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**Asymptotics for the concentrated field between closely located hard inclusions in all dimensions.**
*(English)*
Zbl 1478.78042

Summary: 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 H. Li et al. [Multiscale Model. Simul. 17, No. 3, 899–925 (2019; Zbl 1426.78022)] 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 \(m,n \geq 2\), where \(m\) and \(n\) are the parameters of convexity and dimension, respectively, which is only partially solved in [H. Li, SIAM J. Math. Anal. 52, No. 4, 3350–3375 (2020; Zbl 1447.35140)].

### MSC:

78A48 | Composite media; random media in optics and electromagnetic theory |

35B44 | Blow-up in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35C20 | Asymptotic expansions of solutions to PDEs |

### Keywords:

asymptotic expansions; perfect conductivity problem; \(m\)-convex inclusions; optimal blow-up rate
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\textit{Z. Zhao} and \textit{X. Hao}, Commun. Pure Appl. Anal. 20, No. 6, 2379--2398 (2021; Zbl 1478.78042)

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### References:

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