$\mathrm{\Gamma }$-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations

We present several asymptotic results concerning the non-local Massari Problem for sets with prescribed mean curvature. In particular, we show that the fractional Massari functional $\mathrm{\Gamma }$-converges to the classical one, and this convergence preserves minimizers in the $L_{loc}^1$-topology. This returns useful information about the asymptotic behavior of the solutions of the inhomogeneous Allen-Cahn equation in the forced and the mass-prescribed settings. In this context, a new geometric object, which we refer to as “non-local hybrid mean curvature”, naturally appears.