Coupled singular perturbations for phase transitions

The $\[$\varGamma(L^{1}(\varOmega;\mathbb{R}^{d}))$$ -limit of the sequence $\[J_{\varepsilon}(u):=\frac{1}{\varepsilon}E_{\varepsilon}(u),\]$ where E_ε is the family of anisotropic singular perturbations E_ε(u):=∫_Ωf(x,u(x),ε∇u(x)) dx of a non-convex functional of vector-valued functions E(u):=∫_Ωf(x,u(x),∇u(x)) dx is obtained where f is a non-negative energy density satisfying f(x,u,0)=0 if and only if u∈{a,b}.