• fractional Allen-Cahn equation
• fractional laplacian
• fractional perimeter
• fractional Sobolev spaces
• Gamma-convergence
• nonlocal operator
• perimeter
• Phase transition

We studied a nonlocal model for phase transition of Allen-Cahn type. At first, we used Direct Method to show the existence of minimizers of a nonlocal energy in the Fractional Sobolev Space. Since minimizers solve the Fractional Allen-Cahn equation, some regularity properties can be deduced. Then, we looked for the reverse implication: are distributional solutions of the Fractional Allen-Cahn equation local minimizers? Since the functional lacks of convexity, the answer is not trivial. In the spirit of the famous conjecture of De Giorgi for the classical Allen-Cahn equation, it seems natural to study the minimization properties of solutions that are monotonically increasing in one direction and with some limiting behaviour. This result is an important tool to construct a global minimizer, namely a profile that minimizes the energy with respect to any compactly supported variation. In view of the conjecture of De Giorgi, it is natural to build such a minimizer in the one-dimensional case and, then, solve the problem in any dimension. Regarding the one-dimensional global minimizer, we can also give precise estimates on the growth at infinity.

These results are the basics to study the Gamma-Convergence of a suitably scaled version of our energy. Qualitatively different phenomena appear, according to the parameters involved. For instance, in some cases the energy due to the interaction coming from far is not negligible and the Gamma-Limit is the nonlocal area functional; in other cases, the contribution of the energy coming from far vanishes and the Gamma-Limit is the classical perimeter. These results have been inspired by the classical theory of Gamma-Convergence for a local phase transition model, due to Modica-Mortola in 1970/80's.