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

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 Γ -convergence the asymptotic behaviour as ε → 0 ε → 0 of the functionals $${aligned} F_ε (u):=∫ _Ω [1/ε W(u)+∑ ₌₁ᵏq_ ε 2 -1|∇ ( )u|_ ^2 ]\,dx, u∈ H^k(Ω ), {aligned}$$ 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 $$q_1,...,qₖ₋₁$$ 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 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.