Fractional minimal surfaces and Allen-Cahn equations

In recent years fractional operators have received considerable attention both in pure and applied mathematics. They appear in biological observations, finance, crystal dislocation, digital image reconstruction and minimal surfaces. In this thesis we study nonlocal minimal surfaces which are boundaries of sets minimizing certain integral norms and can be interpreted as a non-infinitesimal version of classical minimal surfaces. In particular, we consider critical points, with or withouth constraints, of suitable functionals, or approximations through diffuse models as the Allen-Cahn’s. In the first part of the thesis we prove an existence and multiplicity result for critical points of the fractional analogue of the Allen-Cahn equation in bounded domains. We bound the functional using a standard nonlocal tool: we split the domain in two regions and we analyze the three significative interactions. Then, the proof becomes an application of a classical Krasnoselskii’s genus result. Then, we consider a fractional mesoscopic model of phase transition i.e. the fractional Allen-Cahn equation with the addition of a mesoscopic term changing the ‘pure phases’ ±1 in periodic functions. We investigate geometric properties of the interface of the associated minimal solutions. Then we construct minimal interfaces lying to a strip of prescribed direction and universal width. We provide a geometric and variational technique adapted to deal with nonlocal interactions. In the last part of the thesis, we study functionals involving the fractional perimeter. In particular, first we study the localization of sets with constant nonlocal mean curvature and small prescribed volume in an open bounded domain, proving that these sets are ‘sufficiently close’ to critical points of a suitable potential. The proof is an application of the Lyupanov-Schmidt reduction to the fractional perimeter. Finally, we consider the fractional perimeter in a half-space. We prove the existence of a minimal set with fixed volume and some of its properties as intersection with the hyperplane {xN = 0}, symmetry, to be a graph in the xN-direction and smoothness.