Year: 2026
Type: preprint
Abstract: In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $ε$ between the inclusions, which tends to $0$. The purpose of this paper is to provide a simple proof of optimal pointwise estimates for the insulated conductivity problem in any dimension, including the case of flat inclusions. Our approach is based on two fundamental tools: the maximum principle and the Hopf lemma. A key feature of this method is that it avoids the flattening techniques commonly used in the literature, such as those in \citet{dong2021optimal,dong2022gradient}, which require transforming the narrow region into an n-dimensional cuboid. We show that the solution of the insulated problem is $α$-order ($α\in[0,1)$) polynomial growth for $n\geq2$ near the origin. Moreover, when the boundaries near the origin are flat, we prove that the gradient of the solution remains uniformly bounded.
Source: arXiv (Cornell University)
Author: Ma, Linjie
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4259381229, G5984103438, G7148679336
OpenAlex ID: W7155042406
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