On local and global minimizers of some non-convex variational problems

Many physical systems are modeled mathematically as variational problems, where the observed configurations are expected to be local or global minimizers of a suitable energy. Such an energy can be very complicated, as well as the physical phenomenon under investigation. Thus, as a starting point, it is useful to focus on some simple models, which however capture the main features. In this thesis we concentrate on two kinds of energies, that can be both viewed as nonlocal variants of the perimeter functional. The nonlocality consists in a bulk term, that in one case is given by an elastic energy, while in the other by a long-range interaction of Coulumbic type. The physical systems modeled by these energies displays a rich variety of observable patterns, as well as the formation of morphological instabilities of interfaces between different phases. These phenomena can be mathematically understood as the competition between the local geometric part of the energy, i.e., the perimeter, and the nonlocal one. Indeed, while the first one prefers configurations in which the interfaces are regular and as small as possible, the latter, instead, favors more irregular and oscillating patterns. Thus, finding global or local minima of these energies is a highly nontrivial task, and indeed many big issues about them are still open. The aim of this thesis is to give a contribution to the investigation of such issues.