Singular perturbation models in phase transitions for second-order materials

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 7→ Z Ω » W (u)− q |∇u| + ∇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 7→ Fe(u, Ω) := Z Ω » 1 e W (u)− qe|∇u| + e|∇u| – dx. Here, the asymptotic behavior of {Fe} is studied using Γ-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.