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.
