article

Title: Singular perturbation models in phase transitions for second-order materials

Authors: M. Chermisi, G. Dal Maso, I. Fonseca and G. Leoni

Issue: Volume 60 (2011), Issue 2, 367-410

Abstract:

$A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy, precisely, \[ u\mapsto\int_{\Omega}\Big[W(u)-q|\nabla u|^2+|\nabla^2u|^2\Big]\dx. \] When the stiffness coefficient $-q$ is negative, one expects curvature instabilities of the membrane and, in turn, these instabilities generate a pattern of domains that differ both in composition and in local curvature. Scaling arguments motivate the study of the family of singular perturbed energies \[ u\mapsto F_{\varepsilon}(u,\Omega):=\int_{\Omega}\left[\frac{1}{\varepsilon}W(u)-q\varepsilon|\nabla u|^2+\varepsilon^3|\nabla^2u|^2\right]\dx. \] Here, the asymptotic behavior of $\{F_{\varepsilon}\}$ is studied using $\Gamma$-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.$