Gradient estimates for the conductivity problem with imperfect bonding interfaces
Year: 2025
Type: article
Abstract: We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a new dichotomy for the field concentration depending on the bonding parameter 𝛾. Specifically, we show that the gradient of solution is uniformly bounded independent of 𝜀 (the distance between two inclusions) when 𝛾 is sufficiently small. However, the gradient may blow up when 𝛾 is large. Moreover, we identify the threshold of 𝛾 and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.
Authors: Hongjie Dong, Zhuolun Yang, Hanye Zhu
Institutions: Brown University, The Ohio State University, Duke University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
Funder: National Science Foundation
Awards: Quantitative Analysis on High-Contrast Composite Materials, Small Scale and Singularity Formation in Fluids, Problems in Regularity Theory of Partial Differential Equations, Regularity Questions in Linear and Nonlinear Partial Differential Equations, Small Scale and Singularity Formation in Fluids
OpenAlex ID: W4415894296