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Volume 40, Issue 3
The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients

Yuanyuan Hou & Haigang Li

Anal. Theory Appl., 40 (2024), pp. 240-310.

Published online: 2024-10

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  • Abstract

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially “flat”. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order $m$ and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.

  • AMS Subject Headings

35Q74, 35B65, 74G70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-40-240, author = {Hou , Yuanyuan and Li , Haigang}, title = {The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients}, journal = {Analysis in Theory and Applications}, year = {2024}, volume = {40}, number = {3}, pages = {240--310}, abstract = {

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially “flat”. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order $m$ and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2023-0019}, url = {http://global-sci.org/intro/article_detail/ata/23467.html} }
TY - JOUR T1 - The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients AU - Hou , Yuanyuan AU - Li , Haigang JO - Analysis in Theory and Applications VL - 3 SP - 240 EP - 310 PY - 2024 DA - 2024/10 SN - 40 DO - http://doi.org/10.4208/ata.OA-2023-0019 UR - https://global-sci.org/intro/article_detail/ata/23467.html KW - Gradient estimates, convex inclusions, blowup analysis, linear elasticity, composite materials. AB -

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially “flat”. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order $m$ and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.

Hou , Yuanyuan and Li , Haigang. (2024). The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients. Analysis in Theory and Applications. 40 (3). 240-310. doi:10.4208/ata.OA-2023-0019
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