Doubling inequalities for the Lam\'e system with rough coefficients

TL;DR

This paper establishes local and global doubling inequalities for solutions to the Lamé system with rough coefficients, advancing understanding of unique continuation and inverse problems in elasticity.

Contribution

It introduces new local and global doubling inequalities for the Lamé system with Lipschitz and bounded coefficients, using Carleman estimates.

Findings

Proved local doubling inequality for solutions with rough coefficients.

Established global doubling inequality applicable to inverse problems.

Enhanced understanding of unique continuation properties in elasticity systems.

Abstract

In this paper we study the local behavior of a solution to the Lam\'e system when the Lam\'e coefficients $\lambda$ and $\mu$ satisfy that $\mu$ is Lipschitz and $\lambda$ is essentially bounded in dimension $n\ge 2$. One of the main results is the \emph{local} doubling inequality for the solution of the Lam\'e system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the \emph{global} doubling inequality, which is useful in some inverse problems.