Duality between range and no-response tests and its application for inverse problems

In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $ \Omega\setminus\overline D $ with an unknown obstacle $ D\Subset\Omega $ whose boundary $ \partial D $ is visible from the boundary $ \partial\Omega $ of $ \Omega $ and a measurement is given as a set of Cauchy data on $ \partial\Omega $. Here the Cauchy data is given by a unique solution $ u $ of the boundary value problem for the Laplace equation in $ \Omega\setminus\overline D $ with homogeneous and inhomogeneous Dirichlet boundary condition on $ \partial D $ and $ \partial\Omega $, respectively. These testing methods are domain sampling methods to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of $ u $ to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of $ D $ using the RT gives the reconstruction of $ D $ using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on $ \partial\Omega $ is not identically zero and the solution to the associated forward problem does not have any analytic extension across $ \partial D $. Moreover, we will show that these methods can still give the reconstruction of $ D $ if we a priori knows that $ D $ is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to $ \partial\Omega $.