Reconstruction of an obstacle inside a planar domain from boundary measurements (Q1922998)

Let \(D\), \(\Omega\) be two bounded \(C^{1,1}\) domains in the plane with \(\overline D\subset\Omega\). We define the Dirichlet-to-Neumann map associated with \(D\) by \[ \Lambda_Df={\partial u\over\partial\nu}\Biggl|_{\partial\Omega}, \] where \(f\in H^{1/2}(\partial\Omega)\), \(\nu\) is the outward normal, and \(u\) is the \(H^1\) solution of the boundary value problem \[ \Delta u=0\quad\text{in }\Omega\backslash\overline D,\quad u|_{\partial\Omega}= f,\quad{\partial u\over\partial\nu}\Biggl|_{\partial D}= 0. \] The problem under consideration is to reconstruct a two-dimensional obstacle \(D\) from the knowledge of its Dirichlet-to-Neumann map \(\Lambda_D\) of a domain enclosing \(D\). To do so, the author first proves the existence and uniqueness of exponentially growing solutions to an exterior problem which is of independent interest.