A general class of phase transition models with weighted interface energy

We study a family of singular perturbation problems of the kind $$ \inf \left\{\frac{1}{\varepsilon}\int_\Omega f(u , \varepsilon \nabla u, \varepsilon \rho) \, dx \ :\ \int_\Omega u = m_0 \ ,\ \int_\Omega \rho = m_1\right\}\ ,$$ where $u$ represents a fluid density and the nonnegative energy density $f$ vanishes only for $u=\alpha$ or $u=\beta$. The novelty of the model is the additional variable $\rho\ge 0$ which is also unknown and interplays with the gradient of $u$ in the formation of interfaces. Under mild assumptions on $f$, we characterize the limit energy as $\e\to 0$ and find for each $f$ a transition energy (well defined when $u\in BV(\Omega;\{\alpha,\beta\})$ and $\rho$ is a measure) which depends on the $n-1$ dimensional density of the measure $\rho$ on the jump set of $u$. An explicit formula is also given.