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Volume 37, Issue 1
Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators

Yanyan Li & Zhuolun Yang

Anal. Theory Appl., 37 (2021), pp. 114-128.

Published online: 2021-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. When the distance of inclusions, denoted by $\varepsilon$, goes to 0, the gradient of solutions may blow up. When two inclusions are strictly convex, it was known that an upper bound of the blow-up rate is of order $\varepsilon^{-1/2}$ for $n = 2$, and is of order $\varepsilon^{-1/2 + \beta}$ for some $\beta > 0$ when dimension $n \ge 3$. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.

  • AMS Subject Headings

35B44, 35J25, 35J57, 74B05, 74G70, 78A48

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COPYRIGHT: © Global Science Press

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@Article{ATA-37-114, author = {Li , Yanyan and Yang , Zhuolun}, title = {Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators}, journal = {Analysis in Theory and Applications}, year = {2021}, volume = {37}, number = {1}, pages = {114--128}, abstract = {

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. When the distance of inclusions, denoted by $\varepsilon$, goes to 0, the gradient of solutions may blow up. When two inclusions are strictly convex, it was known that an upper bound of the blow-up rate is of order $\varepsilon^{-1/2}$ for $n = 2$, and is of order $\varepsilon^{-1/2 + \beta}$ for some $\beta > 0$ when dimension $n \ge 3$. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2021.pr80.12}, url = {http://global-sci.org/intro/article_detail/ata/18767.html} }
TY - JOUR T1 - Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators AU - Li , Yanyan AU - Yang , Zhuolun JO - Analysis in Theory and Applications VL - 1 SP - 114 EP - 128 PY - 2021 DA - 2021/04 SN - 37 DO - http://doi.org/10.4208/ata.2021.pr80.12 UR - https://global-sci.org/intro/article_detail/ata/18767.html KW - Conductivity problem, harmonic functions, maximum principle, gradient estimates. AB -

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. When the distance of inclusions, denoted by $\varepsilon$, goes to 0, the gradient of solutions may blow up. When two inclusions are strictly convex, it was known that an upper bound of the blow-up rate is of order $\varepsilon^{-1/2}$ for $n = 2$, and is of order $\varepsilon^{-1/2 + \beta}$ for some $\beta > 0$ when dimension $n \ge 3$. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.

Li , Yanyan and Yang , Zhuolun. (2021). Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators. Analysis in Theory and Applications. 37 (1). 114-128. doi:10.4208/ata.2021.pr80.12
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