Regularity for free boundary surfaces minimizing degenerate area functionals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912258031878144 |
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| author | Gasparetto, Carlo Paiano, Filippo Velichkov, Bozhidar |
| author_facet | Gasparetto, Carlo Paiano, Filippo Velichkov, Bozhidar |
| contents | We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminusΩ)^a$ near $\partialΩ$ and $a>0$.
This implies that the boundaries of almost-minimizers are $C^{1,γ_0}$-surfaces that touch $\partial Ω$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound
$d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity for free boundary surfaces minimizing degenerate area functionals Gasparetto, Carlo Paiano, Filippo Velichkov, Bozhidar Analysis of PDEs Differential Geometry 49Q05, 49Q10, 35R35 We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminusΩ)^a$ near $\partialΩ$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,γ_0}$-surfaces that touch $\partial Ω$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$. |
| title | Regularity for free boundary surfaces minimizing degenerate area functionals |
| topic | Analysis of PDEs Differential Geometry 49Q05, 49Q10, 35R35 |
| url | https://arxiv.org/abs/2503.02535 |