Gradient asymptotics of solutions to the Lam\'{e} systems in the presence of two nearly touching $C^{1,\gamma}$-inclusions in all dimensions

TL;DR

This paper derives precise asymptotic formulas for the gradient blow-up of solutions to Lamé systems with nearly touching rigid inclusions, confirming the optimality of blow-up rates in all dimensions.

Contribution

It provides the first complete asymptotic characterization of gradient behavior near nearly touching inclusions in Lamé systems across all dimensions.

Findings

Gradient blow-up rates are optimal in all dimensions.

Asymptotic expressions improve previous results by Chen-Li.

Results apply to $C^{1,eta}$-inclusions in elasticity problems.

Abstract

In this paper, we establish the asymptotic expressions for the gradient of a solution to the Lam\'{e} systems with partially infinity coefficients as two rigid $C^{1,\gamma}$-inclusions are very close but not touching. The novelty of these asymptotics, which improve and make complete the previous results of Chen-Li (JFA 2021), lies in that they show the optimality of the gradient blow-up rate in dimensions greater than two.