Singular perturbations models in phase transitions for anisotropic higher-order materials
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| Format: | Preprint |
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2025
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| author | Brusca, Giuseppe Cosma Donati, Davide Trifone, Chiara |
| author_facet | Brusca, Giuseppe Cosma Donati, Davide Trifone, Chiara |
| contents | 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 $\varepsilon\to 0$ of the functionals \begin{equation*}
F_\varepsilon(u):=\int_Ω\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(Ω), \end{equation*} 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 [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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular perturbations models in phase transitions for anisotropic higher-order materials Brusca, Giuseppe Cosma Donati, Davide Trifone, Chiara Analysis of PDEs 49J45, 26B30, 74N15, 74G65 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 $\varepsilon\to 0$ of the functionals \begin{equation*} F_\varepsilon(u):=\int_Ω\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(Ω), \end{equation*} 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 [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. |
| title | Singular perturbations models in phase transitions for anisotropic higher-order materials |
| topic | Analysis of PDEs 49J45, 26B30, 74N15, 74G65 |
| url | https://arxiv.org/abs/2503.13035 |