that functional and discuss some results. I will finally give a generalization of those results for a more general class of supremal functionals.

18 May 2020

• Title: Variational models for line defects in materials

• Abstract: The purpose of the seminar is to derive a 3d variational model for line defects in materials, known as dislocations, starting from a geometrically nonlinear elastic energy with quadratic growth. Precisely we obtain, through a Γ-convergence result, that the energy stored by a distribution of dislocations in a crystal is the contribution of a volume term representing the elastic energy and a line tension term representing the plastic energy.

11 May 2020

• Title: Free boundary minimal surfaces in the unit ball

• Abstract: Minimal surfaces are surfaces that locally minimize the area. They are a central topic in Geometric Analysis and are studied in several different settings. In this talk, we will concentrate on surfaces in the three-dimensional Euclidean unit ball that have boundary constrained to move in the boundary of the ball (i.e., the unit sphere). We will explain what it means for these surfaces to locally minimize the area, namely to be a free boundary minimal surface, and we will review some recent results about the construction of these objects.

16 Dec 2019

• Title: An entropic story of optimal mass transportation

• Abstract: Adding an entropy term to the optimal transportation cost functional is a means of regularizing the standard Monge-Kantorovich problem. In this talk I will introduce both the standard and the modified formulations. I will also discuss the Gamma-convergence of the regularized cost functionals to the Monge-Kantorovich costs.

09 Dec 2019

• Title: Nonexistence of minimizers for a nonlocal functional under volume constraints

• Abstract: In this talk, we present the minimizing problem for a nonlocal functional, especially a nonlocal perimeter. A nonlocal perimeter is defined by the fractional Sobolev semi-norm of a characteristic function and it is closely related to the classical perimeter. This topic is now widely studied by many authors in the context of calculus of variations, especially on minimizing problems. In the first part of this talk, we briefly review some of the previous works and see what the nonlocal perimeter looks like. In the second part, let us present our recent results on the nonexistence of mininizers for the nonlocal functional containing not only a nonlocal perimeter but also a Riesz and a background potential under a volume constraint. We will observe that, if a set has the volume larger than some number, then it cannot be a minimizer of that functional.

02 Dec 2019

• Title: Direct methods in Shape Optimization and applications to the Robin-Laplacian eigenvalues.

• Abstract: In this seminar we present some useful tools of the variational methods used in shape optimization problems and the main frameworks where these ideas are applied. We focus above all on shape optimization of spectral functionals; in particular, we present some of our results of and some open problems concerning the Robin-Laplacian eigenvalues.

18 Nov 2019

• Title: The effects of perturbations on the minimizing movements scheme

• Abstract: The concept of minimizing movement has been introduced by De Giorgi to study gradient-flow type motions in very general settings. It consists in defining a time scale and a corresponding time-discrete motion by solving an iterative minimization scheme, then obtain a continuous energy-decreasing curve by refining the time scale. In recent years, minimizing movements have been applied to families of Gamma-converging energies.
  In the first part of the talk, we will give a general presentation of the method of minimizing movements along families of functionals, first recalling the classical definition and showing its connection with the notion of curve of maximal slope (according to the work of Ambrosio, Gigli and Savaré).
  In the second part, we will precise what we mean by a perturbation of such scheme and analyze its effects on the limit motions. Finally, we will apply the scheme to an anti-ferromagnetic energy on spin-systems showing that its limit motion can be seen as a particular geometric flow.