Asymptotic analysis of a second-order singular perturbation model for phase transitions

Abstract

We consider the problem of the asymptotic description of a family of energies introduced by Coleman and Mizel in the theory of nonlinear second-order materials depending on an extra parameter k. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k_0 such that, for k<k_0, these energies Gamma-converge to a sharp interface functional. Moreover, for a special choice of the potential term in the energy, we provide an upper bound on the values of k such that minimizers cannot develop oscillations on some fine scale, thus improving previous estimates by Mizel, Peletier and Troy.