Liouville-type theorem for the Lamé system with singular coefficients

Abstract:

In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in the plane. Let $u$ be a real-valued two-vector in $\mathbb {R}^2$ satisfying $\nabla u\in L^p(\mathbb {R}^2)$ for some $p>2$ and the equation $\operatorname {div}\left (\mu \left [\nabla u+(\nabla u)^T\right ]\right ) +\nabla (\lambda \operatorname {div} u)=0$ in $\mathbb {R}^2$. When $\|\nabla \mu \|_{L^2(\mathbb {R}^2)}$ is not large, we show that $u\equiv \text {constant}$ in $\mathbb {R}^2$. As by-products, we prove the weak unique continuation property and the uniqueness of the Cauchy problem for the Lamé system with small $\|\mu \|_{W^{1,2}}$.