Gamma-Convergence of Higher-Order Phase Transition Models

2025/03/11 by Brazke, Denis, Götzmann, Gianna, Knüpfer, Hans
#49J45 #49J53 #82B26 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2503.08309

Abstract

We investigate the asymptotic behavior as ε → 0 of singularly perturbed phase transition models of order n ≥ 2, given by G_ε^λ,n[u] := ∫_I \frac 1ε W(u) -λε^2n-3 (u^(n-1))² + ε^2n-1 (u⁽ⁿ⁾)² dx, u ∈ W^n,2(I), 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_ε^λ,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.

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