A general class of phase transition models with weighted interface energy

We study a family of singular perturbation problems of the kindinf{1ε∫Ωf(u,ε∇u,ερ)dx:∫Ωu=m0,∫Ωρ=m1}, where u represents a fluid density and the non-negative energy density f   vanishes only for u=αu=α or u=βu=β. The novelty of the model is the additional variable ρ⩾0ρ⩾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 ε→0ε→0 and find for each f   a transition energy (well defined when u∈BV(Ω;{α,β})u∈BV(Ω;{α,β}) and ρ   is a measure) which depends on the n−1n−1 dimensional density of the measure ρ on the jump set of u. An explicit formula is also given.