Gamma-Convergence of Higher-Order Phase Transition Models

We investigate the asymptotic behavior as ε → 0 of singularly perturbed phase transition models of order n ≥ 2, given by {align} G_ελ,n[u] := ∫_I 1ε W(u) -λε²ⁿ⁻³ (u⁽ⁿ⁻¹⁾)^2 + ε²ⁿ⁻¹ (u⁽ⁿ⁾)^2 \ dx, u ∈ Wn,2(I), {align} where λ >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 Γ-limit of G_ελ,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.