Analysis for non-local phase transitions close to the critical exponent s=1/2

We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in Hs close to the critical exponent s=12. This is done by computing a scaling factor λ(ε,s), continuous in both variables, such that (Formula presented.) Γ-converge, for any choice of sε→12 as ε→0, to the sharp-interface functional found by Alberti, Bouchitté and Seppecher in [1] with the scaling |logε|-1. Moreover, we prove that all the values s∈[12,1) are regular points for the functional Fεs in the sense of equivalence by Γ-convergence (see [5]), and that the Γ-limits as ε→0 are continuous with respect to s. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on [12,1).