Local stability for soft obstacles by a single measurement
The authors consider an inverse scattering problem arising in target identification. Let \(\kappa > 0\), \(\omega \in S^2\) be given and let \(D\) be a bounded domain with connected complement in \(\mathbb R^3\). Consider the boundary value problem \[ \begin{cases} \Delta u + \kappa^2 u = 0, & {\text{ in }} \, \mathbb{R}^3 \setminus \overline{D}, \\ u = 0, & {\text{ on }} \, \partial D, \end{cases} \] where \(u = u^s + \exp(i\kappa x \cdot \omega)\), with \(u^s\) total field and \(\exp(i\kappa x \cdot \omega)\) representing the incident plane waves. \(u^s\) is required to satisfy the Sommerfeld radiation condition \[ \lim_{r \to \infty} r (\frac{\partial u^s}{\partial r} - i \kappa u^s) = 0, \quad r = \|x\|. \] It is well known that \(u\) and \(u^s\) are uniquely determined (in a certain weak sense). Moreover, \(u^s\) has the following asymptotic behaviour: \[ u^s(x) = \frac{\exp(i\kappa r)}{r} \{u_\infty(\hat x) + O(\frac{1}{r})\} \quad (r \to \infty), \] with \(\hat x = \frac{x}{\|x\|}\), and \(u_\infty\) is the so called far field pattern of the scattered wave. The main concern of the paper is the inverse problem of detecting \(D\) by the knowledge of \(u_\infty\). The main result is the following (improving previous results of V. Isakov): let \(D_1\) and \(D_2\) be domains with sufficiently smooth boundary, and such that \(D_- \subset D_i \subset D_+\) (\(i \in \{1,2\}\)), \(Vol(D_+ \setminus D_-) < \frac{4\pi^4}{3} \kappa^{-3}\). Let \(u_i\) be the solution corresponding to \(D_i\) and let \(u_{i,\infty}\) be the far field pattern. Then there exists \(\epsilon_0 > 0\), such that, if \[ \|u_{1,\infty}- u_{2,\infty}\|_{L^2(\partial B_1(0))} \leq \epsilon < \epsilon_0, \] one has \(d_{\mathcal H}(\partial D_1, \partial D_2) \leq C |\log(\epsilon)|^{-\theta}\), with suitable positive constants \(C\) and \(\theta\). Here \[ d_{\mathcal H}(\partial D_1, \partial D_2) = \max \{\sup_{x \in \partial D_1} d(x, \partial D_2), \sup_{x \in \partial D_2} d(x, \partial D_1)\}. \]
scientific article; zbMATH DE number 5500623