Nonlocal equations and applications

In this thesis we deal with a series of mathematical models and problems related to nonlocal partial differential equations. This thesis is split into four parts. In Part I, we treat the biological theory of the Lévy flight foraging hypothesis. In Part II, we introduce the concept of nonlocal Willmore energy of a surface and we discuss the existence of minimizers in the class of boundaries of convex sets for the critical and subcritical case. In Part III, we prove density estimates for minimizers of a nonlocal variationalfunctional characterized by a double-well potential with slow growth from its minima. In Part IV, we prove a geometrical algorithm to explicitly write the solution ofa Dirichlet problem for s-harmonic maps in balls.