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

We study the asymptotic behavior, as ε tends to zero, of the functionals K_ε^k introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i. e., where k > 0 and W:ℝ→[0,+∞) is a double-well potential with two potential wells of level zero at a,b ∈ ℝ. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k₀ such that, for k < k₀, and for a class of potentials W, (F_ε^k)Γ(L¹)-converges to where m_k is a constant depending on W and k. Moreover, in the special case of the classical potential we provide an upper bound on the values of k such that the minimizers of F_ε^k 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.