We investigate the asymptotic behavior as 0 of singularly perturbed phase transition models of order n 2, given by align G_^, nu: = I 1 W (u) -^2n-3 (u^ (n-1) ) ² + ^2n-1 (u^ (n) ) ² \ dx, u W^n, 2 (I), align where >0 is fixed, I R is an open bounded interval, and W C⁰ (R) is a suitable double-well potential. We find that there exists a positive critical parameter depending on W and n, such that the -limit of G_^, n with respect to the L¹-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.
Brazke et al. (Tue,) studied this question.