In this work in collaboration with Vincent Millot and Rémy Rodiac, we address the question of the convergence of critical points of the Ambrosio-Tortorelli functional, in the sense of inner variations, to those of the Mumford-Shah ones. We extend earlier results by Francfort, Le and Serfaty in the 1-dimensional case to any arbitrary dimension upon the additional assumption of the convergence of the energies. As a byproduct, we also obtain the convergence of the second inner variation, which implies the convergence of stable critical points. The proof rests on elliptic PDE and geometric measure theoretic arguments. Thanks to elliptic regularity estimates, we derive the first inner variations of the Ambrosio-Tortorelli functional which have a varifold structure. Then, we characterize the limit varifold as the rectifiable varifold associated to the jump set.[-]

'Effective' properties of heterogeneous elastic materials are now well understood, following years of progress in homogenization and calculus of variations. For brittle materials, the situation is quite different. For instance, empirical and experimental evidences of toughening point to a coupling between elastic and fracture properties of heterogeneous materials, which contradicts some mathematical results. In this talk, I will present an attempt at formalizing a concept of effective toughness, to compute it, and to design materials with extreme fracture properties.[-]

Nonlocal interaction energies are continuum models for large systems of particles, where typically each particle interacts not only with its immediate neighbors, but also with particles that are far away. Examples of these energies arise in many different applications, such as biology (population dynamics), physics (Ginzburg-Landau vortices), and material science (dislocation theory). A fundamental question is understanding the optimal arrangement of particles at equilibrium, which are described, at least in average, by minimizers of the energy. In this talk I will focus on a class of nonlocal energies that are perturbations of the Coulomb energy and I will show how their minimizers can be explicitly characterized. This is based on joint works with J. Mateu, L. Rondi, L. Scardia, and J. Verdera.[-]

Spin models are lattice models that describe magnetic properties of materials. In this talk we will examine a 2-dimensional planar spin model (known as the J1-J2-J3 model) which exhibits frustration. Frustration is the phenomenon due to conflicting interatomic ferromagnetic/antiferromagnetic interactions that prevent the energy of every pair of interacting spins to be simultaneously minimized. The frustration mechanism leads to complex geometric patterns in the material. We study these complex geometric patterns by carrying out a discrete-to-continuum variational analysis as the lattice spacing tends to zero, finding the energetic regime for which many chiralphases can coexist. In particular, we will show that the surface tension between the chiral phases is captured by a continuum energy obtained by suitably selecting solutions to the eikonal equation. The results presented in the seminar are based on works in collaboration with M. Cicalese and M. Forster.[-]