Gamma-convergence of nonlocal perimeter functionals

Given Omega subset of R-n open, connected and with Lipschitz boundary, and s is an element of (0, 1), we consider the functional J(s)(E, Omega) = integral(E boolean AND Omega)integral(Ec boolean AND Omega) dxdy/vertical bar x - y vertical bar(n+s) + integral(E boolean AND Omega)integral(Ec boolean AND Omega c) dxdy/vertical bar x - y vertical bar(n+s) + integral(E boolean AND Omega c)integral(Ec boolean AND Omega) dxdy/vertical bar x - y vertical bar(n+s), where E subset of R-n is an arbitrary measurable set. We prove that the functionals (1 - s)J(s)(., Omega) are equi- coercive in L-loc(1)(Omega) as s up arrow 1 and that Gamma - lim(s up arrow 1)(1 - s)J(s) (E, Omega) = omega Pn-1(E, Omega), for every E subset of R-n measurable, where P(E, Omega) denotes the perimeter of E in Omega in the sense of De Giorgi. We also prove that as s up arrow 1 limit points of local minimizers of (1 - s)J(s)(., Omega) are local minimizers of P(., Omega).