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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.09690 |
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| _version_ | 1866918381507051520 |
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| author | Fusco, Giuliana Heilmann, Tim |
| author_facet | Fusco, Giuliana Heilmann, Tim |
| contents | We study functionals \begin{equation*}
F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) \, dx
+ \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω
\frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx
+ \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω
\frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy -
ρ(x) \right| \,dx \end{equation*} for a double-well potential $W$ and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm and surfactants Fusco, Giuliana Heilmann, Tim Analysis of PDEs 49J10, 49J45 We study functionals \begin{equation*} F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy - ρ(x) \right| \,dx \end{equation*} for a double-well potential $W$ and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface. |
| title | $Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm and surfactants |
| topic | Analysis of PDEs 49J10, 49J45 |
| url | https://arxiv.org/abs/2603.09690 |