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arXiv · 2204.07290

Gradient estimates for the insulated conductivity problem with inclusions of the general $m$-convex shapes

Zhiwen Zhao
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Abstract

In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized $m$-convex inclusions under these two cases with $m\geq2$, which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non-zero eigenvalue of an elliptic operator of divergence form on $\mathbb{S}^{d-2}$. Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.

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Zhiwen Zhao. 2022-07-24. Gradient estimates for the insulated conductivity problem with inclusions of the general $m$-convex shapes. https://arxiv.org/abs/2204.07290

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