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 ρ is a measure) which depends on the $n-1$ dimensional density of the measure ρ on the jump set of u. An explicit formula is also given.