Borderline regularity in singular free boundary problems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918127880634368 |
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| author | Araújo, Damião J. Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel |
| author_facet | Araújo, Damião J. Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel |
| contents | In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term $σ$. When $σ$ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if $σ$ is continuous, then minimizers are of class $C^1$ along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Borderline regularity in singular free boundary problems Araújo, Damião J. Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel Analysis of PDEs Primary 35B65. Secondary 35R35, 35A21 In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term $σ$. When $σ$ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if $σ$ is continuous, then minimizers are of class $C^1$ along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential. |
| title | Borderline regularity in singular free boundary problems |
| topic | Analysis of PDEs Primary 35B65. Secondary 35R35, 35A21 |
| url | https://arxiv.org/abs/2508.14736 |