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

We study the asymptotic behavior, as e tends to zero, of the functionals F(epsilon)(k) introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i.e., F(epsilon)(k)(u) := integral(I) (W(u)/epsilon - k epsilon (u')(2) + epsilon(3)(u '')(2))dx, u is an element of W(2,2) (I), where k > 0 and W : R. [ 0,+infinity) is a double-well potential with two potential wells of level zero at a, b is an element of R. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k(0) such that, for k < k(0), and for a class of potentials W, (F(epsilon)(k)) Gamma(L(1))-converges to F(k)(u) := m(k) # (S(u)), u is an element of BV (I; {a,b}), where m(k) is a constant depending on W and k. Moreover, in the special case of the classical potential W(s) = (s(2)-1)(2)/2, we provide an upper bound on the values of k such that the minimizers of F(epsilon)(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.