Singular perturbation models in phase transitions for second-order materials M. ChermisiG. Dal MasoIrene FonsecaGiovanni Leoni 49J4526B3046E35singular perturbations$\Gamma$-convergencehigher order derivativesinterpolation inequalitiespattern formation 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, \[ u\mapsto\int_{\Omega}\Big[W(u)-q|\nabla u|^2+|\nabla^2u|^2\Big]\dx. \] 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 \[ u\mapsto F_{\varepsilon}(u,\Omega):=\int_{\Omega}\left[\frac{1}{\varepsilon}W(u)-q\varepsilon|\nabla u|^2+\varepsilon^3|\nabla^2u|^2\right]\dx. \] 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4346 10.1512/iumj.2011.60.4346 en Indiana Univ. Math. J. 60 (2011) 367 - 410 state-of-the-art mathematics http://iumj.org/access/