Gamma-Convergence of Higher-Order Phase Transition Models
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915558153256960 |
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| author | Brazke, Denis Götzmann, Gianna Knüpfer, Hans |
| author_facet | Brazke, Denis Götzmann, Gianna Knüpfer, Hans |
| contents | We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align}
G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align}
where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gamma-Convergence of Higher-Order Phase Transition Models Brazke, Denis Götzmann, Gianna Knüpfer, Hans Analysis of PDEs Functional Analysis 49J45, 49J53, 82B26 We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities. |
| title | Gamma-Convergence of Higher-Order Phase Transition Models |
| topic | Analysis of PDEs Functional Analysis 49J45, 49J53, 82B26 |
| url | https://arxiv.org/abs/2503.08309 |