Analysis for non-local phase transitions close to the critical exponent $s=\frac12$

We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $H^s$ close to the critical exponent $s=\frac12$. This is done by computing a scaling factor $\lambda(\varepsilon,s)$, continuous in both variables, such that $$\mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{\lambda(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+\lambda(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2$$ $\Gamma$-converge, for any choice of $s_\varepsilon \to \frac12$ as $\varepsilon\to 0$, to the sharp-interface functional found by Alberti, Bouchitt\'e and Seppecher with the scaling ${|\log\varepsilon|^{-1}}$. Moreover, we prove that all the values $s\in [\frac12,1 )$ are regular points for the functional $\mathcal{F}^{s}_\varepsilon$ in the sense of equivalence by $\Gamma$-convergence introduced by Braides and Truskinovsky, and that the $\Gamma$-limits as $\varepsilon\to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac12,1)$.