Second Order Expansion for the Nonlocal Perimeter Functional

Abstract The seminal results of Bourgain et al. (Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001) Dávila (Calc Var Differ Equ 15(4):519–527, 2002) show that the classical perimeter can be approximated by a family nonlocal functionals. We consider corresponding second order expansion for functional. In special case, considered energies is also relevant variational model thin ferromagnetic films. derive $$\Gamma $$ ? –limit these functionals as $$\epsilon \rightarrow 0$$ ? ? 0 . existence minimizers with prescribed volume fraction. For small fraction, unique, up to translation, minimizer limit energy given ball. analysis based on systematic exploitation associated symmetrized autocorrelation function.