Gradient estimates for the insulated conductivity problem with partially flat inclusions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918126536359936 |
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| author | Dong, Hongjie Yang, Zhuolun Zhu, Hanye |
| author_facet | Dong, Hongjie Yang, Zhuolun Zhu, Hanye |
| contents | We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between inclusions approaches 0. The optimal blow-up rate was proved in [10] and was achieved in the presence of a uniform background gradient field. In this paper, we demonstrate that when the inclusions are partially flat, the gradient of solutions does not blow up under any uniform background fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for the insulated conductivity problem with partially flat inclusions Dong, Hongjie Yang, Zhuolun Zhu, Hanye Analysis of PDEs 35B44, 35J25, 35Q74, 74E30, 74G70 We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between inclusions approaches 0. The optimal blow-up rate was proved in [10] and was achieved in the presence of a uniform background gradient field. In this paper, we demonstrate that when the inclusions are partially flat, the gradient of solutions does not blow up under any uniform background fields. |
| title | Gradient estimates for the insulated conductivity problem with partially flat inclusions |
| topic | Analysis of PDEs 35B44, 35J25, 35Q74, 74E30, 74G70 |
| url | https://arxiv.org/abs/2508.12700 |