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

Abstract

We study the asymptotic behavior, as ε tends to zero, of the functionals F k ε 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 k 0 such that, for k < k 0 , and for a class of potentials W , (F k ε ) (L 1 )-converges to F k (u) := m k #(S(u)), u ∈ 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 k ε 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.

Domain Formation in Membranes Near the Onset of Instability

“…Recall that the last term in (1.3) renders the problem nonlocal. A local approximation of (1.3) was studied in [7] and [8]. We refer to the derivation of (6.20) in the Appendix for the precise connection between the models.…”

“…At this point one can use a long-wavelength approximation as suggested for example in [22] resulting in an approximation energy 20) which was studied in [7,8]. Returning to the full energy in (6.19), we have…”

Second-Order Edge-Penalization in the Ambrosio--Tortorelli functional

“…These considerations also motivate the analysis carried out in the present paper. More precisely we are interested in replacing the term ε|∇v| 2 in (1.2) by a second-order term depending on the Hessian or on the Laplacian of v. These second-order penalizations are strongly related to some second-order Cahn-Hilliard-type functionals used to approximate the perimeter [17,24] (see also the more recent [10,12]). At a first glance a higher-order approximation seems counterintuitive, since one expects convergence to the first-order perimeter term in the limit.…”

Coupled second order singular perturbations for phase transitions

“…(see Coleman, Marcus & Mizel [12] and Seul & Andelman [33]) by considering a perturbed second order energy of the form f ε (x, u, ∇u, ∇ 2 u) = W (u) − qε 2 |∇u| 2 + ε 4 |∇ 2 u| 2 , q > 0 (see also Fonseca & Mantegazza [21] for the case q = 0, Cicalese, Spadaro & Zeppieri [11] for q > 0 in the one-dimensional case, and Hilhorst, Peletier & Schätzle [29] for the case q < 0 where |∇ 2 u| 2 is replaced by | u| 2 ).…”