Singular Perturbation Models in Phase Transitions for Second Order Materials

M. Chermisi

Department of Mathematical Sciences

Carnegie Mellon University Pittsburgh, PA, U.S.A. chermisi@andrew.cmu.edu

G. Dal Maso SISSA, Trieste, Ital 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,

u ↦ ∫_{Ω} [W (u) − q |∇u|^{2} + |∇^{2}u|^{2} ] 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 ↦ F_{ε}(u, Ω) := ∫_{Ω} [\frac{1}{ε}W (u) − qε|∇u|^{2} + ε^{3}|∇^{2}u|^{2}] dx.

Here, the asymptotic behavior of {F_{ε}} is studied using Γ-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.