A model for phase transitions with competing terms

Abstract

In this paper, we study, via Γ-convergence techniques, the asymptotic behaviour of a family of coupled singular perturbations of a non-convex functional of the type  

$${\int }_{\mathrm{\Omega }}f\left(u\right(x),\mathrm{\nabla }u(x),\rho (x\left)\right)\mathrm{d}x$$

as a variational model to address two-phase transition problems under the volume constraints ${\int }_{\mathrm{\Omega }}u\left(x\right)\mathrm{d}x=V_f$⁠, ${\int }_{\mathrm{\Omega }}\rho \left(x\right)\mathrm{d}x=V_s$⁠, and where the additional unknown ρ interplays with $\mathrm{\nabla }u$ in the formation of interfaces.