Analysis for non-local phase transitions close to the critical exponent $s=\frac{1}{2}$

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 $\lambda (\epsilon ,s)$, continuous in both variables, such that $${\mathcal{F}}_{\epsilon }^{s_{\epsilon }}(u)=\frac{\lambda (\epsilon ,s_{\epsilon })}{\epsilon }\int W(u)dt+\lambda (\epsilon ,s_{\epsilon }){\epsilon }^{(2s_{\epsilon }-1{)}^{+}}[u{]}_{H^{s_{\epsilon }}}^2$$ $\mathrm{\Gamma }$-converge, for any choice of $s_{\epsilon }\to \frac{1}{2}$ as $\epsilon \to 0$, to the sharp-interface functional found by Alberti, Bouchitté and Seppecher with the scaling $\mathrm{\mid }\log \epsilon {\mathrm{\mid }}^{-1}$. Moreover, we prove that all the values $s\in [\frac{1}{2},1)$ are regular points for the functional ${\mathcal{F}}_{\epsilon }^s$ in the sense of equivalence by $\mathrm{\Gamma }$-convergence introduced by Braides and Truskinovsky, and that the $\mathrm{\Gamma }$-limits as $\epsilon \to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac{1}{2},1)$.