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{\bf \large Singular Perturbation Models in Phase Transitions for Second Order
Materials}

M. Chermisi\\
Departamento de Matematica\\
Instituto Superior Tecnico\\
1049-001 Lisboa, Portugal\\
Milena.Chermisi@math.ist.utl.pt 


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G. Dal Maso\\
SISSA, Trieste, Italy\\
dalmaso@sissa.it

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I. Fonseca\\
Department of Mathematical Sciences\\
 Carnegie Mellon University\\
 Pittsburgh, PA, U.S.A.\\
fonseca@andrew.cmu.edu

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G. Leoni\\
Department of Mathematical Sciences\\
 Carnegie Mellon University\\
 Pittsburgh, PA, U.S.A.\\
giovanni@andrew.cmu.edu
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\noindent {\bf 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,
\[
u\mapsto\int_{\Omega}\bigg[W\left(  u\right)  -q\left\vert \nabla u\right\vert
^{2}+\left\vert \nabla^{2}u\right\vert ^{2}\bigg]\,dx.
\]
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.

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