On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914071980277760 |
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| author | Almi, Stefano Davoli, Elisa Kubin, Anna Tasso, Emanuele |
| author_facet | Almi, Stefano Davoli, Elisa Kubin, Anna Tasso, Emanuele |
| contents | We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case Almi, Stefano Davoli, Elisa Kubin, Anna Tasso, Emanuele Analysis of PDEs 49Q20, 49J45, 26B30 We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space. |
| title | On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case |
| topic | Analysis of PDEs 49Q20, 49J45, 26B30 |
| url | https://arxiv.org/abs/2410.23908 |