We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in \(H^s\) close to the critical exponent \(s=\frac{1}{2}\) . This is done by computing a scaling factor \(\lambda (\varepsilon ,s)\) , continuous in both variables, such that \(\begin{aligned} {\mathcal {F}}^{s_\varepsilon }_\varepsilon (u)=\frac{\lambda (\varepsilon ,s_\varepsilon )}{\varepsilon }\int W(u)\,dt+\lambda (\varepsilon ,s_\varepsilon )\varepsilon ^{(2s_\varepsilon -1)^+} {[}u ]_{{s_\varepsilon }}^2 \end{aligned}\) \(\Gamma \) -converge, for any choice of \(s_\varepsilon \rightarrow \frac{1}{2}\) as \(\varepsilon \rightarrow 0\) , to the sharp-interface functional found by Alberti, Bouchitté and Seppecher in [1] with the scaling \({|\log \varepsilon |^{-1}}\) . Moreover, we prove that all the values \(s\in [\frac{1}{2},1 )\) are regular points for the functional \({\mathcal {F}}^{s}_\varepsilon \) in the sense of equivalence by \(\Gamma \) -convergence (see [5]), and that the \(\Gamma \) -limits as \(\varepsilon \rightarrow 0\) are continuous with respect to s. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on \([\frac{1}{2},1)\) .
