*Published Paper*

**Inserted:** 20 dec 2002

**Last Updated:** 29 nov 2003

**Journal:** J. Convex Anal.

**Volume:** 10

**Number:** 1

**Pages:** 1-34

**Year:** 2003

**Abstract:**

We study the $\Gamma-$convergence as $\e\to0^+$ of the family of degenerate functionals \[ Q_\e(u)=\e\int_\O\langle ADu,Du\rangle\, dx +\frac{1}{\e}\int_\O W(u)\,dx \] where $A(x)$ is a symmetric, \it non negative \rm $n\times n$ matrix on $\O$ (i.e. $\langle A(x)\xi,\xi\rangle\geq0$ for all $x\in\O$ and $\xi\in\Rn$) with regular entries and $W:\R\to[0,+\infty)$ 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 $\Ga-$limit functional exploiting some tools of Analysis in Carnot-Carathéodory spaces. We extend some previous results obtained for the non degenerate perturbations $Q_\e$ in the classical gradient theory of phase transiti

**Keywords:**
phase transitions, $\Gamma$-convergence, Carnot-Carathéodory spaces