Published Paper

Inserted: 16 mar 2025
Last Updated: 12 sep 2025

Journal: Calc. Var. Partial Differential Equations
Volume: 64
Number: 243
Year: 2025
Doi: https://doi.org/10.1007/s00526-025-03120-4

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 $\Gamma$-convergence the asymptotic behaviour as $\varepsilon\to0$ of the functionals \[ F_\varepsilon(u):=\int_\Omega \Bigl[\frac{1}{\varepsilon}W(u)+\sum_{\ell=1}^{k}q_\ell\varepsilon^{2\ell-1}[\nabla^{(\ell)}u]_\ell^2\Bigr]\,dx, \qquad u\in H^k(\Omega), \] for fixed $k>1$ integer, addressing also to 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 \cite{BDS}. 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.

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