A general class of phase transition models with weighted interface energy

Abstract

We study a family of singular perturbation problems of the kind

where $u$ represents a fluid density and the non-negative energy density $f$ vanishes only for $u=\alpha$ or $u=\beta$. The novelty of the model is the additional variable $\rho ⩾0$ 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 $\varepsilon \to 0$ and find for each $f$ a transition energy (well defined when $u\in \mathrm{BV}(\Omega ;\{\alpha ,\beta \})$ 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.