A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies

We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order 2 s ∈ ( 0 , 1 ) 2s \in (0,1) and a first-order asymptotic development in the Γ \Gamma -sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order μ > 2 − 2 s \mu > 2-2s . In view of this range value for μ \mu , it would be interesting to develop a new asymptotic development for the Γ \Gamma -convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.