Degenerate Perturbations of a Two-Phase Transition Model

We study the

-convergence as

\varepsilon \to 0^+ ε → 0 +

of the family of degenerate functionals

Q_\varepsilon(u) = \varepsilon \int_\Omega \langle ADu, Du\rangle \, dx + \frac{1}{\varepsilon} \int_\Omega W(u) \, dx Q ε ( u ) = ε ∫ Ω ⟨ A D u , D u ⟩   d x + 1 ε ∫ Ω W ( u )   d x

, where A(x) is a symmetric, non negative

n\times n n × n

matrix on

\Omega Ω

(i.e.

\langle A(x)\xi, \xi\rangle \ge 0 ⟨ A ( x ) ξ , ξ ⟩ ≥ 0

for all

x \in \Omega x ∈ Ω

and

\xi \in \mathbb{R}^n ξ ∈ R n

) with regular entries and

W : \mathbb{R} \to [0, +\infty) W : R → [ 0 , + ∞ )

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 Γ

-limit functional exploiting some tools of analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations

Q_\varepsilon Q ε

in the classical gradient theory of phase transitions.