Optimal higher derivative estimates for solutions of the Lam\'e system with closely spaced hard inclusions

TL;DR

This paper provides precise estimates for the growth of higher derivatives of solutions to the Lamé system with closely spaced hard inclusions, revealing the singular behavior as the gap narrows.

Contribution

It offers the first sharp characterization of higher derivative singularities in the Lamé system with hard inclusions, especially in symmetric domains.

Findings

Higher derivatives blow up as inclusions approach each other.

Derived sharp bounds for stress concentration in narrow regions.

Results are validated in two and three dimensions with symmetry.

Abstract

We investigate higher derivative estimates for the Lam\'e system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. As the distance $\varepsilon$ between two closely spaced hard inclusions approaches zero, the stress in the narrow regions between the inclusions increases significantly. This stress is captured by the gradient of the solution. The key contribution of this paper is a detailed characterization of this singularity, achieved by deriving higher derivative estimates for solutions to the Lam\'e system with partially infinite coefficients. These upper bounds are shown to be sharp in two and three dimensions when the domain exhibits certain symmetries. To the best of our knowledge, this is the first work to precisely quantify the singular behavior of higher derivatives in the Lam\'e system with hard inclusions.