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Publication 10-CNA-01

Singular perturbation models in phase transitions for second order materials

M. Chermisi
Department of Mathematics
Instituto Superior Técnico
Lisbon, Portugal
chermisi@math.ist.utl.pt

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

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

G. Leoni
Department of Mathematical Sciences
Carnegie-Mellon University
Pittsburgh, PA, U.S.A.
giovanni@andrew.cmu.edu

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$ 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

$\displaystyle u\mapsto F_{\varepsilon}(u,\Omega):=\int_{\Omega}\,\left[ \frac {... ...vert\nabla u\vert^{2}+\varepsilon^{3}\vert\nabla ^{2}u\vert^{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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