Degenerate Perturbations of a Two-Phase Transition Model

We study the G-convergence as e approaches 0+ of the family of degenerate functionals Qe(u) = e Integral over W of <ADu, Du> dx + (1/e) Integral over W of W (u) dx, where A(x) is a symmetric, non negative n times n matrix on W (i.e. <A(x) ξ, ξ> ≥ 0 for all x in W and x in Rn) with regular entries and W: R to [0, +infinity) is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix A, we obtain a minimal interface criterion for the G-limit functional exploiting some tools of analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations Qe in the classical gradient theory of phase transitions.