Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908585561161728 |
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| author | Friedrich, Manuel Labourie, Camille Stinson, Kerrek |
| author_facet | Friedrich, Manuel Labourie, Camille Stinson, Kerrek |
| contents | We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than $1$. This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_08670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane Friedrich, Manuel Labourie, Camille Stinson, Kerrek Analysis of PDEs 49J45, 70G75, 74B20, 74G65, 74R10, 74A30 We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than $1$. This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional. |
| title | Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane |
| topic | Analysis of PDEs 49J45, 70G75, 74B20, 74G65, 74R10, 74A30 |
| url | https://arxiv.org/abs/2510.08670 |