Variational problems with linear growth arise naturally in the Calcu- lus of Variations from the study of singular perturbation problems associated to a large number of physical and mathematical applications. These problems must be posed over the class of functions of bounded variation, and their analysis is signifi- cantly more involved than that which is… Read More »
We study a model for transition layers, called Nel walls, in thin films of ferromagnetic materials. The magnetisation is represented by a map from a line to the unit circle in this model, and there is an energy functional consisting of an Allen-Cahn type term and a nonlocal term penalising a fractional Sobolev norm of… Read More »
In 1939, Frenkel and Kontorova proposed a model for the motion of a dislocation (an imperfection in a crystal). The model is simple, a chain of atoms following Newton’s equation of motion. The atoms interact with their nearest neighbours via a harmonic spring and are exposed to a periodic (non-convex) on-site potential. Despite the simplicity, the model has… Read More »
Nonlinear diffusion ∂tρ = ∆(Φ(ρ)) is considered as the hydrodynamic limit of the zero- range process. It is shown that for suitable choices of Φ, a metric can be defined with respect to which the non-linear diffusion is the gradient flow of the thermodynamic entropy of the zero-range process. Hence we call this metric the… Read More »
In this talk we will present a methodology for the approximation of solutions to problems arising from calculus of variations in L-infity. We make use of L-p approximations and present a variety of theoretical and numerical results to this end.
We will discuss about a p-Laplacian problem involving a nonlinearity of critical growth. Although the problem is variational, standard variational techniques do not directly apply because of the possible lack of compactness of the minimising sequences, due to the combined effect of dilations and translations. I will present a recent result in collaboration with B.… Read More »
Large-interaction limits of certain systems of elliptic and parabolic PDE, such as, for instance, population systems with large competition, both provide a powerful mathematical tool that can be exploited to obtain information about systems that are otherwise difficult to analyse, and correspond to important biological and physical phenomena such as spatial segregation, phase separation, or… Read More »
Mean field games have been introduced by J.-M. Lasry and P.-L. Lions as an effective model for very many competing rational agents. They are a system of Hamilton-Jacobi equations and Kolmogorov-Fokker-Planck equations. One of the challenges is the fact that these two types of equations have different “natural” notions of generalized solutions. We investigate dynamical mean field… Read More »
We want to present some new results related to Multi-marginal Optimal Transport Theory for Coulomb and repulsive harmonic costs. In particular, we are inquiring about the existence of Monge-type solutions in the multi-marginal case. We intend to give a brief survey on the general theory and discuss some of its main issues. Most of the… Read More »
Calculus of Variations in $L^\infty$ has a long history, the scalar case of which was initiated by G.Aronsson in the 1960s and is under active research ever since. Aronsson’s motivation to study this problem was related to the optimisation of Lipschitz Extensions of functions. Mathematically, minimising the supremum is very challenging because the equations are… Read More »