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

TL;DR

This paper studies gradient behavior in $p$-Laplacian equations for perfect conductors with different boundary regularities, showing boundedness for flat boundaries and precise blow-up rates for $C^{1,eta}$ boundaries.

Contribution

It provides new gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,eta}$ inclusions, including bounds and asymptotic expansions.

Findings

Gradient remains bounded for partially flat boundaries.

Optimal blow-up rates are established for $C^{1,eta}$ boundaries.

Asymptotic expansions are derived in special cases.

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.