| Abstract: |
| We study the higher derivative estimates for the Lam\`e system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. The stress in the narrow regions between two closely spaced hard inclusions significantly increases as $\varepsilon$, the distance between inclusions, approaches to 0. The stress is represented by the gradient of the solution. The novelty of this paper is that to fully characterize this singularity, we derive higher derivative estimates for solutions to the Lam\`e system with partially infinite coefficients. These upper bounds are shown to be sharp in dimensions two and three when the domain exhibits certain symmetry. To the best of our knowledge, this work is the first to quantify precisely this singular behavior of the higher derivatives for the Lam\`e system with hard inclusions. |
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