A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912385938227200 |
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| author | Bellettini, Giovanni Kholmatov, Shokhrukh Yu. |
| author_facet | Bellettini, Giovanni Kholmatov, Shokhrukh Yu. |
| contents | We prove a $C^{1,1}$-regularity of minimizers of the functional $$ \int_I \sqrt{1+|Du|^2} + \int_I |u-g|ds,\quad u\in BV(I), $$ provided $I\subset\mathbb{R}$ is a bounded open interval and $\|g\|_\infty$ is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D Bellettini, Giovanni Kholmatov, Shokhrukh Yu. Analysis of PDEs Classical Analysis and ODEs 49Q15, 49Q20, 26A06, 26A45 We prove a $C^{1,1}$-regularity of minimizers of the functional $$ \int_I \sqrt{1+|Du|^2} + \int_I |u-g|ds,\quad u\in BV(I), $$ provided $I\subset\mathbb{R}$ is a bounded open interval and $\|g\|_\infty$ is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting. |
| title | A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D |
| topic | Analysis of PDEs Classical Analysis and ODEs 49Q15, 49Q20, 26A06, 26A45 |
| url | https://arxiv.org/abs/2505.15586 |