Higher-Order Singular Perturbation Models for Phase Transitions

Variational models of phase transitions take into account double-well energies singularly perturbed by gradient terms, such as the Cahn--Hilliard free energy. The derivation by \Gamma-convergence of a sharp-interface limit for such energies is a classical result by Modica and Mortola. We consider a singular perturbation of a double-well energy by derivatives of order k and show that we still can describe the limit as in the case k = 1 with a suitable interfacial energy density, in accordance with the case k = 1 and with the case k = 2 previously analyzed by Fonseca and Mantegazza. The main issue is the derivation of an optimal-profile problem on the real line describing the interfacial energy density, which must be conveniently approximated by minimum problems on finite intervals with homogeneous condition on the derivatives at the endpoints up to order k-1. To that end, a careful study of sets where sequences of functions with equibounded energy are ``close to the wells"" and have ``small derivatives"" in terms of interpolation inequalities and energy estimates must be carried out.