Preprint

Inserted: 18 jul 2026
Last Updated: 4 nov 2026

Year: 2026

Abstract:

We study functionals \[ \begin{aligned} F_{\varepsilon}(u,\rho) := {}& \frac{1}{\varepsilon} \int_{\Omega}W(u) \, dx + \frac{1}{\lvert\ln\varepsilon\rvert} \int_{\Omega} \int_{\Omega} \frac{(u(y) - u(x))^{2}}{\lvert y - x\rvert^{{N+1}}} \, dy \,dx \cr &+ \frac{1}{\lvert\ln\varepsilon\rvert} \int_{\Omega} \left\lvert \int_{\Omega} \frac{(u(y) - u(x))^{2}}{\lvert y - x\rvert^{{N+1}}} \, dy - \rho(x) \right\rvert dx \end{aligned} \] 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.