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PreprintCC BY 4.0arXiv

The conductivity problem with imperfect bonding interfaces and finite internal conductivities

  • Hongjie Dong
  • Zhuolun Yang
  • Hanye Zhu
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Abstract

We study the field concentration phenomenon between two closely spaced inclusions with imperfect bonding interfaces of low conductivity type. The inclusions are assumed to have finite conductivities. The problem is governed by a system of elliptic equations coupled with Robin-type boundary conditions. While it is known that finite-conductivity inclusions with ideal interfaces yield bounded gradients, in this paper we show that with imperfect bonding interfaces, the gradient of the solution may blow up as ε (the distance between two inclusions) tends to zero when the bonding parameter γ is large, while it is uniformly bounded independently of ε when the bonding parameter γ is small. Moreover, we identify the threshold of γ and the optimal blow-up rates under certain symmetry assumptions. Compared to the case when the inclusions are perfect conductors, we find a novel logarithmic blow-up phenomenon at the critical value of γ.

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Cite

BibTeXRIS
BibTeX
@misc{dong2026conductivity,
  title = {{The conductivity problem with imperfect bonding interfaces and finite internal conductivities}},
  author = {Hongjie Dong and Zhuolun Yang and Hanye Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05708},
  eprint = {2610.05708},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05708},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The conductivity problem with imperfect bonding interfaces and finite internal conductivities
AU  - Hongjie Dong
AU  - Zhuolun Yang
AU  - Hanye Zhu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05708
UR  - https://arxiv.org/abs/2610.05708
AB  - We study the field concentration phenomenon between two closely spaced inclusions with imperfect bonding interfaces of low conductivity type. The inclusions are assumed to have finite conductivities. The problem is governed by a system of elliptic equations coupled with Robin-type boundary conditions. While it is known that finite-conductivity inclusions with ideal interfaces yield bounded gradients, in this paper we show that with imperfect bonding interfaces, the gradient of the solution may blow up as ε (the distance between two inclusions) tends to zero when the bonding parameter γ is large, while it is uniformly bounded independently of ε when the bonding parameter γ is small. Moreover, we identify the threshold of γ and the optimal blow-up rates under certain symmetry assumptions. Compared to the case when the inclusions are perfect conductors, we find a novel logarithmic blow-up phenomenon at the critical value of γ.
ER  -