Singular perturbations models in phase transitions for anisotropic higher-order materials

2025/03/17 by Brusca, Giuseppe Cosma, Donati, Davide, Trifone, Chiara
#26B30 #49J45 #74G65 #74N15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.13035

Abstract

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by Γ-convergence the asymptotic behaviour as ε→ 0 of the functionals F_ε(u):=∫_Ω[(1)/(ε)W(u)+∑_ℓ=1^kq_ℓε^2ℓ-1|∇^(ℓ)u|_ℓ²] dx, u∈ H^k(Ω), for fixed k>1 integer, addressing also to the case in which the coefficients q₁,...,q_k-1 are negative and |⋅|_ℓ is any norm on the space of symmetric ℓ-tensors for each ℓ∈\1,...,k\. The negativity of the coefficients leads to the lack of a priori bounds on the functionals; such issue is overcome by proving a nonlinear interpolation inequality. With this inequality at our disposal, a compactness result is achieved by resorting to the recent paper [10]. A further difficulty is the presence of general tensor norms which carry anisotropies, making standard slicing arguments not suitable. We prove that the Γ-limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.

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