$\mathrm{\Gamma }$-convergence for nonlocal phase transitions involving the $H^{1\mathrm{/}2}$ norm

We study functionals \begin{equation*} F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) , dx + \varepsilon |u|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$ when $\epsilon \ln ({\lambda }_{\epsilon })\to k$ as $\epsilon \to 0^{+}$ and show compactness in the space of $BV$ functions on $\mathrm{\Omega }$ and the $\mathrm{\Gamma }$-convergence to the classical surface tension functional.