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

Abstract

In this paper, we establish the asymptotic expressions for the gradient of a solution to the Lamé 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.