Non-local models in homogenization and phase transitions

In this thesis work, we collect some results about models for homogenization and phase transitions. These classical topics in the Calculus of Variations are investigated via Gamma-convergence in different non-local frameworks or with the use of higher-order singular perturbations. In the first chapter we address the multiscale analysis of periodically oscillating energies of fractional and convolution-type and of general non-local energies on perforated domains, with particular emphasis in determining regimes for which the non-local-to-local passage occurs before the homogenization process. In spite of the similarities, the study of the energies of fractional and convolution-type requires different methods and their asymptotic analysis displays distinct outcomes. The problem of perforated domains is studied at the critical exponent: this instance incorporates several technical difficulties due to scaling invariance and leads to the emergence of a convex strange term. The second chapter is devoted to the study of some diffuse-interface models. First, we investigate equi-coerciveness with respect to the Hausdorff convergence of the graphs in a phase-field model involving a fractional singular perturbation. Then, we turn our attention to the study of higher-order singular perturbations models for phase transitions. We obtain an approximation of the perimeter in the vein of Modica and Mortola replacing the rescaled Dirichlet energy with a singular perturbation of arbitrarily high integer order. We further extend the analysis by accounting for several anisotropic perturbations of intermediate order that contribute negatively to the energy.