A Born approximation from backscattering data for live loads in Lamé system (Q2633992)

Let \(\Delta^*=\mu\Delta+(\lambda+\mu)\nabla\nabla\cdot\) denote the operator associated with an isotropic and homogeneous elastic medium which fills \(\mathbb{R}^n\) where \(\lambda\) and \(\mu\) are the Lamé parameters. The paper concerns the recovery of a compactly supported matrix-valued potential \(Q\) from scattering solutions \(u=u_i+v\) of \(\Delta^* u+\omega^2 u=Qu\). The incoming solutions \(u_i\) solve the equation with \(Q=0\) and the scattered solutions \(v\) satisfy the outgoing Kupradze radiation condition. The authors construct from backscattering data a Born-type approximation \(Q_b\) to \(Q\). The main result of the paper is Theorem 1.1, which states that, if the dimension \(n=2\), then \(Q_b\) recovers the singularities of \(Q\). More precisely, for particular \(0<\beta<\alpha\), it is shown that \(Q-Q_b\) has Sobolev regularity order \(\alpha\) if \(Q\) is known to have Sobolev regularity order \(\beta\).