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 \(\Gamma \) -convergence the asymptotic behaviour as \(\varepsilon \rightarrow 0\) of the functionals \(\begin{aligned} F_\varepsilon (u):=\int _\Omega \Bigl [\frac{1}{\varepsilon }W(u)+\sum _{\ell =1}^{k}q_\ell \varepsilon ^{2\ell -1}|\nabla ^{(\ell )}u|_\ell ^2\Bigr ]\,dx, \qquad u\in H^k(\Omega ), \end{aligned}\) for fixed \(k>1\) integer, addressing also the case in which the coefficients \(q_1,...,q_{k-1}\) are negative and \(|\cdot |_\ell \) is any norm on the space of symmetric \(\ell \) -tensors for each \(\ell \in \{1,...,k\}\) . 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 \(\Gamma \) -limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.
