Calculus of Variations and Geometric Measure Theory

S. Conti - I. Fonseca - G. Leoni

A $\Gamma$-convergence result for the two-gradient theory of phase transitions

created on 18 Jun 2001
modified on 10 Jul 2002


Published Paper

Inserted: 18 jun 2001
Last Updated: 10 jul 2002

Journal: Comm. Pure Appl. Math.
Volume: 55
Number: 7
Pages: 857-936
Year: 2002


The generalization to gradient vector fields of the classical double-well, singularly perturbed functionals, \[I_{\varepsilon}\left( u;\Omega\right) :=\int_{\Omega}W(\nabla u)/\varepsilon+\varepsilon
^{2}\,\,dx,\] 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 $I_{\varepsilon}$ $\Gamma$-converge to $I\left( u;\Omega\right) =K^{\ast}\mathcal{H}^{N-1}(S\left( \nabla u\right) \cap\Omega)$ if $u\in W^{1,1}(\Omega;R^{d})$, $\nabla u\in BV\left( \Omega;\left\{A,B\right\} \right) $, and to $+\infty$ otherwise, where $K^{\ast}$ is the (constant) interfacial energy per unit area.

Keywords: phase transitions, $\Gamma$-convergence, singular perturbations, double-well potential