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

We study functionals \begin{equation*} F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) , dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} , dy ,dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} , dy - ρ(x) \right| ,dx \end{equation*} for a double-well potential $W$ and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of $BV$ functions on $\mathrm{\Omega }$ and the $\mathrm{\Gamma }$-convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface.