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

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 0 ε → 0 of the functionals aligned F_ (u): = _ 1 W (u) + =₁^kq_ ^2 -1| ^ () u|_ ² \, dx, u Hᵏ (), aligned F ε (u): = ∫ Ω 1 ε W (u) + ∑ ℓ = 1 k q ℓ ε 2 ℓ - 1 | ∇ (ℓ) u | ℓ 2 d x, u ∈ H k (Ω), for fixed k>1 k > 1 integer, addressing also the case in which the coefficients q₁,. . . , q₊-₁ q 1,. . . , q k - 1 are negative and | |_ | · | ℓ is any norm on the space of symmetric ℓ -tensors for each \1,. . . , k\ 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.