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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2103.14143 |
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| _version_ | 1866912154116947968 |
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| author | Weinkove, Ben |
| author_facet | Weinkove, Ben |
| contents | We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\varepsilon$ apart. The solution $u$ represents the electric potential. In dimensions $n \ge 3$ it is an open problem to find the optimal bound on the gradient of $u$, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order $\varepsilon^{-(1-γ)/2}$ for some $γ>0$. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for $n \ge 4$. Our method gives effective lower bounds on the best constant $γ$, which in particular approach $1$ as $n$ tends to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_14143 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The insulated conductivity problem, effective gradient estimates and the maximum principle Weinkove, Ben Analysis of PDEs We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\varepsilon$ apart. The solution $u$ represents the electric potential. In dimensions $n \ge 3$ it is an open problem to find the optimal bound on the gradient of $u$, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order $\varepsilon^{-(1-γ)/2}$ for some $γ>0$. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for $n \ge 4$. Our method gives effective lower bounds on the best constant $γ$, which in particular approach $1$ as $n$ tends to infinity. |
| title | The insulated conductivity problem, effective gradient estimates and the maximum principle |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2103.14143 |