Some new applications of Γ-convergence to free discontinuity problems, phase transitions, and nonlocal models

In this thesis, we study variational models arising in three different settings: free discontinuity problems, higher-order models for phase transitions, and nonlocal variational problems. First, we study free discontinuity functionals whose surface terms are of the type commonly used in the variational theory of cohesive fracture mechanics. Since these surface terms are, generally, linear near the origin and bounded at infinity, the natural setting of the corresponding energies is that of spaces of generalised functions of bounded variation and generalised functions of bounded deformation. We investigate the fine properties of these spaces and, in particular, contribute to the general picture of generalised functions of bounded deformation by introducing a matrix-valued measure generalising the distributional symmetric gradient. We also address homogenisation problems for free discontinuity functionals. In the full gradient setting, we consider functionals with linear growth in the bulk part, a Cantor contribution, and cohesive surface terms, and prove deterministic and stochastic homogenisation results. In the framework of linearised elasticity, we study instead functionals defined on functions of bounded deformation whose bulk and surface densities have linear growth, and obtain similar deterministic and stochastic homogenisation results. Finally, we study the relaxation of energies defined on structured deformations, where the macroscopic deformation is coupled with a matrix-valued field accounting for the cumulative effect of submacroscopic slips, separations, and other disarrangements. We extend the classical integral representation theory to this cohesive setting, obtaining an explicit representation for the relaxed energies. The second part of the thesis concerns higher-order singular perturbation problems for phase transitions. We consider Modica-Mortola type energies, in which the usual first-order gradient term is replaced by a higher-order perturbation of arbitrarily prescribed order. We show that sequences with equibounded energy converge to sharp interfaces, and that the Γ-limit is given by a perimeter functional. We then study related anisotropic models involving general tensor norms and derivatives of intermediate orders, whose coefficients may be negative. For this broader class of energies, we establish equi-coercivity and Γ-convergence to an anisotropic perimeter functional, with an interfacial density determined by the orientation of the limit interface. The third and final part concerns the passage from nonlocal to local models driven by concentration phenomena. We first consider fractional quadratic energies with oscillating coefficients, and hence involving two parameters, one governing the concentration induced by the fractional kernel, while the other determining the frequency of the oscillations. We study the Γ-limit of these energies as the two parameters vary simultaneously and analyse how the interaction between these two scales influences the resulting local energy. We then turn to convolution-type functionals depending on finite differences. In this setting, the concentration of the interaction kernels produces free discontinuity energies with bulk and surface terms, whose densities we identify.