Preprint

Inserted: 18 jul 2026
Last Updated: 18 jul 2026

Year: 2026

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.