Nonlocal equations and applications

This thesis is split into four parts, all related to nonlocal PDEs. In Part I we study the Lévy flight foraging hypothesis. We assume that the forager diffuses according to a fractional heat equation and we define some efficiency functionals to test which distribution optimizes the predator's hunt. In Part II we define the nonlocal Willmore energy, and we study the existence of minimizers for convex sets. In Part III, we prove density estimates for a nonlocal variational problem with degenerate double-well potential. In Part IV, we prove a geometrical algorithm to obtain explicit solutions to the fractional Dirichlet problem.