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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.11054 |
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| _version_ | 1866908628990033920 |
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| author | Heilmann, Tim |
| author_facet | Heilmann, Tim |
| contents | We study functionals \begin{equation*}
F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) \, dx +
\varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\|\cdot\|_{H^{1/2}}$ when $\varepsilon \ln(λ_\varepsilon) \rightarrow k$ as $\varepsilon \rightarrow 0^+$ and show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to the classical surface tension functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm Heilmann, Tim Analysis of PDEs 49J10, 49J45 We study functionals \begin{equation*} F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\|\cdot\|_{H^{1/2}}$ when $\varepsilon \ln(λ_\varepsilon) \rightarrow k$ as $\varepsilon \rightarrow 0^+$ and show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to the classical surface tension functional. |
| title | $Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm |
| topic | Analysis of PDEs 49J10, 49J45 |
| url | https://arxiv.org/abs/2507.11054 |