Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917622201712640 |
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| author | Knees, Dorothee Owczarek, Sebastian Neff, Patrizio |
| author_facet | Knees, Dorothee Owczarek, Sebastian Neff, Patrizio |
| contents | In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model.
As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary $ε> 0$, the displacement vector $u$ belongs to $H^{\frac{3}{2}-ε}(Ω)$, and the microdistortion tensor $P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$ while $\Curl P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains Knees, Dorothee Owczarek, Sebastian Neff, Patrizio Analysis of PDEs In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary $ε> 0$, the displacement vector $u$ belongs to $H^{\frac{3}{2}-ε}(Ω)$, and the microdistortion tensor $P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$ while $\Curl P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$. |
| title | Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.17451 |