Singular perturbation models in phase transitions for second order materials

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, \[ u\mapsto\int_{\Omega}\bigg[W\left( u\right) -q\left\vert \nabla u\right\vert ^{2}+\left\vert \nabla^{2}u\right\vert ^{2}\bigg]\,dx. \] 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 \[ u\mapsto F_{\varepsilon}(u,\Omega):=\int_{\Omega}\,\left[ \frac {1}{\varepsilon}W(u)-q\varepsilon|\nabla u|^{2}+\varepsilon^{3}|\nabla ^{2}u|^{2}\right] \,\,dx. \] 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.