For the Lamé system with two hard inclusions a distance ε apart, all m-th derivatives blow up at the optimal rate (ε+...)^(-m/2), with log factors in 3D and matching lower bounds in symmetric cases.
Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow
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abstract
In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].
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Optimal higher derivative estimates for solutions of the Lam\'e system with closely spaced hard inclusions
For the Lamé system with two hard inclusions a distance ε apart, all m-th derivatives blow up at the optimal rate (ε+...)^(-m/2), with log factors in 3D and matching lower bounds in symmetric cases.