Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,\gamma}$ inclusions

Hongjie Dong; Longjuan Xu

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arXiv · 2601.09435

Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,\gamma}$ inclusions

Hongjie Dong·Longjuan Xu

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Abstract

In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with $C^{1,\gamma}$ boundaries ($\gamma\in(0,1)$), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases.

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Hongjie Dong, Longjuan Xu. 2026-01-14. Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,\gamma}$ inclusions. https://arxiv.org/abs/2601.09435

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