# American Institute of Mathematical Sciences

June  2021, 20(6): 2379-2398. doi: 10.3934/cpaa.2021086

## Asymptotics for the concentrated field between closely located hard inclusions in all dimensions

 a. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China b. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands

* Corresponding author

Received  December 2020 Revised  April 2021 Published  June 2021 Early access  May 2021

Fund Project: Z. W. Zhao was partially supported by NSFC (11971061) and BJNSF (1202013)

When hard inclusions are frequently spaced very closely, the electric field, which is the gradient of the solution to the perfect conductivity equation, may be arbitrarily large as the distance between two inclusions goes to zero. In this paper, our objectives are two-fold: first, we extend the asymptotic expansions of [26] to the higher dimensions greater than three by capturing the blow-up factors in all dimensions, which consist of some certain integrals of the solutions to the case when two inclusions are touching; second, our results answer the optimality of the blow-up rate for any $m,n\geq2$, where $m$ and $n$ are the parameters of convexity and dimension, respectively, which is only partially solved in [29].

Citation: Zhiwen Zhao, Xia Hao. Asymptotics for the concentrated field between closely located hard inclusions in all dimensions. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2379-2398. doi: 10.3934/cpaa.2021086
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##### References:
Two close-to-touching disks
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