Upper and lower bounds for stress concentration in linear elasticity when 𝐶^{1,𝛼} inclusions are close to boundary

We establish boundary gradient estimates for both Lamé systems with partially infinite coefficients and perfect conductivity problem. The inclusion and the matrix domain are both assumed to be of C 1 , α C^{1, \alpha } , weaker than C 2 , α C^{2, \alpha } assumptions in the previous work by Bao-Ju-Li [Adv. Math. 314 (2017), pp. 583–629]. When the inclusion is located close to the boundary of matrix domain, we give the specific examples of boundary data to obtain the lower bound gradient estimates in all dimensions, which guarantee the blow-up occurs and indicate that the blow-up rates of the gradients with respect to the distance between the interfacial surfaces are optimal.