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

Abstract

We study functionals \begin{equation*} F_\varepsilon (u,\rho) := \frac{1}{\varepsilon} \int_\Omega W(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_\Omega \int_\Omega \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx + \frac{1}{|\ln(\varepsilon)|} \int_\Omega \left| \int_\Omega \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy - \rho(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 $\Omega$ and the $\Gamma$-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.