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Carnegie Mellon Center for Nonlinear Analysis
Singular Perturbation Models in Phase Transitions for Second Order Materials


Milena Chermisi
New Jersey Institute of Technology
Department of Mathematical Sciences

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

$\displaystyle 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$ of the squared gradient is negative, one expects curvature instabilities of the membrane and, in turn, this 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

$\displaystyle u \mapsto
F_\varepsilon(u, \Omega) : = \int_\Omega \frac{1}{\vare...
...epsilon2 \vert\nabla u\vert^2
+\varepsilon4 \vert\nabla2 u\vert^2\right]\, dx.

Here, the asymptotic behavior of $ \{F_{\varepsilon}\}$ is studied using Γ-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained. Joint collaboration with joint work with G. Dal Maso (SISSA), I. Fonseca (CMU), and G. Leoni (CMU).