Reconstruction of discontinuous parameters in a second order impedance boundary operator
Let \(\Omega\) be a doubly connected bounded domain in \(\mathbb R^2\) with \(C^{1,\alpha}\) boundary, \(\alpha\in (0,1)\). Denote by \(\Gamma\) and \(\Sigma\) respectively the interior and exterior boundary of \(\Omega\). Let \(\phi \in L^2(\Sigma)\) denote the current flux and let \(q\in L^{\infty}(\Gamma)\) be a Robin parameter such that \(q\geq \gamma\) with some \(\gamma>0\). Let \(\eta_*>0\) and \(\eta_{\text{{ad}}}=\{ \eta\in L^{\infty}(\Gamma): \eta(\tau)\geq \eta_*, \tau \in \Gamma \}\). For every \(\eta\in \eta_{\text{{ad}}}\), let \(u_{\eta}\) be the solution to the boundary value problem \(\Delta u=0\) in \(\Omega\), \(\partial u/\partial n=\phi\) on \(\Sigma\), \(\partial u/\partial n+qu-\partial/\partial \tau(\eta \partial u/\partial \tau)=0\) on \(\Gamma\), where \(\partial u/\partial \tau\) denotes the tangential derivative of \(u|_{\Gamma}\) and \(n\) is the outward unit vector. The paper is concerned with the inverse problem of recovering the function \(\eta\in \eta_{\text{{ad}}}\) from the prescribed flux \(\phi\) and the measurement \(u|_{\eta}\) on \(\Sigma\). The authors study the identifiability and reconstruction in the case of piecewise continuous parameters. Results of numerical tests are also discussed.
scientific article; zbMATH DE number 6652343