Regularity and stability for the scattering map of a linearized inverse medium problem (Q1576981)

The authors study the scattering problem for the 2-D Helmholtz equation \[ \Delta\phi(x)+ k^2(1+ q(x))\phi(x)= 0,\qquad x\in\mathbb{R}^2, \] where the wave number \(k>0\) and \(q\) is supported in a bounded domain \(\Omega\subset \mathbb{R}^2\). Then \(\phi= \phi_0+\psi\), where \(\phi_0\) is the incident field satisfying \[ \Delta\phi_0(x)+ k^2\phi_0(x)= 0,\qquad x\in\mathbb{R}^2, \] and the scattered field \(\psi\) satisfies the Sommerfeld radiation condition. The authors consider the (scatterer to near-field) map \[ M(q)\phi_0= \psi|_{\partial\Omega}, \] and investigate its properties, such as regluarity, stability and Fréchet differentiability. A lower bound of the Fréchet derivative gives rise to stability properties of the inverse scattering problem, which consists in the reconstruction of \(q\) from near-field data.