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-convergence the asymptotic behaviour as ε→0 of the functionals (Formula presented.) for fixed k>1 integer, addressing also the case in which the coefficients q1,..,qk-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.