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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/2601.10875 |
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| _version_ | 1866918291795083264 |
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| author | Babadjian, Jean-François Rakovsky, Martin Rodiac, Rémy |
| author_facet | Babadjian, Jean-François Rakovsky, Martin Rodiac, Rémy |
| contents | We consider a family $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ converges in $L^2(Ω)$ to a limit $(u, 1)$ as $\varepsilon \to 0$, where $u$ is in $SBV^2(Ω)$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_\varepsilon,v_\varepsilon)$ converges to the Mumford-Shah energy of $u$, then the first inner variation converges as well. In particular, $u$ is a critical point of the Mumford-Shah functional in the sense of inner variations. In this work, focusing on the two-dimensional setting, we extend this result under the sole convergence of the phase-field energy to the length energy term in the Mumford-Shah functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10875 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy Babadjian, Jean-François Rakovsky, Martin Rodiac, Rémy Analysis of PDEs We consider a family $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ converges in $L^2(Ω)$ to a limit $(u, 1)$ as $\varepsilon \to 0$, where $u$ is in $SBV^2(Ω)$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_\varepsilon,v_\varepsilon)$ converges to the Mumford-Shah energy of $u$, then the first inner variation converges as well. In particular, $u$ is a critical point of the Mumford-Shah functional in the sense of inner variations. In this work, focusing on the two-dimensional setting, we extend this result under the sole convergence of the phase-field energy to the length energy term in the Mumford-Shah functional. |
| title | Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.10875 |