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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2402.02793 |
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| _version_ | 1866916783128051712 |
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| author | Hanke, Martin |
| author_facet | Hanke, Martin |
| contents | We consider the conductivity problem for a homogeneous body with an inclusion of a different, but known, conductivity. Our interest concerns the associated shape derivative, i.e., the derivative of the corresponding electrostatic potential with respect to the shape of the inclusion. For a smooth inclusion it is known that the shape derivative is the solution of a specific inhomogeneous transmission problem. We show that this characterization of the shape derivative is also valid when the inclusion is a polygonal domain, but due to singularities at the vertices of the polygon, the shape derivative fails to belong to $H^1$ in this case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the shape derivative of polygonal inclusions in the conductivity problem Hanke, Martin Analysis of PDEs We consider the conductivity problem for a homogeneous body with an inclusion of a different, but known, conductivity. Our interest concerns the associated shape derivative, i.e., the derivative of the corresponding electrostatic potential with respect to the shape of the inclusion. For a smooth inclusion it is known that the shape derivative is the solution of a specific inhomogeneous transmission problem. We show that this characterization of the shape derivative is also valid when the inclusion is a polygonal domain, but due to singularities at the vertices of the polygon, the shape derivative fails to belong to $H^1$ in this case. |
| title | On the shape derivative of polygonal inclusions in the conductivity problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.02793 |