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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2604.17314 |
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| _version_ | 1866911624971943936 |
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| author | Ma, Linjie |
| author_facet | Ma, Linjie |
| contents | In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $ε$ between the inclusions, which tends to $0$. The purpose of this paper is to provide a simple proof of optimal pointwise estimates for the insulated conductivity problem in any dimension, including the case of flat inclusions. Our approach is based on two fundamental tools: the maximum principle and the Hopf lemma. A key feature of this method is that it avoids the flattening techniques commonly used in the literature, such as those in \citet{dong2021optimal,dong2022gradient}, which require transforming the narrow region into an n-dimensional cuboid. We show that the solution of the insulated problem is $α$-order ($α\in[0,1)$) polynomial growth for $n\geq2$ near the origin. Moreover, when the boundaries near the origin are flat, we prove that the gradient of the solution remains uniformly bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A simple proof for the insulated conductivity problem and application to flat boundaries Ma, Linjie Analysis of PDEs In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $ε$ between the inclusions, which tends to $0$. The purpose of this paper is to provide a simple proof of optimal pointwise estimates for the insulated conductivity problem in any dimension, including the case of flat inclusions. Our approach is based on two fundamental tools: the maximum principle and the Hopf lemma. A key feature of this method is that it avoids the flattening techniques commonly used in the literature, such as those in \citet{dong2021optimal,dong2022gradient}, which require transforming the narrow region into an n-dimensional cuboid. We show that the solution of the insulated problem is $α$-order ($α\in[0,1)$) polynomial growth for $n\geq2$ near the origin. Moreover, when the boundaries near the origin are flat, we prove that the gradient of the solution remains uniformly bounded. |
| title | A simple proof for the insulated conductivity problem and application to flat boundaries |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.17314 |