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MiS Preprint
34/2001
A Gamma-convergence result for the two-gradient theory of phase transitions
Sergio Conti, Irene Fonseca and Giovanni Leoni
Abstract
The generalization to gradient vector fields of the classical double-well, singularly perturbed functionals, $$I_\epsilon (u;\Omega) := \int_\Omega \frac{1}{\epsilon} W (\nabla u) + \epsilon |\nabla^2 u|^2 dx $$ where $W(\xi) =0$ if and only if $\xi =A$ or $\xi =B$, and $A-B$ is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that $I_\epsilon \Gamma$ -converge to $$I(u;\Omega) = \{ \begin{array} c K \times H^{N-1} (S(\nabla u)\cap \Omega) & if u \in W^{1,1} (\Omega ; \mathbb(R)^d ),\nabla u \in BV (\Omega ; \{ A,B \}), \\ +\infty & otherwise, \end{array}$$ where $K^*$ is the (constant) interfacial energy per unit area.
