Estimates and Asymptotics for the stress concentration between closely spaced stiff $C^{1, \gamma}$ inclusions in linear elasticity

This paper is concerned with the stress concentration phenomenon in elastic composite materials when the inclusions are very closely spaced. We investigate the gradient blow-up estimates for the Lam\'{e} system of linear elasticity with partially infinite coefficients to show the dependence of the degree of stress enhancement on the distance between inclusions in a composite containing densely placed stiff inclusions. In this paper we assume that the inclusions to be of $C^{1, \gamma}$, weaker than the previous $C^{2, \gamma}$ assumption. To overcome this new difficulty, we make use of $W^{1, p}$ estimates for elliptic system with right hand side in divergence form, instead of a direct $W^{2, p}$ argument for $C^{2, \gamma}$ inclusion case, and combine with the Campanato's approach to establish the optimal gradient estimates, including upper and lower bounds. Moreover, an asymptotic formula of the gradient near the blow-up point is obtained for some symmetric $C^{1, \gamma}$ inclusions.