Degenerate perturbations of a two-phase transition model

We study the Gamma-convergence as epsilon --> 0(+) of the family of degenerate functionals Q(epsilon)(u) = epsilon integral(Omega)<ADu, Du>dx + 1/epsilon integral(Omega)W(u)dx where A(x) is a symmetric, non negative n x n matrix on Omega (i.e. <A(x)xi, xi> greater than or equal to 0 for all x is an element of Omega and xi is an element of R-n) with regular entries and W : R --> [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 Gamma-limit functional exploiting some tools of Analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations Q(epsilon) in the classical gradient theory of phase transitions.