Geometric patterns and microstructures in the study of material defects and composites

The main focus of this PhD thesis is the study of microstructures and geometric patterns in materials, in the framework of the Calculus of Variations. My PhD research, carried out in collaboration with my supervisor Mariapia Palombaro and Marcello Ponsiglione, led to the production of three papers [21, 22, 23]. Papers [21, 22] have already been published, while [23] is currently in preparation. This thesis is divided into two main parts. In the first part we present the results obtained in [22, 23]. In these two works geometric patterns have to be understood as patterns of dislocations in crystals. The second part is devoted to [21], where suitable microgeometries are needed as a mean to produce gradients that display critical integrability properties.