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

We study the asymptotic behavior, as tends to zero, of the functionals introduced by Coleman and Mizel in the theory of nonlinear second-order materials. By proving a new nonlinear interpolation inequality, we show a Γ-convergence result. Moreover, in the special case of the classical potential, we provide an upper bound on the values of k such that the minimizers of the functional cannot develop oscillations on some fine scale and a lower bound on the values for which oscillations occur, the latter improving a previous estimate by Mizel, Peletier and Troy.